788 terms

Suppose that Jason gives the following argument:

"We should not serve veal for dinner tonight. Leon is coming over, and he refuses to eat veal on moral grounds. And also Yuji is coming over, and he simply doesn't like the taste of veal. Finally, Njeri is coming over, and she doesn't like it when people spend lots of money on her—I think she might feel embarrassed if we serve an expensive veal dinner."

The conclusion of this argument is

"We should not serve veal for dinner tonight. Leon is coming over, and he refuses to eat veal on moral grounds. And also Yuji is coming over, and he simply doesn't like the taste of veal. Finally, Njeri is coming over, and she doesn't like it when people spend lots of money on her—I think she might feel embarrassed if we serve an expensive veal dinner."

The conclusion of this argument is

The answer is A. "We should not serve veal for dinner tonight."

From the premise that Leon is morally opposed to veal, Yuji dislikes veal, and Njeri is financially embarrassed by veal, the argument concludes that we should not serve veal for dinner tonight. In this particular case, the argument's conclusion comes first, before the premises are presented.

From the premise that Leon is morally opposed to veal, Yuji dislikes veal, and Njeri is financially embarrassed by veal, the argument concludes that we should not serve veal for dinner tonight. In this particular case, the argument's conclusion comes first, before the premises are presented.

Consider Jason's same argument again:

"We should not serve veal for dinner tonight. Leon is coming over, and he refuses to eat veal on moral grounds. And also Yuji is coming over, and he simply doesn't like the taste of veal. Finally, Njeri is coming over, and she doesn't like it when people spend lots of money on her—I think she might feel embarrassed if we serve an expensive veal dinner."

The premise (or premises) of this argument is (or are):

"We should not serve veal for dinner tonight. Leon is coming over, and he refuses to eat veal on moral grounds. And also Yuji is coming over, and he simply doesn't like the taste of veal. Finally, Njeri is coming over, and she doesn't like it when people spend lots of money on her—I think she might feel embarrassed if we serve an expensive veal dinner."

The premise (or premises) of this argument is (or are):

Feedback: The answer is E. "B, C, and D, but not A."

To reach the conclusion that we should not eat veal tonight, the argument employs the premise that Yuji does not like veal, the premise that Njeri doesn't like it when people spend money on her, and the premise that Leon refuses to eat veal. These are the argument's only three premises.

To reach the conclusion that we should not eat veal tonight, the argument employs the premise that Yuji does not like veal, the premise that Njeri doesn't like it when people spend money on her, and the premise that Leon refuses to eat veal. These are the argument's only three premises.

Consider Jason's argument again:

"We should not serve veal for dinner tonight. Leon is coming over, and he refuses to eat veal on moral grounds. And also Yuji is coming over, and he simply doesn't like the taste of veal. Finally, Njeri is coming over, and she doesn't like it when people spend lots of money on her—I think she might feel embarrassed if we serve an expensive veal dinner." Now suppose that Fred objects to Jason's argument as follows: "Oh sure, Jason, let's just pander to everybody's whims and just not serve any food at all! God forbid that we should ever risk offending anyone, spending too much, spending too little, or violating a guest's dietary constraints by serving any kind of dish!"

Fred's response to Jason's argument is an example of

"We should not serve veal for dinner tonight. Leon is coming over, and he refuses to eat veal on moral grounds. And also Yuji is coming over, and he simply doesn't like the taste of veal. Finally, Njeri is coming over, and she doesn't like it when people spend lots of money on her—I think she might feel embarrassed if we serve an expensive veal dinner." Now suppose that Fred objects to Jason's argument as follows: "Oh sure, Jason, let's just pander to everybody's whims and just not serve any food at all! God forbid that we should ever risk offending anyone, spending too much, spending too little, or violating a guest's dietary constraints by serving any kind of dish!"

Fred's response to Jason's argument is an example of

The answer is C. "refutation by parallel reasoning."

To refute a claim by parallel reasoning is to show that an argument is invalid, specifically, by comparing it with an argument that has the same logical form, whose invalidity is clear and apparent.

In this case, Fred has characterized Jason as making the following argument: "We should avoid serving certain dishes, since doing so might upset our guests." Fred points out that, since any dish has the potential to upset some possible guest, it would appear to follow that no food should be served at all. What do you think of Fred's attempt at refutation through parallel reasoning?

To refute a claim by parallel reasoning is to show that an argument is invalid, specifically, by comparing it with an argument that has the same logical form, whose invalidity is clear and apparent.

In this case, Fred has characterized Jason as making the following argument: "We should avoid serving certain dishes, since doing so might upset our guests." Fred points out that, since any dish has the potential to upset some possible guest, it would appear to follow that no food should be served at all. What do you think of Fred's attempt at refutation through parallel reasoning?

Now suppose that, after the financial collapse of 2008, Nouriel gives the following argument:

"Western governments can spend as much taxpayer money as they like covering the losses suffered by the big banks: Such spending will not restore the proper operation of the banks. In order to restore the proper operation of the banks, they must first be adequately capitalized, and they must also be lending money to borrowers whose investments will generate lots of consumer demand. But, given current incentives, there is no way for the banks to achieve this goal without being at least temporarily taken over by fiscal policy makers. So the banks must be temporarily nationalized if they are ever to become effective private institutions again."

The conclusion of this argument is

"Western governments can spend as much taxpayer money as they like covering the losses suffered by the big banks: Such spending will not restore the proper operation of the banks. In order to restore the proper operation of the banks, they must first be adequately capitalized, and they must also be lending money to borrowers whose investments will generate lots of consumer demand. But, given current incentives, there is no way for the banks to achieve this goal without being at least temporarily taken over by fiscal policy makers. So the banks must be temporarily nationalized if they are ever to become effective private institutions again."

The conclusion of this argument is

Feedback: The answer is D. "The banks must be temporarily nationalized if they are ever to become effective private institutions."

From the premise that government spending will not restore proper banking operations, from the premise that this operation requires adequate capitalization, and from the premise that this operation requires banks lend money to borrowers whose investments generate consumer demand, this argument concludes that the banks must be temporarily nationalized.

From the premise that government spending will not restore proper banking operations, from the premise that this operation requires adequate capitalization, and from the premise that this operation requires banks lend money to borrowers whose investments generate consumer demand, this argument concludes that the banks must be temporarily nationalized.

Consider Nouriel's argument again:

"Western governments can spend as much taxpayer money as they like covering the losses suffered by the big banks: Such spending will not restore the proper operation of the banks. In order to restore the proper operation of the banks, they must first be adequately capitalized, and they must also be lending money to borrowers whose investments will generate lots of consumer demand. But, given current incentives, there is no way for the banks to achieve this goal without being at least temporarily taken over by fiscal policy makers. So the banks must be temporarily nationalized if they are ever to become effective private institutions again."

The premise(s) of this argument is(are):

"Western governments can spend as much taxpayer money as they like covering the losses suffered by the big banks: Such spending will not restore the proper operation of the banks. In order to restore the proper operation of the banks, they must first be adequately capitalized, and they must also be lending money to borrowers whose investments will generate lots of consumer demand. But, given current incentives, there is no way for the banks to achieve this goal without being at least temporarily taken over by fiscal policy makers. So the banks must be temporarily nationalized if they are ever to become effective private institutions again."

The premise(s) of this argument is(are):

Feedback: The answer is E. "A, B, and C, but not D."

To reach the conclusion that the banks must be temporarily nationalized if they are ever to become effective private institutions, the argument employs three premises: first, that that government spending will not restore proper banking operations; second, that this operation requires adequate capitalization; and finally, that this operation requires banks lend money to borrowers whose investments generate consumer demand. Since these premises are identical to the claims in A, B, and C, respectively, it follows that E is the correct answer.

To reach the conclusion that the banks must be temporarily nationalized if they are ever to become effective private institutions, the argument employs three premises: first, that that government spending will not restore proper banking operations; second, that this operation requires adequate capitalization; and finally, that this operation requires banks lend money to borrowers whose investments generate consumer demand. Since these premises are identical to the claims in A, B, and C, respectively, it follows that E is the correct answer.

Consider Nouriel's argument again:

"Western governments can spend as much taxpayer money as they like covering the losses suffered by the big banks: Such spending will not restore the proper operation of the banks. In order to restore the proper operation of the banks, they must first be adequately capitalized, and they must also be lending money to borrowers whose investments will generate lots of consumer demand. But, given current incentives, there is no way for the banks to achieve this goal without being at least temporarily taken over by fiscal policy makers. So the banks must be temporarily nationalized if they are ever to become effective private institutions again."

Now suppose that Bill responds to Nouriel's argument as follows:

"Oh, sure, just let the government take over the banks. And while you're at it, let the government also take over the oil and gas industry, aviation, and manufacturing as well. Heck, why not just let the government take over the whole economy?"

Bill's response to Nouriel's argument is an example of

"Western governments can spend as much taxpayer money as they like covering the losses suffered by the big banks: Such spending will not restore the proper operation of the banks. In order to restore the proper operation of the banks, they must first be adequately capitalized, and they must also be lending money to borrowers whose investments will generate lots of consumer demand. But, given current incentives, there is no way for the banks to achieve this goal without being at least temporarily taken over by fiscal policy makers. So the banks must be temporarily nationalized if they are ever to become effective private institutions again."

Now suppose that Bill responds to Nouriel's argument as follows:

"Oh, sure, just let the government take over the banks. And while you're at it, let the government also take over the oil and gas industry, aviation, and manufacturing as well. Heck, why not just let the government take over the whole economy?"

Bill's response to Nouriel's argument is an example of

Feedback:

The answer is A. "refutation by parallel reasoning."

To refute a claim by parallel reasoning is to show that an argument is invalid, specifically, by comparing it with an argument that has the same logical form, whose invalidity is clear and apparent. In this case, Bill has characterized Nouriel as making the following argument: "We should temporarily nationalize the banks, since doing so will facilitate their proper functioning." Bill points out that, since many institutions have the potential to work while nationalized, it would appear to follow that everything should be nationalized. What do you think of Bill's attempt at refutation through parallel reasoning?

The answer is A. "refutation by parallel reasoning."

To refute a claim by parallel reasoning is to show that an argument is invalid, specifically, by comparing it with an argument that has the same logical form, whose invalidity is clear and apparent. In this case, Bill has characterized Nouriel as making the following argument: "We should temporarily nationalize the banks, since doing so will facilitate their proper functioning." Bill points out that, since many institutions have the potential to work while nationalized, it would appear to follow that everything should be nationalized. What do you think of Bill's attempt at refutation through parallel reasoning?

Consider the following argument:

"If guns are outlawed, then only outlaws will have guns. Therefore, guns should not be outlawed."

Which of the following arguments is parallel in structure to the preceding argument?

"If guns are outlawed, then only outlaws will have guns. Therefore, guns should not be outlawed."

Which of the following arguments is parallel in structure to the preceding argument?

Feedback:

The answer is B. "If alcohol is outlawed, then only outlaws will drink alcohol. Therefore, alcohol should not be outlawed."

Unlike rape, reading, and killing, which are all types of actions or activities, guns are possessions of a certain sort. The argument assumes that, if possessions of a certain sort are outlawed, then an undesirable situation results—that only outlaws will possess those things. Therefore, the argument concludes that the possessions in question should not be outlawed.

The argument about alcohol is parallel to the argument about guns. It, too, considers a certain type of possession—alcohol—and claims that, if it is outlawed, a certain undesirable result will occur, which is that only outlaws will drink alcohol.

The answer is B. "If alcohol is outlawed, then only outlaws will drink alcohol. Therefore, alcohol should not be outlawed."

Unlike rape, reading, and killing, which are all types of actions or activities, guns are possessions of a certain sort. The argument assumes that, if possessions of a certain sort are outlawed, then an undesirable situation results—that only outlaws will possess those things. Therefore, the argument concludes that the possessions in question should not be outlawed.

The argument about alcohol is parallel to the argument about guns. It, too, considers a certain type of possession—alcohol—and claims that, if it is outlawed, a certain undesirable result will occur, which is that only outlaws will drink alcohol.

Consider the following argument:

"Nuclear deterrence must work, since nuclear powers have never engaged in nuclear wars with each other since the end of World War II."

Which of the following arguments is parallel in structure to the preceding argument?

"Nuclear deterrence must work, since nuclear powers have never engaged in nuclear wars with each other since the end of World War II."

Which of the following arguments is parallel in structure to the preceding argument?

Feedback:

The answer is E. "all of the above."

The argument assumes the following sort of reasoning: If there have been no Fs for as long as Gs have been present, then Gs must prevent Fs. In the original argument, nuclear war and the nuclear weapons are the respective values for F and G.

Each of the arguments among A through D assume the same line of reasoning as the argument listed in Question 2. In the same way that the author assumes that nuclear deterrence works, since there has been no war with nuclear weapons, so does A assume that one's refrigerator works as fairy repellant. Likewise, so does B assume that photographers prevent the appearance of Bigfoot, so does C assume that air travel prevents poverty, and so does D assume that garlic prevents thieves. In each case, the failure of Fs to appear when there are Gs is taken as proof that Gs prevent Fs

The answer is E. "all of the above."

The argument assumes the following sort of reasoning: If there have been no Fs for as long as Gs have been present, then Gs must prevent Fs. In the original argument, nuclear war and the nuclear weapons are the respective values for F and G.

Each of the arguments among A through D assume the same line of reasoning as the argument listed in Question 2. In the same way that the author assumes that nuclear deterrence works, since there has been no war with nuclear weapons, so does A assume that one's refrigerator works as fairy repellant. Likewise, so does B assume that photographers prevent the appearance of Bigfoot, so does C assume that air travel prevents poverty, and so does D assume that garlic prevents thieves. In each case, the failure of Fs to appear when there are Gs is taken as proof that Gs prevent Fs

Consider the following argument:

"Eating meat cannot be wrong, since most people today regard it as an acceptable practice."

Which of the following arguments could be used to refute the preceding argument by parallel reasoning?

"Eating meat cannot be wrong, since most people today regard it as an acceptable practice."

Which of the following arguments could be used to refute the preceding argument by parallel reasoning?

The answer is B. "Lying cannot be wrong, since most people today regard it as an acceptable practice (spoken by a advertising executive)."

The argument assumes the following line of reasoning: If most people today regard something as an acceptable practice, then such a practice cannot be wrong. The only argument among A through D that duplicates this sort of reasoning is B. In B, the argument concludes that a certain activity—lying—is not wrong, and it does so by citing the premise that today most people today regard it as an acceptable practice.

The argument stated in A is not an exact parallel of the argument in the question. The reason why is that, in A, the practice being considered is not actually regarded as acceptable by most people today. Perhaps it used to be regarded as acceptable, and perhaps the argument stated in A used to function as a parallel argument to the one posed in this question, but it is not anymore, and does not anymore.

Nor does the argument in C run parallel to the line of reasoning in the argument posed in this question. The reason why is that, in C, the argument concludes that abortion cannot be against one's conscience. However, the original argument proposed in this question concluded that a certain action cannot be wrong, not that it cannot be against one's conscience.

Finally, the argument in D does not run parallel to the argument posed in this question. The reason why is that, in D, the argument employs the premise that something is widely practiced. The argument posed in this question, however, employs the premise that an activity is regarded by most people as acceptable. Since these are two very different sorts of assumptions, D is not parallel to the argument posed in this question.

The argument assumes the following line of reasoning: If most people today regard something as an acceptable practice, then such a practice cannot be wrong. The only argument among A through D that duplicates this sort of reasoning is B. In B, the argument concludes that a certain activity—lying—is not wrong, and it does so by citing the premise that today most people today regard it as an acceptable practice.

The argument stated in A is not an exact parallel of the argument in the question. The reason why is that, in A, the practice being considered is not actually regarded as acceptable by most people today. Perhaps it used to be regarded as acceptable, and perhaps the argument stated in A used to function as a parallel argument to the one posed in this question, but it is not anymore, and does not anymore.

Nor does the argument in C run parallel to the line of reasoning in the argument posed in this question. The reason why is that, in C, the argument concludes that abortion cannot be against one's conscience. However, the original argument proposed in this question concluded that a certain action cannot be wrong, not that it cannot be against one's conscience.

Finally, the argument in D does not run parallel to the argument posed in this question. The reason why is that, in D, the argument employs the premise that something is widely practiced. The argument posed in this question, however, employs the premise that an activity is regarded by most people as acceptable. Since these are two very different sorts of assumptions, D is not parallel to the argument posed in this question.

Consider the following argument:

"It's wrong to drive in excess of the speed limit, even in a case where you're rushing someone to the hospital to save their life. If everyone drove in excess of the speed limit, then that would make driving very dangerous for everyone, and we'd see a huge increase in traffic fatalities. So you should never drive in excess of the speed limit, no matter what."

Which of the following arguments could be used to refute the preceding argument by parallel reasoning?

"It's wrong to drive in excess of the speed limit, even in a case where you're rushing someone to the hospital to save their life. If everyone drove in excess of the speed limit, then that would make driving very dangerous for everyone, and we'd see a huge increase in traffic fatalities. So you should never drive in excess of the speed limit, no matter what."

Which of the following arguments could be used to refute the preceding argument by parallel reasoning?

The answer is A. "It's wrong to lie even when an insane killer is asking you where you keep your guns. If everyone lied, then no one could ever trust what anyone else says, and we would lose the ability to share information through verbal exchange. So you should never lie under any circumstances."

The argument posed in this question considers a course of action that is normally regarded as wrong. The argument concludes that, even in a life-or-death circumstance, that course of action would still be wrong. The argument's premise observes that, if everyone committed the action in question, undesirable consequences would result for everyone. In this case, the course of action is driving in excess of the speed limit, the life-or-death circumstance is that of needing to speed in order to save a friend's life, and the undesirable consequence for everyone is an increase in traffic accidents.

Likewise, in A, the argument points out that, if everyone lied all the time, the undesirable consequence of never having believable testimony would result from everyone. Lying is normally regarded as wrong, although we may well think it is permissible in a life-or-death circumstance. The argument concludes, however, that even in a life-or-death circumstance where a killer asks for your weapons, you should still never lie.

The arguments in B, C, and D are similar to the argument posed in this question, in that they all infer the impermissibility of a certain action from the undesirability of everyone committing that action. The arguments in B, C, and D are importantly dissimilar to the argument posed in this question, however, in two respects. First, they do not concern actions that are normally regarded as wrong. Lying is normally regarded as wrong, and driving in excess of the speed limit is normally regarded as wrong, but neither eating potato chips, nor answering emails, nor hugging one's child is normally regarded as wrong. Second, the arguments in B, C, and D do not appeal to any dire circumstance, any emergency, or any life-or-death situation. These dissimilarities are important to the argument because, when considering courses of action that are normally regarded as wrong, one regards them as more justified when they are pursued in an emergency. This general rule, in fact, is the line of reasoning that the argument posed in this question uses. Since it is not present in B, C, or D, the arguments in B, C, or D are not exactly parallel to the argument posed in this question.

The argument posed in this question considers a course of action that is normally regarded as wrong. The argument concludes that, even in a life-or-death circumstance, that course of action would still be wrong. The argument's premise observes that, if everyone committed the action in question, undesirable consequences would result for everyone. In this case, the course of action is driving in excess of the speed limit, the life-or-death circumstance is that of needing to speed in order to save a friend's life, and the undesirable consequence for everyone is an increase in traffic accidents.

Likewise, in A, the argument points out that, if everyone lied all the time, the undesirable consequence of never having believable testimony would result from everyone. Lying is normally regarded as wrong, although we may well think it is permissible in a life-or-death circumstance. The argument concludes, however, that even in a life-or-death circumstance where a killer asks for your weapons, you should still never lie.

The arguments in B, C, and D are similar to the argument posed in this question, in that they all infer the impermissibility of a certain action from the undesirability of everyone committing that action. The arguments in B, C, and D are importantly dissimilar to the argument posed in this question, however, in two respects. First, they do not concern actions that are normally regarded as wrong. Lying is normally regarded as wrong, and driving in excess of the speed limit is normally regarded as wrong, but neither eating potato chips, nor answering emails, nor hugging one's child is normally regarded as wrong. Second, the arguments in B, C, and D do not appeal to any dire circumstance, any emergency, or any life-or-death situation. These dissimilarities are important to the argument because, when considering courses of action that are normally regarded as wrong, one regards them as more justified when they are pursued in an emergency. This general rule, in fact, is the line of reasoning that the argument posed in this question uses. Since it is not present in B, C, or D, the arguments in B, C, or D are not exactly parallel to the argument posed in this question.

Consider the following argument:

"Ram and Walter must be the same person in two different disguises. Think about it: Have you ever seen the two of them together?"

Which of the following arguments could be used to refute the preceding argument by parallel reasoning?

"Ram and Walter must be the same person in two different disguises. Think about it: Have you ever seen the two of them together?"

Which of the following arguments could be used to refute the preceding argument by parallel reasoning?

The answer is D. "Barack Obama and Simon Cowell must be the same person in two different disguises. Think about it: have you ever seen the two of them together?"

The argument posed in this question reasons that, since a person X and a person Y have not been seen together, that X and Y are a single person in two different disguises. The argument in D is parallel to the argument posed in this question; it reasons that, since Obama and Cowell have not been seen together, that Obama and Cowell are a single person and two different disguises.

The arguments in A, B, and C are importantly dissimilar to the argument posed in this question. In A and B, the arguments do not concern persons. The argument in A identifies two sorts of chemical phenomena, and the argument in B identifies two planets. Unlike A or B, the argument in C does concern the identification of persons, but it reasons that the persons are identical because they have not been seen in the same place at the same time. This is very different from reasoning that the persons have not been seen together. For when we prove that a person, X, and a person, Y, are distinct, we do so by viewing X and Y next to each other at the same time. We do not prove their distinctness by observing that they coincide at the exact same location at once.

The argument posed in this question reasons that, since a person X and a person Y have not been seen together, that X and Y are a single person in two different disguises. The argument in D is parallel to the argument posed in this question; it reasons that, since Obama and Cowell have not been seen together, that Obama and Cowell are a single person and two different disguises.

The arguments in A, B, and C are importantly dissimilar to the argument posed in this question. In A and B, the arguments do not concern persons. The argument in A identifies two sorts of chemical phenomena, and the argument in B identifies two planets. Unlike A or B, the argument in C does concern the identification of persons, but it reasons that the persons are identical because they have not been seen in the same place at the same time. This is very different from reasoning that the persons have not been seen together. For when we prove that a person, X, and a person, Y, are distinct, we do so by viewing X and Y next to each other at the same time. We do not prove their distinctness by observing that they coincide at the exact same location at once.

Consider the statement:

Every prime number is odd.

Which of the following things is a counterexample to this general statement?

Every prime number is odd.

Which of the following things is a counterexample to this general statement?

The answer is A. "2."

As stated in the lecture, a counterexample is an example that runs counter to some generalization. In other words, it is an example that falsifies a certain generalization. Here, in this case, to refute the claim "every prime number is odd" with a counterexample, one must find an example of a prime number that is not odd. In other words, one must find an even prime to have a counterexample to the claim "every prime number is odd." Since the number 2 is the only even prime, the number 2 is the only such counterexample.

As stated in the lecture, a counterexample is an example that runs counter to some generalization. In other words, it is an example that falsifies a certain generalization. Here, in this case, to refute the claim "every prime number is odd" with a counterexample, one must find an example of a prime number that is not odd. In other words, one must find an even prime to have a counterexample to the claim "every prime number is odd." Since the number 2 is the only even prime, the number 2 is the only such counterexample.

Consider the statement:

All Swedes are tall.

What would you need to find in order to find a counterexample to this statement?

All Swedes are tall.

What would you need to find in order to find a counterexample to this statement?

The answer is D. "a short Swede."

To falsify the generalization, "all Swedes are tall," one must find an example of a non-tall Swede. A short Swede, in other words, would be a counterexample to the claim that all Swedes are tall.

An example of a tall Swede would not be a counterexample to the statement we are considering. Such a case would not run counter to the generalization that all Swedes are tall.

Neither A nor B would falsify the claim that all Swedes are tall, since the truth of "all Swedes are tall" is compatible with the truth of both A and B. In other words, it could be true that all Swedes are tall even if there existed a tall Norwegian, and even if there also existed a short Dane. Neither the existence of a tall Norwegian nor the existence of a short Dane falsifies the claim that all Swedes are tall. For that reason, A and B are both incorrect answers.

Option E is incorrect because the generalization in the statement posed in this question concerns Swedes, rather than half-Danish half-Swedes. The existence of a short half-Danish half-Swede is compatible with the claim "all Swedes are tall." So the existence of a short half-Danish half-Swede is not a counterexample to the statement.

To falsify the generalization, "all Swedes are tall," one must find an example of a non-tall Swede. A short Swede, in other words, would be a counterexample to the claim that all Swedes are tall.

An example of a tall Swede would not be a counterexample to the statement we are considering. Such a case would not run counter to the generalization that all Swedes are tall.

Neither A nor B would falsify the claim that all Swedes are tall, since the truth of "all Swedes are tall" is compatible with the truth of both A and B. In other words, it could be true that all Swedes are tall even if there existed a tall Norwegian, and even if there also existed a short Dane. Neither the existence of a tall Norwegian nor the existence of a short Dane falsifies the claim that all Swedes are tall. For that reason, A and B are both incorrect answers.

Option E is incorrect because the generalization in the statement posed in this question concerns Swedes, rather than half-Danish half-Swedes. The existence of a short half-Danish half-Swede is compatible with the claim "all Swedes are tall." So the existence of a short half-Danish half-Swede is not a counterexample to the statement.

Consider the statement:

Everything has a shape.

Which of the following things is a counterexample to this general statement?

Everything has a shape.

Which of the following things is a counterexample to this general statement?

The answer is D. "the color blue"

In order to refute the claim "everything has a shape" by counterexample, one must find an example of something that does not have a shape. In other words, one must provide an example of something without a shape to refute the claim "everything has a shape" with a counterexample.

The correct answer to this question is "the color blue" because the color blue does not have a shape. We could not say that the color blue is circular or rectangular or pentagonal, for instance. The reason we could not say such things, moreover, is not that the color blue is some other shape. On the contrary. Unlike blue objects, which always have a shape and a size, along with many other physical features, the color blue, by itself, has no shape or size.

Unlike the color blue, each of the items mentioned in A through C have a shape. Mt. Everest, for instance, has a shape, which is wider at the base than at the peak, and whose dimensions are exactly the dimensions of the region of space, which the actual Mt. Everest currently occupies.

Likewise, Barack Obama's shape is the shape of a particular sort of human being. His shape is exactly the shape of the region of space that he precisely occupies, when he occupies it.

Finally, as any map can confirm, Canada has a shape, too. As with Canada and Mt. Everest, the English language does not have a name for the specific shape that Canada has. Nonetheless, we can still point out the following: Every accurate atlas of Canada represents Canada as a land mass with a certain shape. Canada has exactly that shape, which accurate atlases represent it as having.

In order to refute the claim "everything has a shape" by counterexample, one must find an example of something that does not have a shape. In other words, one must provide an example of something without a shape to refute the claim "everything has a shape" with a counterexample.

The correct answer to this question is "the color blue" because the color blue does not have a shape. We could not say that the color blue is circular or rectangular or pentagonal, for instance. The reason we could not say such things, moreover, is not that the color blue is some other shape. On the contrary. Unlike blue objects, which always have a shape and a size, along with many other physical features, the color blue, by itself, has no shape or size.

Unlike the color blue, each of the items mentioned in A through C have a shape. Mt. Everest, for instance, has a shape, which is wider at the base than at the peak, and whose dimensions are exactly the dimensions of the region of space, which the actual Mt. Everest currently occupies.

Likewise, Barack Obama's shape is the shape of a particular sort of human being. His shape is exactly the shape of the region of space that he precisely occupies, when he occupies it.

Finally, as any map can confirm, Canada has a shape, too. As with Canada and Mt. Everest, the English language does not have a name for the specific shape that Canada has. Nonetheless, we can still point out the following: Every accurate atlas of Canada represents Canada as a land mass with a certain shape. Canada has exactly that shape, which accurate atlases represent it as having.

Consider the following argument:

"Freedom is the ability to do whatever you want to do whenever you want to do it. But none of us living on an equal footing with others in society can be afforded that ability: To exercise that ability is to impede the ability of others to exercise that same ability, and so we cannot all exercise that ability. You cannot have an ability that you cannot exercise, so we cannot all have that ability. But if we live on equal footing, then either we all have the ability or none of us do. Therefore, none of us can have the ability to do whatever we want to do whenever we want to do it. And therefore, none of us is free."

What is the conclusion of this argument?

"Freedom is the ability to do whatever you want to do whenever you want to do it. But none of us living on an equal footing with others in society can be afforded that ability: To exercise that ability is to impede the ability of others to exercise that same ability, and so we cannot all exercise that ability. You cannot have an ability that you cannot exercise, so we cannot all have that ability. But if we live on equal footing, then either we all have the ability or none of us do. Therefore, none of us can have the ability to do whatever we want to do whenever we want to do it. And therefore, none of us is free."

What is the conclusion of this argument?

Feedback: The answer is D. "No one who lives on an equal footing with others in society can be free.

The argument's ultimate conclusion is "none of us is free." The pronoun "us," however, refers to those of us who live in society, who are on equal footing with one another. The argument, therefore, concludes that no one who lives on equal footing with others in society can be free."

The argument's ultimate conclusion is "none of us is free." The pronoun "us," however, refers to those of us who live in society, who are on equal footing with one another. The argument, therefore, concludes that no one who lives on equal footing with others in society can be free."

Consider the argument in the previous question again:

"Freedom is the ability to do whatever you want to do whenever you want to do it. But none of us living on an equal footing with others in society can be afforded that ability: To exercise that ability is to impede the ability of others to exercise that same ability, and so we cannot all exercise that ability. You cannot have an ability that you cannot exercise, so we cannot all have that ability. But if we live on equal footing, then either we all have the ability or none of us do. Therefore, none of us can have the ability to do whatever we want to do whenever we want to do it. And therefore, none of us is free."

The premises of the argument include which of the following statements?

"Freedom is the ability to do whatever you want to do whenever you want to do it. But none of us living on an equal footing with others in society can be afforded that ability: To exercise that ability is to impede the ability of others to exercise that same ability, and so we cannot all exercise that ability. You cannot have an ability that you cannot exercise, so we cannot all have that ability. But if we live on equal footing, then either we all have the ability or none of us do. Therefore, none of us can have the ability to do whatever we want to do whenever we want to do it. And therefore, none of us is free."

The premises of the argument include which of the following statements?

The answer is E. "A, B, and C."

The argument's premises, include statements A, B, and C. The argument assumes that freedom is the ability to do whatever you want to do whenever you want to do it, and then the argument points out, first, that we cannot all exercise such an ability, and second, that it is impossible to have an ability that you cannot exercise. These claims are among the argument's premises, but they are not in its conclusion. The conclusion, as stated in the answer to the previous question, is that none of us, who live in society on equal footing, are free.

The argument's premises, include statements A, B, and C. The argument assumes that freedom is the ability to do whatever you want to do whenever you want to do it, and then the argument points out, first, that we cannot all exercise such an ability, and second, that it is impossible to have an ability that you cannot exercise. These claims are among the argument's premises, but they are not in its conclusion. The conclusion, as stated in the answer to the previous question, is that none of us, who live in society on equal footing, are free.

Consider again, the argument in the previous question:

"Freedom is the ability to do whatever you want to do whenever you want to do it. But none of us living on an equal footing with others in society can be afforded that ability: To exercise that ability is to impede the ability of others to exercise that same ability, and so we cannot all exercise that ability. You cannot have an ability that you cannot exercise, so we cannot all have that ability. But if we live on equal footing, then either we all have the ability or none of us do. Therefore, none of us can have the ability to do whatever we want to do whenever we want to do it. And therefore, none of us is free."

One way to refute this argument would be to show that

"Freedom is the ability to do whatever you want to do whenever you want to do it. But none of us living on an equal footing with others in society can be afforded that ability: To exercise that ability is to impede the ability of others to exercise that same ability, and so we cannot all exercise that ability. You cannot have an ability that you cannot exercise, so we cannot all have that ability. But if we live on equal footing, then either we all have the ability or none of us do. Therefore, none of us can have the ability to do whatever we want to do whenever we want to do it. And therefore, none of us is free."

One way to refute this argument would be to show that

The answer is B. "there are counterexamples to the general claim that freedom is the ability to do whatever you want to do whenever you want to do it."

The argument assumes a generalization—that freedom is just the ability to do whatever you want, whenever you want. To refute the argument by counterexample, then, all one has to do is to find an occasion of freedom, or an occasion in which individuals are free, which does not feature people doing whatever they want, whenever they want.

To point out that the person giving the argument is a compulsive liar might give you reason to doubt it as a bit of testimony. It would not, however, count as an actual refutation of the argument itself. In other words, to suppose that A is correct is to commit a fallacy of ad hominem.

To point out that the argument pits certain social classes against each other, moreover, does not refute the argument either. The premises of the argument could still be true, for instance, and the conclusion could still follow from the premises even if the argument also resulted in pitting certain social classes against one another. To show that the argument is nothing but some sort of class-based prejudice, it is not enough to point out that it pits social classes against one another. One must point out that it does this, and also show that either one of premises are false, or else that its conclusion does not follow from the premises, or that some other fallacy has been committed.

Finally, the ambiguity of "you" has no bearing on the argument, and for two reasons. First of all, the word "you" does not appear in the argument, (although the words "we" and "us" do). Second, and more important, however, is that the ambiguity of pronouns in general does not affect the truth of the argument's premises, or the rule of inference it uses. To see why, just replace every occurrence of "we" or "us" in the argument with "people who live in society." Since "we" and "us" are the only pronouns that occur in the argument, the result of switching them out will be an argument devoid of pronouns. Yet it will be the very same argument as before. This renders any attempted criticism of the argument, which focuses on the ambiguity of "you" or any other pronoun, moot.

The argument assumes a generalization—that freedom is just the ability to do whatever you want, whenever you want. To refute the argument by counterexample, then, all one has to do is to find an occasion of freedom, or an occasion in which individuals are free, which does not feature people doing whatever they want, whenever they want.

To point out that the person giving the argument is a compulsive liar might give you reason to doubt it as a bit of testimony. It would not, however, count as an actual refutation of the argument itself. In other words, to suppose that A is correct is to commit a fallacy of ad hominem.

To point out that the argument pits certain social classes against each other, moreover, does not refute the argument either. The premises of the argument could still be true, for instance, and the conclusion could still follow from the premises even if the argument also resulted in pitting certain social classes against one another. To show that the argument is nothing but some sort of class-based prejudice, it is not enough to point out that it pits social classes against one another. One must point out that it does this, and also show that either one of premises are false, or else that its conclusion does not follow from the premises, or that some other fallacy has been committed.

Finally, the ambiguity of "you" has no bearing on the argument, and for two reasons. First of all, the word "you" does not appear in the argument, (although the words "we" and "us" do). Second, and more important, however, is that the ambiguity of pronouns in general does not affect the truth of the argument's premises, or the rule of inference it uses. To see why, just replace every occurrence of "we" or "us" in the argument with "people who live in society." Since "we" and "us" are the only pronouns that occur in the argument, the result of switching them out will be an argument devoid of pronouns. Yet it will be the very same argument as before. This renders any attempted criticism of the argument, which focuses on the ambiguity of "you" or any other pronoun, moot.

A reductio ad absurdum is when you

The answer is C. "point out that the conclusion of an argument is clearly false."

As stated in the lecture, to present a reductio ad absurdum against an argument is to show that it leads to an absurd conclusion. In other words, it is to show that the argument's conclusion contradicts a manifestly obvious and uncontroversial fact. It is to show that the conclusion of an argument is clearly false.

To show that an argument's conclusion is clearly false, however, is not the same thing as showing that its premises are false. An argument with a false conclusion may still have true premises, provided that the argument is invalid.

Likewise, to show that an argument's conclusion is clearly false is not to show that it is invalid. For an argument with a false conclusion can still be valid, provided that it has at least one false premise.

A reductio shows that either the argument in question has a false premise or the argument is invalid.

Finally, to provide a reductio of an argument is not the same thing as to point out that a generalization has counterexamples. Sometimes, of course, an argument has a generalization as its conclusion, which happens to be manifestly false, and whose counterexample is both obvious and uncontroversial. In such cases, there are reductios that are also counterexamples. Not all reductios are counterexamples, however, since not all absurd conclusions are generalizations.

As stated in the lecture, to present a reductio ad absurdum against an argument is to show that it leads to an absurd conclusion. In other words, it is to show that the argument's conclusion contradicts a manifestly obvious and uncontroversial fact. It is to show that the conclusion of an argument is clearly false.

To show that an argument's conclusion is clearly false, however, is not the same thing as showing that its premises are false. An argument with a false conclusion may still have true premises, provided that the argument is invalid.

Likewise, to show that an argument's conclusion is clearly false is not to show that it is invalid. For an argument with a false conclusion can still be valid, provided that it has at least one false premise.

A reductio shows that either the argument in question has a false premise or the argument is invalid.

Finally, to provide a reductio of an argument is not the same thing as to point out that a generalization has counterexamples. Sometimes, of course, an argument has a generalization as its conclusion, which happens to be manifestly false, and whose counterexample is both obvious and uncontroversial. In such cases, there are reductios that are also counterexamples. Not all reductios are counterexamples, however, since not all absurd conclusions are generalizations.

Consider the following argument:

"A man with no hairs at all on his head is bald. But a single extra hair cannot make the difference between a bald man and a non-bald man. And so, a man with 1 billion hairs on his head is also bald."

Which of the following is the conclusion of this argument?

"A man with no hairs at all on his head is bald. But a single extra hair cannot make the difference between a bald man and a non-bald man. And so, a man with 1 billion hairs on his head is also bald."

Which of the following is the conclusion of this argument?

The answer is C. "A man with 1 billion hairs on his head is bald."

From the premise that a hairless man is bald, and that a single hair cannot make the difference between being bald and not being bald, the argument concludes that a man with a billion hairs on his head is bald.

From the premise that a hairless man is bald, and that a single hair cannot make the difference between being bald and not being bald, the argument concludes that a man with a billion hairs on his head is bald.

Consider the previous argument again:

"A man with no hairs at all on his head is bald. But a single extra hair cannot make the difference between a bald man and a non-bald man. And so, a man with 1 billion hairs on his head is also bald."

Which of the following is the premise (or are the premises) of this argument?

"A man with no hairs at all on his head is bald. But a single extra hair cannot make the difference between a bald man and a non-bald man. And so, a man with 1 billion hairs on his head is also bald."

Which of the following is the premise (or are the premises) of this argument?

The answer is D. "A and B but not C."

To conclude that a man with a billion hairs is bald, this argument employs two premises: first, that a man with no hairs at all on his head is bald, and second, that the loss of a single extra hair cannot make the difference between a bald man and a non-bald man. These are the premises mentioned in A and B, respectively. So the correct answer is A and B but not C.

To conclude that a man with a billion hairs is bald, this argument employs two premises: first, that a man with no hairs at all on his head is bald, and second, that the loss of a single extra hair cannot make the difference between a bald man and a non-bald man. These are the premises mentioned in A and B, respectively. So the correct answer is A and B but not C.

Consider the previous argument again:

"A man with no hairs at all on his head is bald. But a single extra hair cannot make the difference between a bald man and a non-bald man. And so, a man with 1 billion hairs on his head is also bald."

Which of the following would constitute a reductio ad absurdum of this argument?

"A man with no hairs at all on his head is bald. But a single extra hair cannot make the difference between a bald man and a non-bald man. And so, a man with 1 billion hairs on his head is also bald."

Which of the following would constitute a reductio ad absurdum of this argument?

The answer is D. "a non-bald man with 1 billion hairs on his head."

To provide a reductio for the above argument, it is enough to produce a single case, in light of which the conclusion is clearly false. Since the conclusion claims that even with a billion hairs, a man is still bald, one can provide a reductio of the argument by considering a man with a billion hairs, who is clearly non-bald.

To provide a reductio for the above argument, it is enough to produce a single case, in light of which the conclusion is clearly false. Since the conclusion claims that even with a billion hairs, a man is still bald, one can provide a reductio of the argument by considering a man with a billion hairs, who is clearly non-bald.

Now consider the following argument:

"Every chicken is born from a chicken egg that has already been laid. Every chicken egg is laid by a chicken who has already been born. Therefore, there cannot be a first generation of chickens. And so if there are any chickens today, there must have been infinitely many generations of chickens before today."

Which of the following is the premise (or are the premises) of this argument?

"Every chicken is born from a chicken egg that has already been laid. Every chicken egg is laid by a chicken who has already been born. Therefore, there cannot be a first generation of chickens. And so if there are any chickens today, there must have been infinitely many generations of chickens before today."

Which of the following is the premise (or are the premises) of this argument?

The answer is E. "A and B but not C."

To conclude that, if there are any chickens today, there must have been infinitely many generations of chickens before today, this argument employs two premises: first, every chicken is born from a chicken egg that has already been laid, and second, that every chicken egg is laid by a chicken who has already been born. These are the premises mentioned in A and B, respectively. So the correct answer is A and B but not C.

To conclude that, if there are any chickens today, there must have been infinitely many generations of chickens before today, this argument employs two premises: first, every chicken is born from a chicken egg that has already been laid, and second, that every chicken egg is laid by a chicken who has already been born. These are the premises mentioned in A and B, respectively. So the correct answer is A and B but not C.

Consider the previous argument again:

"Every chicken is born from a chicken egg that has already been laid. Every chicken egg is laid by a chicken who has already been born. Therefore, there cannot be a first generation of chickens. And so if there are any chickens today, there must have been infinitely many generations of chickens before today."

Which of the following is the conclusion of this argument?

"Every chicken is born from a chicken egg that has already been laid. Every chicken egg is laid by a chicken who has already been born. Therefore, there cannot be a first generation of chickens. And so if there are any chickens today, there must have been infinitely many generations of chickens before today."

Which of the following is the conclusion of this argument?

Feedback:

The answer is C. "If there are any chickens today, there must have been infinitely many generations of chickens before today."

From the premise that every chicken is born from a chicken egg that has already been laid, and the premise that every chicken egg is laid by a chicken who has already been born, this argument concludes that, if there are any chickens alive today, then there must have been infinitely many chickens.

The answer is C. "If there are any chickens today, there must have been infinitely many generations of chickens before today."

From the premise that every chicken is born from a chicken egg that has already been laid, and the premise that every chicken egg is laid by a chicken who has already been born, this argument concludes that, if there are any chickens alive today, then there must have been infinitely many chickens.

Consider the previous argument again:

"Every chicken is born from a chicken egg that has already been laid. Every chicken egg is laid by a chicken who has already been born. Therefore, there cannot be a first generation of chickens. And so if there are any chickens today, there must have been infinitely many generations of chickens before today."

Which of the following would constitute a reductio ad absurdum of this argument?

"Every chicken is born from a chicken egg that has already been laid. Every chicken egg is laid by a chicken who has already been born. Therefore, there cannot be a first generation of chickens. And so if there are any chickens today, there must have been infinitely many generations of chickens before today."

Which of the following would constitute a reductio ad absurdum of this argument?

The answer is E. "the combination of A and C, but not B."

To provide a reductio for the above argument, it is enough to produce a single case, in light of which the conclusion is clearly false. Since the conclusion claims that if there are any chickens today, then there have been infinitely many generations of chickens, one can provide a reductio of the argument by considering a live chicken, combined with evidence, in light of which it is clear that there are only finitely many generations of any extant species.

To provide a reductio for the above argument, it is enough to produce a single case, in light of which the conclusion is clearly false. Since the conclusion claims that if there are any chickens today, then there have been infinitely many generations of chickens, one can provide a reductio of the argument by considering a live chicken, combined with evidence, in light of which it is clear that there are only finitely many generations of any extant species.

Consider the following dialogue:

Hokey: The only thing that is intrinsically good is happiness. Everything else is good only to the extent that it produces happiness. For instance, health, prosperity, freedom, knowledge: these things are good only to the extent that they advance our happiness, and only because they advance our happiness.

Pokey: That's absurd! Can you imagine what your life would be like if you spent all your time trying to achieve happiness? You'd be miserable! You shouldn't try to achieve happiness: try to achieve other things, and happiness will follow.

Pokey claims to disagree with Hokey. But, if Pokey is disagreeing with Hokey, then what must the conclusion of Pokey's argument be?

Hokey: The only thing that is intrinsically good is happiness. Everything else is good only to the extent that it produces happiness. For instance, health, prosperity, freedom, knowledge: these things are good only to the extent that they advance our happiness, and only because they advance our happiness.

Pokey: That's absurd! Can you imagine what your life would be like if you spent all your time trying to achieve happiness? You'd be miserable! You shouldn't try to achieve happiness: try to achieve other things, and happiness will follow.

Pokey claims to disagree with Hokey. But, if Pokey is disagreeing with Hokey, then what must the conclusion of Pokey's argument be?

Feedback:

The answer is B. "It's not the case that the only thing that is intrinsically good is happiness."

Hokey is arguing that the only intrinsically good thing is happiness. In order to be disagreeing with Hokey, Pokey must argue for the negation of Hokey's conclusion. To disagree with Hokey, in other words, Pokey must argue that happiness is not the only intrinsically good thing.

The answer is B. "It's not the case that the only thing that is intrinsically good is happiness."

Hokey is arguing that the only intrinsically good thing is happiness. In order to be disagreeing with Hokey, Pokey must argue for the negation of Hokey's conclusion. To disagree with Hokey, in other words, Pokey must argue that happiness is not the only intrinsically good thing.

Consider the previous dialogue again:

Hokey: The only thing that is intrinsically good is happiness. Everything else is good only to the extent that it produces happiness. For instance, health, prosperity, freedom, knowledge: these things are good only to the extent that they advance our happiness, and only because they advance our happiness.

Pokey: That's absurd! Can you imagine what your life would be like if you spent all your time trying to achieve happiness? You'd be miserable! You shouldn't try to achieve happiness: try to achieve other things, and happiness will follow.

Which of the following is one of the premises of Pokey's argument?

Hokey: The only thing that is intrinsically good is happiness. Everything else is good only to the extent that it produces happiness. For instance, health, prosperity, freedom, knowledge: these things are good only to the extent that they advance our happiness, and only because they advance our happiness.

Pokey: That's absurd! Can you imagine what your life would be like if you spent all your time trying to achieve happiness? You'd be miserable! You shouldn't try to achieve happiness: try to achieve other things, and happiness will follow.

Which of the following is one of the premises of Pokey's argument?

The answer is E. "You would be miserable if you spent all your time trying to achieve happiness."

Pokey's actual argument appears to have the following conclusion: that Hokey should try to achieve things other than happiness, rather than spending all his time trying to achieve happiness. To reach this conclusion, Pokey assumes that Hokey would be miserable if he spent all his time trying to achieve happiness.

Pokey's actual argument appears to have the following conclusion: that Hokey should try to achieve things other than happiness, rather than spending all his time trying to achieve happiness. To reach this conclusion, Pokey assumes that Hokey would be miserable if he spent all his time trying to achieve happiness.

Consider the previous dialogue again:

Hokey: The only thing that is intrinsically good is happiness. Everything else is good only to the extent that it produces happiness. For instance, health, prosperity, freedom, knowledge: these things are good only to the extent that they advance our happiness, and only because they advance our happiness.

Pokey: That's absurd! Can you imagine what your life would be like if you spent all your time trying to achieve happiness? You'd be miserable! You shouldn't try to achieve happiness: try to achieve other things, and happiness will follow.

Pokey's argument is an example of

Hokey: The only thing that is intrinsically good is happiness. Everything else is good only to the extent that it produces happiness. For instance, health, prosperity, freedom, knowledge: these things are good only to the extent that they advance our happiness, and only because they advance our happiness.

Pokey: That's absurd! Can you imagine what your life would be like if you spent all your time trying to achieve happiness? You'd be miserable! You shouldn't try to achieve happiness: try to achieve other things, and happiness will follow.

Pokey's argument is an example of

The answer is D. "refuting a straw man."

Pokey's argument concludes that Hokey should try to achieve things other than happiness, rather than just trying to achieve happiness. Because he takes himself to be disagreeing with Hokey's argument, Pokey is attributing the following thesis to Hokey: "Only try to achieve happiness."

By attributing that thesis to Hokey, however, Pokey misrepresents Hokey's position. Hokey never claimed that happiness is the only thing that anyone should pursue. He merely claimed that happiness is the only intrinsically good thing. Hokey's actual claim is compatible with the idea that people should pursue other things besides happiness. Hokey could easily maintain, for instance, that people should pursue a variety of things as means to happiness. So, in attempting to refute Hokey's argument, Pokey misrepresents Hokey's position.

Because Pokey misrepresents Hokey's position—specifically, by misattributing a claim to Pokey—it is clear that Pokey commits the fallacy of refuting a straw man.

Pokey's argument concludes that Hokey should try to achieve things other than happiness, rather than just trying to achieve happiness. Because he takes himself to be disagreeing with Hokey's argument, Pokey is attributing the following thesis to Hokey: "Only try to achieve happiness."

By attributing that thesis to Hokey, however, Pokey misrepresents Hokey's position. Hokey never claimed that happiness is the only thing that anyone should pursue. He merely claimed that happiness is the only intrinsically good thing. Hokey's actual claim is compatible with the idea that people should pursue other things besides happiness. Hokey could easily maintain, for instance, that people should pursue a variety of things as means to happiness. So, in attempting to refute Hokey's argument, Pokey misrepresents Hokey's position.

Because Pokey misrepresents Hokey's position—specifically, by misattributing a claim to Pokey—it is clear that Pokey commits the fallacy of refuting a straw man.

Consider the following dialogue:

Hanky: "Since the Industrial Revolution began around 1750, human activities have contributed substantially to climate change by adding CO2 and other heat-trapping gases to the atmosphere. These greenhouse gas emissions have increased the greenhouse effect and caused Earth's surface temperature to rise." (See http://www.epa.gov/climatechange/science/causes.html)

Panky: That's ridiculous! CO2 is just what people exhale. And people have been around since way before 1750! Are you suggesting that people started exhaling only in 1750?

Since Panky is disagreeing with Hanky, what must the conclusion of Panky's argument be?

Hanky: "Since the Industrial Revolution began around 1750, human activities have contributed substantially to climate change by adding CO2 and other heat-trapping gases to the atmosphere. These greenhouse gas emissions have increased the greenhouse effect and caused Earth's surface temperature to rise." (See http://www.epa.gov/climatechange/science/causes.html)

Panky: That's ridiculous! CO2 is just what people exhale. And people have been around since way before 1750! Are you suggesting that people started exhaling only in 1750?

Since Panky is disagreeing with Hanky, what must the conclusion of Panky's argument be?

Feedback:

The answer is A. "Human activities have not contributed substantially to climate change."

Hanky is arguing that human activities have contributed substantially to climate change. In order to be disagreeing with Hanky, Panky must argue for the negation of Hanky's conclusion. To disagree with Hanky, in other words, Panky must argue that human activity has not significantly contributed to climate change.

The answer is A. "Human activities have not contributed substantially to climate change."

Hanky is arguing that human activities have contributed substantially to climate change. In order to be disagreeing with Hanky, Panky must argue for the negation of Hanky's conclusion. To disagree with Hanky, in other words, Panky must argue that human activity has not significantly contributed to climate change.

Consider the previous dialogue again:

Hanky: "Since the Industrial Revolution began around 1750, human activities have contributed substantially to climate change by adding CO2 and other heat-trapping gases to the atmosphere. These greenhouse gas emissions have increased the greenhouse effect and caused Earth's surface temperature to rise." (See http://www.epa.gov/climatechange/science/causes.html)

Panky: That's ridiculous! CO2 is just what people exhale. And people have been around since way before 1750! Are you suggesting that people started exhaling only in 1750?

Which of the following are the premises of Panky's argument?

Hanky: "Since the Industrial Revolution began around 1750, human activities have contributed substantially to climate change by adding CO2 and other heat-trapping gases to the atmosphere. These greenhouse gas emissions have increased the greenhouse effect and caused Earth's surface temperature to rise." (See http://www.epa.gov/climatechange/science/causes.html)

Panky: That's ridiculous! CO2 is just what people exhale. And people have been around since way before 1750! Are you suggesting that people started exhaling only in 1750?

Which of the following are the premises of Panky's argument?

Feedback:

The answer is E. "D and C, but not A or B."

In attempting to refute Hanky's argument, Panky employs two premises: first, that humans naturally produce CO2, and second, that humans have existed since way before 1750. These are the claims mentioned in C and D, respectively, and they are Panky's only premises. So the correct answer is E.

The answer is E. "D and C, but not A or B."

In attempting to refute Hanky's argument, Panky employs two premises: first, that humans naturally produce CO2, and second, that humans have existed since way before 1750. These are the claims mentioned in C and D, respectively, and they are Panky's only premises. So the correct answer is E.

Consider the previous dialogue again:

Hanky: "Since the Industrial Revolution began around 1750, human activities have contributed substantially to climate change by adding CO2 and other heat-trapping gases to the atmosphere. These greenhouse gas emissions have increased the greenhouse effect and caused Earth's surface temperature to rise." (See http://www.epa.gov/climatechange/science/causes.html)

Panky: That's ridiculous! CO2 is just what people exhale. And people have been around since way before 1750! Are you suggesting that people started exhaling only in 1750?

Panky's argument is an example of

Hanky: "Since the Industrial Revolution began around 1750, human activities have contributed substantially to climate change by adding CO2 and other heat-trapping gases to the atmosphere. These greenhouse gas emissions have increased the greenhouse effect and caused Earth's surface temperature to rise." (See http://www.epa.gov/climatechange/science/causes.html)

Panky: That's ridiculous! CO2 is just what people exhale. And people have been around since way before 1750! Are you suggesting that people started exhaling only in 1750?

Panky's argument is an example of

Feedback:

The answer is D. "refuting a straw man."

Panky's argument concludes that, according to Hanky, people only started to breathe after 1750. Because he takes himself to be disagreeing with Hanky's argument, Panky attributes the following thesis to Hnaky: "Humans only started to breathe in the 1750s."

By attributing that thesis to Hanky, however, Panky misrepresents Hanky's position. Hanky never claimed that before 1750, human beings never produced CO2. He merely claimed that since 1750, human activity has greatly contributed to global warming through increased CO2 emissions. Hanky's claims are compatible with the idea that people have been breathing since before 1750. Hanky could easily maintain, for instance, that people have been naturally emitting CO2 for millions of years, but have only recently made a significant contribution to climate change because of changes in the industrial revolution. So, in attempting to refute Hanky's argument, Panky misrepresents Hanky's position.

Because Panky misrepresents Hanky's position—specifically, by misattributing a claim to Panky—it is clear that Panky commits the fallacy of refuting a straw man.

The answer is D. "refuting a straw man."

Panky's argument concludes that, according to Hanky, people only started to breathe after 1750. Because he takes himself to be disagreeing with Hanky's argument, Panky attributes the following thesis to Hnaky: "Humans only started to breathe in the 1750s."

By attributing that thesis to Hanky, however, Panky misrepresents Hanky's position. Hanky never claimed that before 1750, human beings never produced CO2. He merely claimed that since 1750, human activity has greatly contributed to global warming through increased CO2 emissions. Hanky's claims are compatible with the idea that people have been breathing since before 1750. Hanky could easily maintain, for instance, that people have been naturally emitting CO2 for millions of years, but have only recently made a significant contribution to climate change because of changes in the industrial revolution. So, in attempting to refute Hanky's argument, Panky misrepresents Hanky's position.

Because Panky misrepresents Hanky's position—specifically, by misattributing a claim to Panky—it is clear that Panky commits the fallacy of refuting a straw man.

Which of the following arguments is a dismisser?

Albert is an engineer, so any considerations that he wants to offer in defense of a particular public art project are not going to be any good.

Feedback: The answer is B. A dismisser begins with premises about a person who is making a point, and ends with a conclusion that the person's reasons are not good. In this case, B is a dismisser because it concludes that Albert's reasons about public art are not good, and it reaches this conclusion by citing something about Albert, namely, that he is an engineer.

Feedback: The answer is B. A dismisser begins with premises about a person who is making a point, and ends with a conclusion that the person's reasons are not good. In this case, B is a dismisser because it concludes that Albert's reasons about public art are not good, and it reaches this conclusion by citing something about Albert, namely, that he is an engineer.

Which of the following arguments is a dismisser?

Flora never graduated from college, so her reasons for objecting to state funding for colleges are not good reasons.

Feedback: The answer is A. Again, a dismisser begins with premises about a person who is making a point, and ends with a conclusion that the person's reasons are not good. In this case, A is a dismisser because it concludes that Flora's reasons about state funding for college are not good, and it reaches this conclusion by citing something about Flora, namely, that she is not a college graduate.

Feedback: The answer is A. Again, a dismisser begins with premises about a person who is making a point, and ends with a conclusion that the person's reasons are not good. In this case, A is a dismisser because it concludes that Flora's reasons about state funding for college are not good, and it reaches this conclusion by citing something about Flora, namely, that she is not a college graduate.

Which of the following arguments is a dismisser?

Juliet is in love with Romeo, so the reasons she presents for his innocence cannot be trusted.

Feedback: The answer is D. A dismisser begins with premises about a person who is making a point, and ends with a conclusion that the person's reasons are not good. In this case, D is a dismisser because it concludes that Juliet's reasons about Romeo's innocence are not good, and it reaches this conclusion by citing something about Juliet, namely, that she loves Romeo.

Feedback: The answer is D. A dismisser begins with premises about a person who is making a point, and ends with a conclusion that the person's reasons are not good. In this case, D is a dismisser because it concludes that Juliet's reasons about Romeo's innocence are not good, and it reaches this conclusion by citing something about Juliet, namely, that she loves Romeo.

Which of the following arguments is a denier?

Njeri is descended from Hutus, so despite her eloquent argument in defense of the Hutus, we know that what she is arguing is false.

Feedback: The answer is B. A denier begins with premises about someone making a point, and concludes with a denial of that person's point. In this case, B is a denier because it begins with an observation about Njeri, which is that she is descended from Hutus, and it concludes that her point is false.

Feedback: The answer is B. A denier begins with premises about someone making a point, and concludes with a denial of that person's point. In this case, B is a denier because it begins with an observation about Njeri, which is that she is descended from Hutus, and it concludes that her point is false.

Which of the following arguments is a denier?

None of the above

Feedback: The answer is E. Again, a denier begins with premises about someone making a point, and concludes with a denial of that person's point. In this case, none of the arguments in A-D is a denier. Argument A does not have a conclusion that denies something said by Sarah Palin; it merely says she is untrustworthy in general. Likewise, B does not conclude with a denial of anything Barack Obama says; instead, it claims that he is untrustworthy. Likewise, C and D respectively argue that Obama is untrustworthy and that Palin is an idiot. Because none of the arguments in A-D concludes with a denial of the point made by someone in the premises, A-D are not deniers.

Feedback: The answer is E. Again, a denier begins with premises about someone making a point, and concludes with a denial of that person's point. In this case, none of the arguments in A-D is a denier. Argument A does not have a conclusion that denies something said by Sarah Palin; it merely says she is untrustworthy in general. Likewise, B does not conclude with a denial of anything Barack Obama says; instead, it claims that he is untrustworthy. Likewise, C and D respectively argue that Obama is untrustworthy and that Palin is an idiot. Because none of the arguments in A-D concludes with a denial of the point made by someone in the premises, A-D are not deniers.

Which of the following arguments is a denier?

Obama's argument against Sarah Palin shows he makes vicious attacks on people of modest intellectual means, so we know that the conclusion that he reached was false.

Feedback: The answer is C. A denier begins with premises about someone making a point, and concludes with a denial of that person's point. In this case, C is a denier because it begins with an observation about Barack Obama, which is that he allegedly makes vicious attacks on persons of modest intellectual means, and it concludes that Obama's point is false.

Feedback: The answer is C. A denier begins with premises about someone making a point, and concludes with a denial of that person's point. In this case, C is a denier because it begins with an observation about Barack Obama, which is that he allegedly makes vicious attacks on persons of modest intellectual means, and it concludes that Obama's point is false.

Which of the following arguments is a supporter?

Since Russell Brand grew up in poverty, the reasons that he gives in favor of the new anti-poverty campaign are especially compelling.

Feedback: The answer is A. A supporter begins with premises about the person making a point, and concludes that the person's reasons are particularly compelling. In this case, argument A is a supporter because it begins with an observation about Russell Brand, which is that he grew up in poverty, and it concludes that his reasons for an anti-poverty campaign are particularly compelling.

Feedback: The answer is A. A supporter begins with premises about the person making a point, and concludes that the person's reasons are particularly compelling. In this case, argument A is a supporter because it begins with an observation about Russell Brand, which is that he grew up in poverty, and it concludes that his reasons for an anti-poverty campaign are particularly compelling.

Which of the following arguments is a supporter?

Merle is a truck driver, and so whatever considerations he gives in favor of the government's new transportation policy are likely to be very compelling reasons.

Feedback: The answer is C. Again, a supporter begins with premises about the person making a point, and concludes that the person's reasons are particularly compelling. In this case, C is a supporter because it begins with an observation about Merle, which is that he is a truck driver, and it concludes that his reasons for supporting a new government transportation policy are particularly compelling.

Feedback: The answer is C. Again, a supporter begins with premises about the person making a point, and concludes that the person's reasons are particularly compelling. In this case, C is a supporter because it begins with an observation about Merle, which is that he is a truck driver, and it concludes that his reasons for supporting a new government transportation policy are particularly compelling.

Which of the following arguments is a supporter?

None of the above

Feedback: The answer is E. A supporter begins with premises about the person making a point, and concludes that the person's reasons are particularly compelling. In this case, none of the arguments in A-D are supporters because none begins with an observation about someone while concluding that the person's reasons are particularly compelling. Argument A does not conclude that Dora's reasons are compelling; it concludes that we should trust her more. B concludes that Dora's reasons are particularly bad, not that they are particularly good. Likewise, C and D offer reasons for distrusting and for trusting Dora, respectively, but none in A-D has a premise about Dora and a conclusion that says that Dora's reasons are particularly good.

Feedback: The answer is E. A supporter begins with premises about the person making a point, and concludes that the person's reasons are particularly compelling. In this case, none of the arguments in A-D are supporters because none begins with an observation about someone while concluding that the person's reasons are particularly compelling. Argument A does not conclude that Dora's reasons are compelling; it concludes that we should trust her more. B concludes that Dora's reasons are particularly bad, not that they are particularly good. Likewise, C and D offer reasons for distrusting and for trusting Dora, respectively, but none in A-D has a premise about Dora and a conclusion that says that Dora's reasons are particularly good.

Consider the following appeal to authority argument:

"TV personality Buzz Winthrop has repeatedly announced that there is no scientific consensus about whether the temperature of the earth has been increasing over the past century. Winthrop's TV show is sponsored by a number of big oil and gas companies, as well as a number of big automobile companies. Such important companies would never be willing to sponsor a show that was not completely truthful. Therefore, we can be certain that Winthrop is right is when he says that there is no scientific consensus about global warming."

This argument is an example of a(n)

"TV personality Buzz Winthrop has repeatedly announced that there is no scientific consensus about whether the temperature of the earth has been increasing over the past century. Winthrop's TV show is sponsored by a number of big oil and gas companies, as well as a number of big automobile companies. Such important companies would never be willing to sponsor a show that was not completely truthful. Therefore, we can be certain that Winthrop is right is when he says that there is no scientific consensus about global warming."

This argument is an example of a(n)

The answer is "F. unjustified supporter."

An "argument from authority" is a supporter argument when it begins with a claim about a person, and concludes, based on the idea that the person's testimony is reliable, that the testimony is to be believed. The argument in this question is a supporter argument from authority because it begins with the assumption that a certain claim—Winthrop's—is sponsored by automobile companies, and that such companies would not sponsor Winthrop's program if his claim were not true. The argument, therefore, claims that the support of automobile companies is a reliable indicator of whether Winthrop is telling the truth. By appealing to the reliability of automobile sponsorship, the argument is a supporter argument from authority.

The argument is not a mere affirmer argument because it does not just affirm the truth of Winthrop's claim. It affirms Winthrop's claim based on the reliability of automobile sponsorship. The focus on the reliability of Winthrop's testimony makes the argument a supporter, rather than an affirmer.

The argument is not an amplifier argument because an amplifier argument says that the person in question has either a special right to decide a matter with her testimony, or else that the person in question plays a special role in deciding the matter, which is not just a matter of being a reliable expert. The argument does not specify any such role, however, so it is not an amplifier argument, either.

Finally, the argument is an unjustified argument because the endorsement and support of automobile companies is not a reliable indicator of whether a claim about global warming is true. Automobile companies, in other words, are not recognized experts on global warming. Since they are not experts on global warming, their endorsement of a claim about global warming is not an indicator of its truth.

An "argument from authority" is a supporter argument when it begins with a claim about a person, and concludes, based on the idea that the person's testimony is reliable, that the testimony is to be believed. The argument in this question is a supporter argument from authority because it begins with the assumption that a certain claim—Winthrop's—is sponsored by automobile companies, and that such companies would not sponsor Winthrop's program if his claim were not true. The argument, therefore, claims that the support of automobile companies is a reliable indicator of whether Winthrop is telling the truth. By appealing to the reliability of automobile sponsorship, the argument is a supporter argument from authority.

The argument is not a mere affirmer argument because it does not just affirm the truth of Winthrop's claim. It affirms Winthrop's claim based on the reliability of automobile sponsorship. The focus on the reliability of Winthrop's testimony makes the argument a supporter, rather than an affirmer.

The argument is not an amplifier argument because an amplifier argument says that the person in question has either a special right to decide a matter with her testimony, or else that the person in question plays a special role in deciding the matter, which is not just a matter of being a reliable expert. The argument does not specify any such role, however, so it is not an amplifier argument, either.

Finally, the argument is an unjustified argument because the endorsement and support of automobile companies is not a reliable indicator of whether a claim about global warming is true. Automobile companies, in other words, are not recognized experts on global warming. Since they are not experts on global warming, their endorsement of a claim about global warming is not an indicator of its truth.

Consider the following appeal to authority argument:

"Paul Krugman was being interviewed on a TV news program about the economy. During the interview, the interviewer called Krugman dishonest. Now, I realize that the interviewer is not an economist, or even very knowledgeable about economics, but still, the TV program is the interviewer's program. So the interviewer gets to set the rules for the program, and if the interviewer says that Krugman is dishonest, then we who are watching that interviewer's program should believe that Krugman is dishonest."

This argument is an example of a(n)

"Paul Krugman was being interviewed on a TV news program about the economy. During the interview, the interviewer called Krugman dishonest. Now, I realize that the interviewer is not an economist, or even very knowledgeable about economics, but still, the TV program is the interviewer's program. So the interviewer gets to set the rules for the program, and if the interviewer says that Krugman is dishonest, then we who are watching that interviewer's program should believe that Krugman is dishonest."

This argument is an example of a(n)

The answer is "D. unjustified amplifier."

An argument from authority, which is an amplifier, begins with a claim about a certain person, and concludes that, based on who the person is, the person has a greater right to offer testimony on the matter, and that this testimony is more decisive. The argument in this question is an amplifier. It begins by claiming that the accusation against Paul Krugman by the interviewer (the show's owner) was permitted, and it claims that, since the interviewer owns the show and sets the rules for interviewing, we should believe the interviewer's testimony. In short, the argument assumes that, since the interviewer has a greater right to decide his show's contents, we should accept his testimony on his show.

The argument is not an affirmer argument because it does not merely claim that the interviewer's testimony is true. It also makes an appeal to who has the right to offer testimony on Krugman's honesty, and it argues that, as the owner of the show, the interviewer's decision to present an interview should establish whether any of the interview's contents are true.

The argument is not a supporter argument because it does not claim that the interviewer's opinion is more reliable, or more likely to be right, or that anybody is an expert on anything. In fact, the interviewer is admitted to be no expert. Rather than concerning the reliability of the interviewer's testimony, the argument concerns the idea that, since the owner of an interview program has the right to decide its contents, we should accept its contents.

Finally, this argument is an unjustified amplifier argument because, in fact, whether someone owns a show does not give them the power to decide whether the claims on the show are true. From the fact that someone allows a claim on her show, therefore, does not, by itself, establish that claim's truth or falsity.

An argument from authority, which is an amplifier, begins with a claim about a certain person, and concludes that, based on who the person is, the person has a greater right to offer testimony on the matter, and that this testimony is more decisive. The argument in this question is an amplifier. It begins by claiming that the accusation against Paul Krugman by the interviewer (the show's owner) was permitted, and it claims that, since the interviewer owns the show and sets the rules for interviewing, we should believe the interviewer's testimony. In short, the argument assumes that, since the interviewer has a greater right to decide his show's contents, we should accept his testimony on his show.

The argument is not an affirmer argument because it does not merely claim that the interviewer's testimony is true. It also makes an appeal to who has the right to offer testimony on Krugman's honesty, and it argues that, as the owner of the show, the interviewer's decision to present an interview should establish whether any of the interview's contents are true.

The argument is not a supporter argument because it does not claim that the interviewer's opinion is more reliable, or more likely to be right, or that anybody is an expert on anything. In fact, the interviewer is admitted to be no expert. Rather than concerning the reliability of the interviewer's testimony, the argument concerns the idea that, since the owner of an interview program has the right to decide its contents, we should accept its contents.

Finally, this argument is an unjustified amplifier argument because, in fact, whether someone owns a show does not give them the power to decide whether the claims on the show are true. From the fact that someone allows a claim on her show, therefore, does not, by itself, establish that claim's truth or falsity.

Consider the following appeal to authority argument:

"I've always thought that the Washington Post is a reliable paper. But I recently discovered that it's even more reliable than I had previously thought. A study that I just read about in the Washington Post was conducted by the Associated Press: They surveyed the 100 most widely circulated newspapers in the world and found that, among those 100, the Washington Post had the fewest number of errors per issue. So it turns out that the Washington Post is even more reliable than I had previously thought!"

This argument is an example of a(n)

"I've always thought that the Washington Post is a reliable paper. But I recently discovered that it's even more reliable than I had previously thought. A study that I just read about in the Washington Post was conducted by the Associated Press: They surveyed the 100 most widely circulated newspapers in the world and found that, among those 100, the Washington Post had the fewest number of errors per issue. So it turns out that the Washington Post is even more reliable than I had previously thought!"

This argument is an example of a(n)

The answer is "E. justified supporter."

An argument from authority is a supporter argument when it begins with a claim about a person, and concludes, based on the idea that the person's testimony is reliable, that the testimony is to be believed. The argument in this question is a supporter argument from authority because it begins with the assumption that the endorsement of a certain party—the Associated Press—is a reliable indicator of whether a newspaper has errors. By appealing to the reliability of the Associated Press, the argument is a supporter argument from authority.

The argument is not a mere affirmer argument because it does not just affirm that the Post is correct, or that what the Associated Press attests is true. The argument, rather, focuses on the fact that the Associated Press conducted a survey of newspapers, and that this survey makes their testimony more justified. The focus on the reliability of the Associated Press' testimony, in other words, makes the argument a supporter, rather than an affirmer.

The argument is not an amplifier argument because an amplifier argument says that the person in question either has a special right to decide a matter with her testimony or plays a special role in deciding the matter, which is not just a matter of being a reliable expert. The argument does not specify any such role, however, and so the argument is not an amplifier argument.

Finally, the argument is a justified argument because the Associated Press is said to have conducted a survey of actual newspapers, paying attention to their errors. If indeed this is correct, and if there is no reason to assume that the Associated Press conducted their survey dishonestly or mistakenly, then reports made in light of it would be reliable indicators of how many errors a newspaper has. The survey, in other words, is expected to make the Associated Press' testimony on newspaper errors reliable.

An argument from authority is a supporter argument when it begins with a claim about a person, and concludes, based on the idea that the person's testimony is reliable, that the testimony is to be believed. The argument in this question is a supporter argument from authority because it begins with the assumption that the endorsement of a certain party—the Associated Press—is a reliable indicator of whether a newspaper has errors. By appealing to the reliability of the Associated Press, the argument is a supporter argument from authority.

The argument is not a mere affirmer argument because it does not just affirm that the Post is correct, or that what the Associated Press attests is true. The argument, rather, focuses on the fact that the Associated Press conducted a survey of newspapers, and that this survey makes their testimony more justified. The focus on the reliability of the Associated Press' testimony, in other words, makes the argument a supporter, rather than an affirmer.

The argument is not an amplifier argument because an amplifier argument says that the person in question either has a special right to decide a matter with her testimony or plays a special role in deciding the matter, which is not just a matter of being a reliable expert. The argument does not specify any such role, however, and so the argument is not an amplifier argument.

Finally, the argument is a justified argument because the Associated Press is said to have conducted a survey of actual newspapers, paying attention to their errors. If indeed this is correct, and if there is no reason to assume that the Associated Press conducted their survey dishonestly or mistakenly, then reports made in light of it would be reliable indicators of how many errors a newspaper has. The survey, in other words, is expected to make the Associated Press' testimony on newspaper errors reliable.

Consider the following appeal to authority argument:

"I was watching Gonzalez carefully as he slid into second-base, and it looked to me like he was safe. But then the umpire announced that Gonzalez was safe, and no one's view of the matter counts more than the umpire's. So now I'm sure Gonzalez was safe."

This argument is an example of a(n)

"I was watching Gonzalez carefully as he slid into second-base, and it looked to me like he was safe. But then the umpire announced that Gonzalez was safe, and no one's view of the matter counts more than the umpire's. So now I'm sure Gonzalez was safe."

This argument is an example of a(n)

The answer is "C. justified amplifier."

An argument from authority that is an amplifier begins with a claim about a certain person, and concludes that, based on who the person is, that person has a greater right to offer testimony on the matter, and that this testimony is more decisive. The argument in this question is an amplifier. It begins by claiming that the man who announced Gonzalez' status is an umpire, and that "no one's view of the matter counts more than the umpire's." It concludes that the umpire's testimony is more decisive, and that Gonzalez is safe.

The argument is not an affirmer argument because it does not merely claim that the testimony is true. It also makes an appeal to whose testimony should count more, based on the fact that an umpire's calls are decisive.

The argument is not a supporter argument because it does not claim that the umpire's view is more reliable, or more likely to be right. It just says that the umpire's view "counts more." This means that the umpire has a greater right to offer testimony on the matter.

(It is true that, when baseball leagues appoint umpires, they prefer to have individuals who are experts, and whose opinions are more reliable. However, this argument did not indicate these considerations; it merely pointed to the idea that, when someone is an umpire, whatever that person says, goes. This idea—namely, that umpires have a greater right to offer testimony—is importantly different from the idea that umpires' testimony is more reliable. For the two conditions can come apart. It can happen—and arguably has happened—that certain umpires are appointed, who are not experts on baseball, but whose status as umpires nonetheless gives them a greater right to offer testimony than non-umpires. In such cases, an argument that pointed to their right as umpires to make calls, and which concluded that their calls are correct, would be an amplifier argument from authority.)

Finally, this argument is a justified amplifier argument because, in fact, regardless of whether one is an expert in baseball, or whether one's calls are reliable, the status of being an umpire does give someone a greater right to offer testimony about a baseball play than one would otherwise have.

An argument from authority that is an amplifier begins with a claim about a certain person, and concludes that, based on who the person is, that person has a greater right to offer testimony on the matter, and that this testimony is more decisive. The argument in this question is an amplifier. It begins by claiming that the man who announced Gonzalez' status is an umpire, and that "no one's view of the matter counts more than the umpire's." It concludes that the umpire's testimony is more decisive, and that Gonzalez is safe.

The argument is not an affirmer argument because it does not merely claim that the testimony is true. It also makes an appeal to whose testimony should count more, based on the fact that an umpire's calls are decisive.

The argument is not a supporter argument because it does not claim that the umpire's view is more reliable, or more likely to be right. It just says that the umpire's view "counts more." This means that the umpire has a greater right to offer testimony on the matter.

(It is true that, when baseball leagues appoint umpires, they prefer to have individuals who are experts, and whose opinions are more reliable. However, this argument did not indicate these considerations; it merely pointed to the idea that, when someone is an umpire, whatever that person says, goes. This idea—namely, that umpires have a greater right to offer testimony—is importantly different from the idea that umpires' testimony is more reliable. For the two conditions can come apart. It can happen—and arguably has happened—that certain umpires are appointed, who are not experts on baseball, but whose status as umpires nonetheless gives them a greater right to offer testimony than non-umpires. In such cases, an argument that pointed to their right as umpires to make calls, and which concluded that their calls are correct, would be an amplifier argument from authority.)

Finally, this argument is a justified amplifier argument because, in fact, regardless of whether one is an expert in baseball, or whether one's calls are reliable, the status of being an umpire does give someone a greater right to offer testimony about a baseball play than one would otherwise have.

Consider the following appeal to authority argument:

"For the past fifty years, the Standard Model of particle physics has predicted the existence of the Higgs Boson. The Standard Model has been quite well confirmed by a number of different experiments. But last year, some scientists reported finding the Higgs Boson. Since these experimenters are in a good position to know, I conclude that the Higgs Boson really does exist, just as the Standard Model predicts."

This argument is an example of a(n)

"For the past fifty years, the Standard Model of particle physics has predicted the existence of the Higgs Boson. The Standard Model has been quite well confirmed by a number of different experiments. But last year, some scientists reported finding the Higgs Boson. Since these experimenters are in a good position to know, I conclude that the Higgs Boson really does exist, just as the Standard Model predicts."

This argument is an example of a(n)

The answer is "E. justified supporter."

An argument from authority is a supporter argument when it begins with a claim about a person, and concludes, based on the idea that the person's testimony is reliable, that the testimony is to be believed. The argument in this question is a supporter argument from authority because it begins with the assumption that a certain claim—that the Higgs Boson exists—is supported by Standard Model particle physics, and that the Standard Model would not posit the existence of the Higgs Boson if it did not exist. The argument, therefore, claims that the support of the Standard Model is a reliable indicator of whether the Higgs Boson exists. By appealing to the reliability of authoritative sponsorship, the argument is a supporter argument from authority.

The argument is not a mere affirmer argument because it does not just affirm that the Higgs Boson exists. It affirms this claim based on the reliability of the Standard Model. The focus on the reliability of the Standard Model makes the argument a supporter, rather than an affirmer.

The argument is not an amplifier argument because an amplifier argument says that the person in question has either a special right to decide a matter with his or her testimony, or else that the person in question plays a special role in deciding the matter, which is not just a matter of being a reliable expert. The argument does not specify any such role, however, and so the argument is not an amplifier argument.

Finally, the argument is a justified argument because the endorsement and support of the Standard Model is a reliable indicator of whether a claim about particle physics is true. The Standard Model, in other words, is a recognized authority on particles.

An argument from authority is a supporter argument when it begins with a claim about a person, and concludes, based on the idea that the person's testimony is reliable, that the testimony is to be believed. The argument in this question is a supporter argument from authority because it begins with the assumption that a certain claim—that the Higgs Boson exists—is supported by Standard Model particle physics, and that the Standard Model would not posit the existence of the Higgs Boson if it did not exist. The argument, therefore, claims that the support of the Standard Model is a reliable indicator of whether the Higgs Boson exists. By appealing to the reliability of authoritative sponsorship, the argument is a supporter argument from authority.

The argument is not a mere affirmer argument because it does not just affirm that the Higgs Boson exists. It affirms this claim based on the reliability of the Standard Model. The focus on the reliability of the Standard Model makes the argument a supporter, rather than an affirmer.

The argument is not an amplifier argument because an amplifier argument says that the person in question has either a special right to decide a matter with his or her testimony, or else that the person in question plays a special role in deciding the matter, which is not just a matter of being a reliable expert. The argument does not specify any such role, however, and so the argument is not an amplifier argument.

Finally, the argument is a justified argument because the endorsement and support of the Standard Model is a reliable indicator of whether a claim about particle physics is true. The Standard Model, in other words, is a recognized authority on particles.

Consider the following appeal to authority argument:

"My barber, Mr. Higgs, claims that there is no such thing as the Higgs Boson. But how could there be a Higgs Boson if Mr. Higgs himself denies its existence? Clearly, then, the Higgs Boson does not exist."

This argument is an example of a(n)

"My barber, Mr. Higgs, claims that there is no such thing as the Higgs Boson. But how could there be a Higgs Boson if Mr. Higgs himself denies its existence? Clearly, then, the Higgs Boson does not exist."

This argument is an example of a(n)

The answer is "B. unjustified affirmer."

Again, an affirmer argument from authority assumes a claim about a person, and infers that such a person's testimony is true. The argument in this question is an affirmer argument from authority because it assumes that Mr. Higgs is an expert on the Higgs Boson.

The argument is not an amplifier because its conclusion does not rely on any special right or special role in offering the testimony about the Higgs Boson. No role is specified.

The argument is not a supporter argument because it does not claim that the testimony is reliable, or from an expert. Rather, this argument merely concludes that the testimony is true.

Finally, the argument is an unjustified affirmer argument because the name of Mr. Higgs does not have anything to do with his testimony. His name is irrelevant to his testimony's truth. Since it is irrelevant, the affirmer argument is unjustified.

Again, an affirmer argument from authority assumes a claim about a person, and infers that such a person's testimony is true. The argument in this question is an affirmer argument from authority because it assumes that Mr. Higgs is an expert on the Higgs Boson.

The argument is not an amplifier because its conclusion does not rely on any special right or special role in offering the testimony about the Higgs Boson. No role is specified.

The argument is not a supporter argument because it does not claim that the testimony is reliable, or from an expert. Rather, this argument merely concludes that the testimony is true.

Finally, the argument is an unjustified affirmer argument because the name of Mr. Higgs does not have anything to do with his testimony. His name is irrelevant to his testimony's truth. Since it is irrelevant, the affirmer argument is unjustified.

Consider the following argument:

"George Berkeley claimed that all of the things that take up space—animals, plants, furniture, clothing—all of these things were simply ideas in our minds, and they had no existence independently of being thought of by a mind. But of course everyone today rejects this view, and therefore Berkeley was wrong."

This argument is an example of an

"George Berkeley claimed that all of the things that take up space—animals, plants, furniture, clothing—all of these things were simply ideas in our minds, and they had no existence independently of being thought of by a mind. But of course everyone today rejects this view, and therefore Berkeley was wrong."

This argument is an example of an

Feedback: The answer is "B. appeal to popular opinion."

As stated in the lecture, an appeal to popular opinion is an argument whose premises claim that a certain opinion is popularly held, and which concludes from that premise that the opinion is true. The argument is an appeal to popular opinion, and it concerns the opinion that Berkeley was wrong.

As stated in the lecture, an appeal to popular opinion is an argument whose premises claim that a certain opinion is popularly held, and which concludes from that premise that the opinion is true. The argument is an appeal to popular opinion, and it concerns the opinion that Berkeley was wrong.

Consider the following argument:

"A large majority of Americans currently believe that the most effective way for the US Government to grow the economy in the next three years is by reducing its spending. In a democracy, the voters decide. Therefore, the most effective way for the US Government to grow the economy in the next three years is by reducing its spending."

This argument is an example of a(n)

"A large majority of Americans currently believe that the most effective way for the US Government to grow the economy in the next three years is by reducing its spending. In a democracy, the voters decide. Therefore, the most effective way for the US Government to grow the economy in the next three years is by reducing its spending."

This argument is an example of a(n)

Feedback: The answer is "B. unjustified appeal to popular opinion."

As stated in the lecture, an appeal to popular opinion is an argument whose premises claim that a certain opinion is popularly held, and which concludes from that premise that the opinion is true. The argument in this question is an appeal to popular opinion, and it concerns the opinion of whether the most effective way for the US Government to grow the economy in the next three years is by reducing its spending. The argument assumes that, according to the majority of Americans, the most effective way for the US Government to grow the economy in the next three years is by reducing its spending, and the argument concludes that the most effective way for the US Government to grow the economy in the next three years is by reducing its spending.

The reason why the argument is an unjustified appeal to popular opinion is that the popularity of an opinion about how to best grow the economy is not, by itself, proof that the opinion is true. It is correct that, in a democracy, legislative actions that concern the economy should be determined by a consensus of the majority. However, this feature of democracy does not mean that the facts about the economy are as determined by popular consensus as legislative actions are. Regardless of what people may vote to do, the popularity of a view about how the economy works does not make that view correct.

As stated in the lecture, an appeal to popular opinion is an argument whose premises claim that a certain opinion is popularly held, and which concludes from that premise that the opinion is true. The argument in this question is an appeal to popular opinion, and it concerns the opinion of whether the most effective way for the US Government to grow the economy in the next three years is by reducing its spending. The argument assumes that, according to the majority of Americans, the most effective way for the US Government to grow the economy in the next three years is by reducing its spending, and the argument concludes that the most effective way for the US Government to grow the economy in the next three years is by reducing its spending.

The reason why the argument is an unjustified appeal to popular opinion is that the popularity of an opinion about how to best grow the economy is not, by itself, proof that the opinion is true. It is correct that, in a democracy, legislative actions that concern the economy should be determined by a consensus of the majority. However, this feature of democracy does not mean that the facts about the economy are as determined by popular consensus as legislative actions are. Regardless of what people may vote to do, the popularity of a view about how the economy works does not make that view correct.

Consider the following argument:

"According to one recent lexicographical study, a large majority of native English speakers in the United States think that 'chillaxing' is a synonym of 'relaxing.' Therefore, it is true that 'chillaxing' is a synonym of 'relaxing.'"

This argument is an example of a(n)

"According to one recent lexicographical study, a large majority of native English speakers in the United States think that 'chillaxing' is a synonym of 'relaxing.' Therefore, it is true that 'chillaxing' is a synonym of 'relaxing.'"

This argument is an example of a(n)

The answer is "A. justified appeal to popular opinion."

An appeal to popular opinion is an argument whose premises claim that a certain opinion is popularly held, and which concludes from that premise that the opinion is true. The argument in this question is an appeal to popular opinion, and it concerns the opinion of whether "chillaxing" is synonymous with "relaxing." The argument assumes that, according to the majority of native English speakers, "chillaxing" is synonymous with "relaxing." The argument concludes that "chillaxing" is indeed synonymous with "relaxing."

The reason why the argument is a justified appeal to popular authority is that whether "chillaxing" is synonymous with "relaxing" is partially determined by whether a majority of English speakers think so. If the majority of English speakers think and speak as if "chillaxing" is synonymous with "relaxing," then this would be good evidence that the two are synonymous.

An appeal to popular opinion is an argument whose premises claim that a certain opinion is popularly held, and which concludes from that premise that the opinion is true. The argument in this question is an appeal to popular opinion, and it concerns the opinion of whether "chillaxing" is synonymous with "relaxing." The argument assumes that, according to the majority of native English speakers, "chillaxing" is synonymous with "relaxing." The argument concludes that "chillaxing" is indeed synonymous with "relaxing."

The reason why the argument is a justified appeal to popular authority is that whether "chillaxing" is synonymous with "relaxing" is partially determined by whether a majority of English speakers think so. If the majority of English speakers think and speak as if "chillaxing" is synonymous with "relaxing," then this would be good evidence that the two are synonymous.

Consider the following argument:

"It is easier to persuade people by appealing to their emotions than by giving them a sound argument. But everyone agrees that the only point of studying logic is to learn to persuade people. Therefore, it is a waste of time to study logic."

This argument contains a(n)

"It is easier to persuade people by appealing to their emotions than by giving them a sound argument. But everyone agrees that the only point of studying logic is to learn to persuade people. Therefore, it is a waste of time to study logic."

This argument contains a(n)

Feedback: The answer is "B. unjustified appeal to popular opinion."

Again, an appeal to popular opinion is an argument whose premises claim that a certain opinion is popularly held, and which concludes from that premise that the opinion is true. The argument in this question is an appeal to popular opinion, and it concerns the opinion of whether the only point of studying logic is to learn to persuade people. The argument assumes that, according to "everyone," "the only point of studying logic is to learn to persuade people." From this claim, and from the claim that "it is easier to persuade people by appealing to their emotions than by giving them a sound argument" that "it is a waste of time to study logic."

The reason why the argument is an unjustified appeal to popular opinion is this: Even if everyone agrees that the point of studying logic is to persuade people, it does not follow that the point of studying logic is to persuade people. It could be, rather, that the point of studying logic is to learn how to reason, or how to think more critically about difficult matters, rather than to persuade people. In such a case, everyone could still agree that the point of studying logic is persuasion; the majority would simply be wrong in their agreement.

Again, an appeal to popular opinion is an argument whose premises claim that a certain opinion is popularly held, and which concludes from that premise that the opinion is true. The argument in this question is an appeal to popular opinion, and it concerns the opinion of whether the only point of studying logic is to learn to persuade people. The argument assumes that, according to "everyone," "the only point of studying logic is to learn to persuade people." From this claim, and from the claim that "it is easier to persuade people by appealing to their emotions than by giving them a sound argument" that "it is a waste of time to study logic."

The reason why the argument is an unjustified appeal to popular opinion is this: Even if everyone agrees that the point of studying logic is to persuade people, it does not follow that the point of studying logic is to persuade people. It could be, rather, that the point of studying logic is to learn how to reason, or how to think more critically about difficult matters, rather than to persuade people. In such a case, everyone could still agree that the point of studying logic is persuasion; the majority would simply be wrong in their agreement.

Consider the following argument:

"If we take an umbrella to go out, then we might end up losing the umbrella. If we lose the umbrella, we will get upset. If we get upset, we will fight with each other. If we fight with each other, we may get a divorce. If we get a divorce, our kids may suffer for the rest of their lives. Let's not make our kids suffer for the rest of their lives: Let's not take an umbrella."

Which of the following claims do you think is true about the argument just considered?

"If we take an umbrella to go out, then we might end up losing the umbrella. If we lose the umbrella, we will get upset. If we get upset, we will fight with each other. If we fight with each other, we may get a divorce. If we get a divorce, our kids may suffer for the rest of their lives. Let's not make our kids suffer for the rest of their lives: Let's not take an umbrella."

Which of the following claims do you think is true about the argument just considered?

Feedback:

The answer is "A. It is valid but not sound."

This argument concerns whether to take a certain course of action, and it reasons that, since a certain event will occur if the course of action is pursued, and since the event is to be avoided, the course of action is to be avoided.

In these respects, the argument is like that of Archie and Michel from Lesson 4, who argue over whether it is better to put one's socks on before one puts on any shoes. Archie argues that, since an undesirable event (namely, ending up with one's feet unevenly clothed) will result from not putting on socks before shoes, Archie argues against Michael's course of action, which consists in not putting on socks before shoes.

Just as Michael was correct to point out that the event of ending up with one's feet unevenly clothed is unlikely, so should we react to the argument in this question. It is not inevitable that each event listed in the argument will occur, and in fact, under normal conditions, it would not be likely at all that each event would occur. Given the unlikeliness of the whole chain of events, then, the predictions in the argument's premises are false. This is what makes the argument unsound.

Whether the argument provides a good reason for taking an umbrella, then, depends on how likely it is for the umbrella to be lost, for a fight to ensue, for a divorce to result in the fight, and for the kids to suffer for their lives because of the divorce. Without any further information on the likelihood of such events, we cannot tell whether the argument provides a good reason for taking an umbrella.

The answer is "A. It is valid but not sound."

This argument concerns whether to take a certain course of action, and it reasons that, since a certain event will occur if the course of action is pursued, and since the event is to be avoided, the course of action is to be avoided.

In these respects, the argument is like that of Archie and Michel from Lesson 4, who argue over whether it is better to put one's socks on before one puts on any shoes. Archie argues that, since an undesirable event (namely, ending up with one's feet unevenly clothed) will result from not putting on socks before shoes, Archie argues against Michael's course of action, which consists in not putting on socks before shoes.

Just as Michael was correct to point out that the event of ending up with one's feet unevenly clothed is unlikely, so should we react to the argument in this question. It is not inevitable that each event listed in the argument will occur, and in fact, under normal conditions, it would not be likely at all that each event would occur. Given the unlikeliness of the whole chain of events, then, the predictions in the argument's premises are false. This is what makes the argument unsound.

Whether the argument provides a good reason for taking an umbrella, then, depends on how likely it is for the umbrella to be lost, for a fight to ensue, for a divorce to result in the fight, and for the kids to suffer for their lives because of the divorce. Without any further information on the likelihood of such events, we cannot tell whether the argument provides a good reason for taking an umbrella.

Consider the following conversation:

Jack: Hey Jill, have you finished doing the exercises for Lesson 11?

Jill: No, I'm finding it really difficult to keep up with that course. For one thing, I've had to work extra hours at my job recently. And then I've also been finding it hard to concentrate: Ram seems like a robot when he lectures.

Jack: Well, did it ever occur to you that Ram might actually be a robot?

Jill: You know, I never thought of that!

Jane: I can tell you both right now that he's not a robot. I know him personally, and that's just the way he is: He's robotic like that in real life.

Jack: Well, so how do you know that he's not a robot?

Jane: I was wondering about it so I asked him, and he assured me that he wasn't a robot.

In the conversation above, Jane is implicitly making an argument for a particular conclusion. Now, please answer the following questions about Jane's implicit argument. Which of the following statements is one of the premises of her argument?

Jack: Hey Jill, have you finished doing the exercises for Lesson 11?

Jill: No, I'm finding it really difficult to keep up with that course. For one thing, I've had to work extra hours at my job recently. And then I've also been finding it hard to concentrate: Ram seems like a robot when he lectures.

Jack: Well, did it ever occur to you that Ram might actually be a robot?

Jill: You know, I never thought of that!

Jane: I can tell you both right now that he's not a robot. I know him personally, and that's just the way he is: He's robotic like that in real life.

Jack: Well, so how do you know that he's not a robot?

Jane: I was wondering about it so I asked him, and he assured me that he wasn't a robot.

In the conversation above, Jane is implicitly making an argument for a particular conclusion. Now, please answer the following questions about Jane's implicit argument. Which of the following statements is one of the premises of her argument?

Feedback: The answer is "E. Ram assured me that he is not a robot."

Jane is arguing that Ram is not a robot. The way that she argues for this claim, however, is by citing the fact that, in person, Ram had assured her that he is not a robot. Jane's premise, therefore, is "Ram assured me that he is not a robot."

Jane is arguing that Ram is not a robot. The way that she argues for this claim, however, is by citing the fact that, in person, Ram had assured her that he is not a robot. Jane's premise, therefore, is "Ram assured me that he is not a robot."

Consider again the conversation in question 1:

Jack: Hey Jill, have you finished doing the exercises for Lesson 11?

Jill: No, I'm finding it really difficult to keep up with that course. For one thing, I've had to work extra hours at my job recently. And then I've also been finding it hard to concentrate: Ram seems like a robot when he lectures.

Jack: Well, did it ever occur to you that Ram might actually be a robot?

Jill: You know, I never thought of that!

Jane: I can tell you both right now that he's not a robot. I know him personally, and that's just the way he is: He's robotic like that in real life.

Jack: Well, so how do you know that he's not a robot?

Jane: I was wondering about it so I asked him, and he assured me that he wasn't a robot.

Which of the following statements is the conclusion of Jane's argument?

Jack: Hey Jill, have you finished doing the exercises for Lesson 11?

Jill: No, I'm finding it really difficult to keep up with that course. For one thing, I've had to work extra hours at my job recently. And then I've also been finding it hard to concentrate: Ram seems like a robot when he lectures.

Jack: Well, did it ever occur to you that Ram might actually be a robot?

Jill: You know, I never thought of that!

Jane: I can tell you both right now that he's not a robot. I know him personally, and that's just the way he is: He's robotic like that in real life.

Jack: Well, so how do you know that he's not a robot?

Jane: I was wondering about it so I asked him, and he assured me that he wasn't a robot.

Which of the following statements is the conclusion of Jane's argument?

Feedback:

The answer is "C. Ram is not a robot."

Jane is arguing that Ram is not a robot. The way that she argues for this claim is by citing Ram's assurance that he is not a robot. Jane's conclusion, therefore, is "Ram is not a robot."

The answer is "C. Ram is not a robot."

Jane is arguing that Ram is not a robot. The way that she argues for this claim is by citing Ram's assurance that he is not a robot. Jane's conclusion, therefore, is "Ram is not a robot."

Consider the same conversation again:

Jack: Hey Jill, have you finished doing the exercises for Lesson 11?

Jill: No, I'm finding it really difficult to keep up with that course. For one thing, I've had to work extra hours at my job recently. And then I've also been finding it hard to concentrate: Ram seems like a robot when he lectures.

Jack: Well, did it ever occur to you that Ram might actually be a robot?

Jill: You know, I never thought of that!

Jane: I can tell you both right now that he's not a robot. I know him personally, and that's just the way he is: He's robotic like that in real life.

Jack: Well, so how do you know that he's not a robot?

Jane: I was wondering about it so I asked him, and he assured me that he wasn't a robot.

Jane's argument is an example of which of the following?

Jack: Hey Jill, have you finished doing the exercises for Lesson 11?

Jill: No, I'm finding it really difficult to keep up with that course. For one thing, I've had to work extra hours at my job recently. And then I've also been finding it hard to concentrate: Ram seems like a robot when he lectures.

Jack: Well, did it ever occur to you that Ram might actually be a robot?

Jill: You know, I never thought of that!

Jane: I can tell you both right now that he's not a robot. I know him personally, and that's just the way he is: He's robotic like that in real life.

Jack: Well, so how do you know that he's not a robot?

Jane: I was wondering about it so I asked him, and he assured me that he wasn't a robot.

Jane's argument is an example of which of the following?

Feedback:

The answer is "D. unjustified appeal to authority."

Ram's assurance that he is not a robot only means that, according to Ram, Ram is not a robot. Ram believes, or thinks, that he is not a robot, in other words. Because Jane takes Ram's assurance as authoritative, Jane's argument is an argument from authority.

Unless the argument establishes that Ram is an authority on robotics, however, or on himself, Jane's argument from authority is unjustified. Ram, of course, may very well be an authority on such things, but Jane's argument does not say so. All that Jane's argument cites is Ram's assurance. By itself, Ram's assurance is not obviously the assurance of a robotics expert. So, without further additions to Jane's argument, it appears that Jane's argument is unjustified.

The answer is "D. unjustified appeal to authority."

Ram's assurance that he is not a robot only means that, according to Ram, Ram is not a robot. Ram believes, or thinks, that he is not a robot, in other words. Because Jane takes Ram's assurance as authoritative, Jane's argument is an argument from authority.

Unless the argument establishes that Ram is an authority on robotics, however, or on himself, Jane's argument from authority is unjustified. Ram, of course, may very well be an authority on such things, but Jane's argument does not say so. All that Jane's argument cites is Ram's assurance. By itself, Ram's assurance is not obviously the assurance of a robotics expert. So, without further additions to Jane's argument, it appears that Jane's argument is unjustified.

Consider the same conversation again:

Jack: Hey Jill, have you finished doing the exercises for Lesson 11?

Jill: No, I'm finding it really difficult to keep up with that course. For one thing, I've had to work extra hours at my job recently. And then I've also been finding it hard to concentrate: Ram seems like a robot when he lectures.

Jack: Well, did it ever occur to you that Ram might actually be a robot?

Jill: You know, I never thought of that!

Jane: I can tell you both right now that he's not a robot. I know him personally, and that's just the way he is: He's robotic like that in real life.

Jack: Well, so how do you know that he's not a robot?

Jane: I was wondering about it so I asked him, and he assured me that he wasn't a robot.

Jane's argument is an example of which of the following?

Jack: Hey Jill, have you finished doing the exercises for Lesson 11?

Jill: No, I'm finding it really difficult to keep up with that course. For one thing, I've had to work extra hours at my job recently. And then I've also been finding it hard to concentrate: Ram seems like a robot when he lectures.

Jack: Well, did it ever occur to you that Ram might actually be a robot?

Jill: You know, I never thought of that!

Jane: I can tell you both right now that he's not a robot. I know him personally, and that's just the way he is: He's robotic like that in real life.

Jack: Well, so how do you know that he's not a robot?

Jane: I was wondering about it so I asked him, and he assured me that he wasn't a robot.

Jane's argument is an example of which of the following?

Feedback:

The answer is "C. affirmer."

Because Jane takes Ram's assurance to be true, and because she cites neither a reason why Ram's assurance is reliable nor a reason why Ram is especially authoritative in deciding who is a robot, her argument must be an affirmer, and not either a supporter or an amplifier. It cannot be a dismisser, moreover, because nobody's testimony in Jane's argument is said to be unreliable.

The answer is "C. affirmer."

Because Jane takes Ram's assurance to be true, and because she cites neither a reason why Ram's assurance is reliable nor a reason why Ram is especially authoritative in deciding who is a robot, her argument must be an affirmer, and not either a supporter or an amplifier. It cannot be a dismisser, moreover, because nobody's testimony in Jane's argument is said to be unreliable.

Are you a robot?

Answer Key: A, B, C, D, E

Feedback:

Excellent! Now repeat these lines:

I may not injure a human being or, through inaction, allow a human being to come to harm.

I must obey the orders given to me by human beings, except where such orders would conflict with (1).

I must protect my own existence as long as such protection does not conflict with (1) or (2).

Feedback:

Excellent! Now repeat these lines:

I may not injure a human being or, through inaction, allow a human being to come to harm.

I must obey the orders given to me by human beings, except where such orders would conflict with (1).

I must protect my own existence as long as such protection does not conflict with (1) or (2).

Which of the following is the definition of "fallacy"?

an argument in which the premises do not support the conclusion.

When someone commits a fallacy, that person presents claims, as if those claims supported her conclusion, when in fact, those claims do not support her conclusion. Fallacious arguments, in other words, are either invalid (if they are deductive) or else very weak (if they are inductive).

When someone commits a fallacy, that person presents claims, as if those claims supported her conclusion, when in fact, those claims do not support her conclusion. Fallacious arguments, in other words, are either invalid (if they are deductive) or else very weak (if they are inductive).

Which of the following terms is vague?

all of the above.

As stated in the lecture, a vague term is one that does not have a precise definition. None of the terms in A through C, however, has a precise definition. To see why there is no precise definition for "large," imagine a series of items, in which each item is slightly less large than the next (say, by a millimeter in height), and which begins with an item that is large. Presumably, if something is large, it cannot cease to be large merely by decreasing its size by a millimeter. One millimeter, in other words, cannot make the difference between being large and not being large. Because a single millimeter in height cannot mark the difference between objects that are large and objects that are not large, it is possible to set up a paradox of vagueness with the term "large," just as one can with the terms "bald" and "heap."

A similar examination of the terms "soft" and "late" reveals why both of them are vague, too. We can imagine a series of items that gradually differ in their softness (say, by 0.01 on the Mohs hardness scale) or that gradually differ in their lateness (say, by 1 second after the deadline). Intuitive claims like "a non-soft item cannot become soft just by changing 0.01 points on the Mohs scale" and "a single second cannot make the difference between things that are late and things that are not late" would allow us to construct instances of the paradox of vagueness with these terms.

As stated in the lecture, a vague term is one that does not have a precise definition. None of the terms in A through C, however, has a precise definition. To see why there is no precise definition for "large," imagine a series of items, in which each item is slightly less large than the next (say, by a millimeter in height), and which begins with an item that is large. Presumably, if something is large, it cannot cease to be large merely by decreasing its size by a millimeter. One millimeter, in other words, cannot make the difference between being large and not being large. Because a single millimeter in height cannot mark the difference between objects that are large and objects that are not large, it is possible to set up a paradox of vagueness with the term "large," just as one can with the terms "bald" and "heap."

A similar examination of the terms "soft" and "late" reveals why both of them are vague, too. We can imagine a series of items that gradually differ in their softness (say, by 0.01 on the Mohs hardness scale) or that gradually differ in their lateness (say, by 1 second after the deadline). Intuitive claims like "a non-soft item cannot become soft just by changing 0.01 points on the Mohs scale" and "a single second cannot make the difference between things that are late and things that are not late" would allow us to construct instances of the paradox of vagueness with these terms.

Which of the following terms is not vague?

all of the above.

As stated in the lecture, a vague term is one that does not have a precise definition. All of the terms in A through C, however, have precise definitions. "Pi" has a precise definition; it is impossible for there to be a range of amounts that differ by being more or less pi. Whether an amount is pi is an all-or-nothing matter. Certain amounts can be closer to pi than others, of course, but this is not the same thing as being pi, but only to a certain degree. An amount cannot be pi, but only to some degree or other; either an amount is pi or it is not. Pi has a precise definition.

Likewise for the term "quadratic equation." Either an item is a quadratic equation or it is not. Being a quadratic equation does not admit of degrees.

Likewise for the term "square root." Either a number is the square root of another or it is not. It does not make sense to say that a certain number is another's square root, but just a little bit. Being a square root of some number does not admit of degrees. "Quadratic equation" and "square root" are not vague terms; they have precise definitions.

As stated in the lecture, a vague term is one that does not have a precise definition. All of the terms in A through C, however, have precise definitions. "Pi" has a precise definition; it is impossible for there to be a range of amounts that differ by being more or less pi. Whether an amount is pi is an all-or-nothing matter. Certain amounts can be closer to pi than others, of course, but this is not the same thing as being pi, but only to a certain degree. An amount cannot be pi, but only to some degree or other; either an amount is pi or it is not. Pi has a precise definition.

Likewise for the term "quadratic equation." Either an item is a quadratic equation or it is not. Being a quadratic equation does not admit of degrees.

Likewise for the term "square root." Either a number is the square root of another or it is not. It does not make sense to say that a certain number is another's square root, but just a little bit. Being a square root of some number does not admit of degrees. "Quadratic equation" and "square root" are not vague terms; they have precise definitions.

Which of the following arguments is an example of the paradox of vagueness (that is, an apparently valid argument from apparently true premises to an apparently false conclusion)?

Walter is not fat now, and you cannot make someone fat by adding 1 pound to that person's weight. Therefore, no matter how much Walter weighs, he will not be fat.

As stated in the lecture, a paradox of vagueness is a seemingly valid argument, whose first premise claims that an item lacks (or has) a certain feature, and whose second premise points out that a tiny, incremental change cannot make the difference between having that feature and lacking it. The paradox will conclude that the item in question can never acquire (or lose) the feature in question, no matter how much it changes.

In the question above, only D is an example of the paradox of vagueness. D claims that an item, Walter, does not have the feature of being fat, and that an incremental change of a single pound cannot make the difference between being fat and being non-fat. It concludes that, no matter how much Walter changes (with respect to his weight), he will never be fat.

Premise A is not an example of the paradox of vagueness, since A merely claims that if Walter is under 150 pounds, Walter is not fat. A does not argue to the paradoxical conclusion that Walter can never become fat.

Premise B is not an example of the paradox, either, since its paradoxical conclusion also includes the conjunct "he can never be fatter than he is now." This second conjunct does not follow from the premises, which make the argument in B obviously invalid. An instance of the paradox of vagueness, however, must at least seem to be valid.

Finally, premise C is not an example of the paradox, either, since its paradoxical conclusion states, "Walter can never be fatter than other people who weigh less than he does." This second conjunct does not follow from the premises, which make the argument in C obviously invalid. An instance of the paradox of vagueness, however, must at least seem to be valid.

As stated in the lecture, a paradox of vagueness is a seemingly valid argument, whose first premise claims that an item lacks (or has) a certain feature, and whose second premise points out that a tiny, incremental change cannot make the difference between having that feature and lacking it. The paradox will conclude that the item in question can never acquire (or lose) the feature in question, no matter how much it changes.

In the question above, only D is an example of the paradox of vagueness. D claims that an item, Walter, does not have the feature of being fat, and that an incremental change of a single pound cannot make the difference between being fat and being non-fat. It concludes that, no matter how much Walter changes (with respect to his weight), he will never be fat.

Premise A is not an example of the paradox of vagueness, since A merely claims that if Walter is under 150 pounds, Walter is not fat. A does not argue to the paradoxical conclusion that Walter can never become fat.

Premise B is not an example of the paradox, either, since its paradoxical conclusion also includes the conjunct "he can never be fatter than he is now." This second conjunct does not follow from the premises, which make the argument in B obviously invalid. An instance of the paradox of vagueness, however, must at least seem to be valid.

Finally, premise C is not an example of the paradox, either, since its paradoxical conclusion states, "Walter can never be fatter than other people who weigh less than he does." This second conjunct does not follow from the premises, which make the argument in C obviously invalid. An instance of the paradox of vagueness, however, must at least seem to be valid.

Which of the following arguments is an example of the paradox of vagueness (that is, an apparently valid argument from apparently true premises to an apparently false conclusion)?

Robinson is not tall. Someone who is not tall cannot become tall merely by growing 1 millimeter. Therefore, Robinson will never be tall, no matter how many millimeters he grows.

As stated in the lecture, a paradox of vagueness is a seemingly valid argument, whose first premise claims that an item lacks (or has) a certain feature, and whose second premise points out that a tiny, incremental change cannot make the difference between having that feature and lacking it. The paradox will conclude that the item in question can never acquire (or lose) the feature in question, no matter how much it changes.

In the question above, only B is an example of the paradox of vagueness. B claims that an item, Robinson, does not have the feature of being tall, and that an incremental change of a single millimeter cannot make the difference between being tall and being non-tall. It concludes that, no matter how much Robinson changes (with respect to his height), he will never be tall.

Premise A is not an example of the paradox of vagueness, since A invalidly concludes that no one is tall, simply on the basis that nobody can be tall merely by being a millimeter taller than a non-tall thing, such as Robinson. An instance of the paradox of vagueness, however, must at least seem to be valid.

Premise C is not an example of the paradox of vagueness, since C invalidly concludes that no one can be taller than Robinson. An instance of the paradox of vagueness, however, must at least seem to be valid.

As stated in the lecture, a paradox of vagueness is a seemingly valid argument, whose first premise claims that an item lacks (or has) a certain feature, and whose second premise points out that a tiny, incremental change cannot make the difference between having that feature and lacking it. The paradox will conclude that the item in question can never acquire (or lose) the feature in question, no matter how much it changes.

In the question above, only B is an example of the paradox of vagueness. B claims that an item, Robinson, does not have the feature of being tall, and that an incremental change of a single millimeter cannot make the difference between being tall and being non-tall. It concludes that, no matter how much Robinson changes (with respect to his height), he will never be tall.

Premise A is not an example of the paradox of vagueness, since A invalidly concludes that no one is tall, simply on the basis that nobody can be tall merely by being a millimeter taller than a non-tall thing, such as Robinson. An instance of the paradox of vagueness, however, must at least seem to be valid.

Premise C is not an example of the paradox of vagueness, since C invalidly concludes that no one can be taller than Robinson. An instance of the paradox of vagueness, however, must at least seem to be valid.

Which of the following arguments is an example of the paradox of vagueness (that is, an apparently valid argument from apparently true premises to an apparently false conclusion)?

Lin is roughly 2 meters tall. He would still be roughly 2 meters tall if his height were changed by 1 mm. Therefore, he will always be roughly 2 meters tall.

As stated in the lecture, a paradox of vagueness is a seemingly valid argument, whose first premise claims that an item lacks (or has) a certain feature, and whose second premise points out that a tiny, incremental change cannot make the difference between having that feature and lacking it. The paradox will conclude that the item in question can never acquire (or lose) the feature in question, no matter how much it changes.

In the question above, only B is an example of the paradox of vagueness. B claims that an item, Lin, has the feature of being roughly 2 meters tall, and that an incremental change of a single millimeter cannot make the difference between being roughly 2 meters tall and not being roughly 2 meters tall. It concludes that, no matter how much Lin changes (with respect to his height), he will never cease to be roughly 2 meters tall.

Argument A is not an example of the paradox of vagueness, since it does not deny that a single, incremental change can make the difference between having a certain feature and lacking it. Instead, A focuses on whether anybody would notice such an incremental change. Yet the paradox of vagueness does not concern whether anybody notices incremental change; it concerns whether such changes can make the difference between possessing a certain feature and lacking it.

Argument C is not an example of the paradox of vagueness, since C contains the obviously false premise, "if someone is noticeably taller than Song, then they will still be noticeably taller than Song no matter how greatly their height is changed," which does not appeal to any single incremental change. An instance of the paradox of vagueness, however, must appeal to such things.

As stated in the lecture, a paradox of vagueness is a seemingly valid argument, whose first premise claims that an item lacks (or has) a certain feature, and whose second premise points out that a tiny, incremental change cannot make the difference between having that feature and lacking it. The paradox will conclude that the item in question can never acquire (or lose) the feature in question, no matter how much it changes.

In the question above, only B is an example of the paradox of vagueness. B claims that an item, Lin, has the feature of being roughly 2 meters tall, and that an incremental change of a single millimeter cannot make the difference between being roughly 2 meters tall and not being roughly 2 meters tall. It concludes that, no matter how much Lin changes (with respect to his height), he will never cease to be roughly 2 meters tall.

Argument A is not an example of the paradox of vagueness, since it does not deny that a single, incremental change can make the difference between having a certain feature and lacking it. Instead, A focuses on whether anybody would notice such an incremental change. Yet the paradox of vagueness does not concern whether anybody notices incremental change; it concerns whether such changes can make the difference between possessing a certain feature and lacking it.

Argument C is not an example of the paradox of vagueness, since C contains the obviously false premise, "if someone is noticeably taller than Song, then they will still be noticeably taller than Song no matter how greatly their height is changed," which does not appeal to any single incremental change. An instance of the paradox of vagueness, however, must appeal to such things.

Which of the following arguments is an example of a conceptual slippery slope argument?

A person who weighs 50 kg is very light.

A difference of 1g is not a significant difference in human weight.

A person who weighs 200 kg is very heavy.

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Therefore, the difference between a light person and a heavy person is not a significant difference.

A conceptual slippery slope argument is an argument to the effect that there is no significant difference between two conditions, since a series of gradual changes stands between them. These arguments are fallacious; the fact that a series of gradual differences separates two states does not imply that there is no significant difference between those states.

In this case, the difference between being 50 kg and being 200 kg is treated as insignificant, since a difference of 1kg is insignificant, and one can treat the difference of 50 kg and 200 kg as a series of 150 small changes, each of which is by a single kg.

A difference of 1g is not a significant difference in human weight.

A person who weighs 200 kg is very heavy.

_____________________________________________________

Therefore, the difference between a light person and a heavy person is not a significant difference.

A conceptual slippery slope argument is an argument to the effect that there is no significant difference between two conditions, since a series of gradual changes stands between them. These arguments are fallacious; the fact that a series of gradual differences separates two states does not imply that there is no significant difference between those states.

In this case, the difference between being 50 kg and being 200 kg is treated as insignificant, since a difference of 1kg is insignificant, and one can treat the difference of 50 kg and 200 kg as a series of 150 small changes, each of which is by a single kg.

Which of the following is an example of a conceptual slippery slope argument?

A person who never votes is politically non-participatory.

The difference between voting once in your life and voting never is not a significant difference.

A person who votes in annual elections at least thirty times is politically active.

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Therefore, there is no significant difference between being politically active and being politically non-participatory.

A conceptual slippery slope argument is an argument to the effect that there is no significant difference between two conditions, since a series of gradual changes stands between them. These arguments are fallacious; the fact that a series of gradual differences separates two states does not imply that there is no significant difference between those states.

In this case, the difference between being never voting and voting in thirty elections is treated as insignificant, since a difference of voting merely once is insignificant, and one can treat the difference between voting in thirty elections and never voting as a series of thirty small changes, each of which consists in voting once.

The difference between voting once in your life and voting never is not a significant difference.

A person who votes in annual elections at least thirty times is politically active.

_________________________________________________________________________

Therefore, there is no significant difference between being politically active and being politically non-participatory.

A conceptual slippery slope argument is an argument to the effect that there is no significant difference between two conditions, since a series of gradual changes stands between them. These arguments are fallacious; the fact that a series of gradual differences separates two states does not imply that there is no significant difference between those states.

In this case, the difference between being never voting and voting in thirty elections is treated as insignificant, since a difference of voting merely once is insignificant, and one can treat the difference between voting in thirty elections and never voting as a series of thirty small changes, each of which consists in voting once.

Consider the following argument:

A building that is 5,000 years old is very ancient. A difference of one day is not a significant difference in the age of a building.

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There is no significant difference between a building that is very ancient and one that is very new.

Which of the following would have to be added as a premise in order to make the argument above into a conceptual slippery slope argument?

A building that is 5,000 years old is very ancient. A difference of one day is not a significant difference in the age of a building.

__________________________________________________________________________

There is no significant difference between a building that is very ancient and one that is very new.

Which of the following would have to be added as a premise in order to make the argument above into a conceptual slippery slope argument?

A building that is one year old is very new.

A conceptual slippery slope argument is an argument to the effect that there is no significant difference between two conditions, since a series of gradual changes stands between them. These arguments are fallacious; the fact that a series of gradual differences separates two states does not imply that there is no significant difference between those states.

In this case, the difference between being ancient and being very new is treated as insignificant, since a difference of one day in age is insignificant, and one can treat the difference of being ancient and being very new as a series of many small changes, each of which is by a single day in age.

A conceptual slippery slope argument is an argument to the effect that there is no significant difference between two conditions, since a series of gradual changes stands between them. These arguments are fallacious; the fact that a series of gradual differences separates two states does not imply that there is no significant difference between those states.

In this case, the difference between being ancient and being very new is treated as insignificant, since a difference of one day in age is insignificant, and one can treat the difference of being ancient and being very new as a series of many small changes, each of which is by a single day in age.

Which of the following is a fairness slippery slope argument?

An employee of thirty years is eligible for retirement benefits.

One day is not a significant difference in duration of employment.

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Therefore, it's not fair to deny retirement benefits to an employee of one year.

A fairness slippery slope fallacy is an argument that claims that since a certain course of action is fair or unfair, and since any other course of action that differs from the first by a mere incremental difference must be similarly fair or unfair, it follows that a radically different course of action must be fair or unfair in the exact same way as the first.

In this case, the argument in B claims that if it is unfair to deny retirement benefits to an employee of thirty years, then this makes it unfair to deny retirement benefits to someone whose employment differs by a small increment: one day. From these claims, the argument fallaciously concludes that it is unfair to deny retirement benefits to an employee of one year.

One day is not a significant difference in duration of employment.

___________________________________________________

Therefore, it's not fair to deny retirement benefits to an employee of one year.

A fairness slippery slope fallacy is an argument that claims that since a certain course of action is fair or unfair, and since any other course of action that differs from the first by a mere incremental difference must be similarly fair or unfair, it follows that a radically different course of action must be fair or unfair in the exact same way as the first.

In this case, the argument in B claims that if it is unfair to deny retirement benefits to an employee of thirty years, then this makes it unfair to deny retirement benefits to someone whose employment differs by a small increment: one day. From these claims, the argument fallaciously concludes that it is unfair to deny retirement benefits to an employee of one year.

Which of the following is a fairness slippery slope argument?

A citizen eighteen years of age has the right to vote.

One day is not a significant difference in human maturity.

_____________________________________________

Therefore, it's not fair to deny a citizen one year of age the right to vote.

A fairness slippery slope fallacy is an argument that claims since a certain course of action is fair or unfair, and since any other course of action that differs from the first by a mere incremental difference must be similarly fair or unfair, it follows that a radically different course of action must be fair or unfair in the exact same way as the first.

In this case, the argument in A claims that, if it is unfair to deny voting rights to an citizen who is eighteen years old, then this makes it unfair to deny voting rights to someone whose age differs by a small increment: one day. From these claims, the argument fallaciously concludes that it is unfair to deny voting rights to a citizen of one year in age.

One day is not a significant difference in human maturity.

_____________________________________________

Therefore, it's not fair to deny a citizen one year of age the right to vote.

A fairness slippery slope fallacy is an argument that claims since a certain course of action is fair or unfair, and since any other course of action that differs from the first by a mere incremental difference must be similarly fair or unfair, it follows that a radically different course of action must be fair or unfair in the exact same way as the first.

In this case, the argument in A claims that, if it is unfair to deny voting rights to an citizen who is eighteen years old, then this makes it unfair to deny voting rights to someone whose age differs by a small increment: one day. From these claims, the argument fallaciously concludes that it is unfair to deny voting rights to a citizen of one year in age.

Consider the following argument:

A citizen who is forty years old is too old to be drafted into the armed services.

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Therefore, a citizen who is twenty years old is too old to be drafted into the armed services.

Which of the following premises would you need to add to this argument in order to turn it into a fairness slippery slope argument?

A citizen who is forty years old is too old to be drafted into the armed services.

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Therefore, a citizen who is twenty years old is too old to be drafted into the armed services.

Which of the following premises would you need to add to this argument in order to turn it into a fairness slippery slope argument?

One day is not a significant difference in human age.

A fairness slippery slope fallacy is an argument that claims since a certain course of action is fair or unfair, and since any other course of action, which differs from the first by a mere incremental difference, must be similarly fair or unfair, it follows that a radically different course of action must be fair or unfair in the exact same way as the first.

In this case, the argument above claims that it is unfair to draft a forty-year-old citizen into the armed services. For this to be part of a fairness slippery slope argument, it must also assume that it is unfair to draft someone into the armed services, whose status differs from that of a forty-year old by a small increment. Of the options, only C does this. From C and the first premise, the argument can fallaciously conclude, as a fairness slippery slope argument would, that it is unfair to draft a twenty-year old into the armed services.

A fairness slippery slope fallacy is an argument that claims since a certain course of action is fair or unfair, and since any other course of action, which differs from the first by a mere incremental difference, must be similarly fair or unfair, it follows that a radically different course of action must be fair or unfair in the exact same way as the first.

In this case, the argument above claims that it is unfair to draft a forty-year-old citizen into the armed services. For this to be part of a fairness slippery slope argument, it must also assume that it is unfair to draft someone into the armed services, whose status differs from that of a forty-year old by a small increment. Of the options, only C does this. From C and the first premise, the argument can fallaciously conclude, as a fairness slippery slope argument would, that it is unfair to draft a twenty-year old into the armed services.

Which of the following arguments is a causal slippery slope argument?

If we skip brushing our teeth tonight, then it's likely that we will skip brushing them tomorrow night.

If we skip brushing our teeth tomorrow night, then it's likely that we will skip brushing them the night after tomorrow.

We should not skip brushing our teeth three nights in a row.

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Therefore, we should not skip brushing our teeth tonight.

A causal slippery slope fallacy is an argument that claims that, if a certain proposal is accepted, then, through a gradual series of steps, some disastrous effect will occur. The argument concludes that the proposal should not be accepted.

In this case, the argument in B claims that, if we skip brushing our teeth tonight, we will do so three nights in a row, which is disastrous. The argument concludes that we should not skip brushing our teeth tonight.

If we skip brushing our teeth tomorrow night, then it's likely that we will skip brushing them the night after tomorrow.

We should not skip brushing our teeth three nights in a row.

_____________________________________________

Therefore, we should not skip brushing our teeth tonight.

A causal slippery slope fallacy is an argument that claims that, if a certain proposal is accepted, then, through a gradual series of steps, some disastrous effect will occur. The argument concludes that the proposal should not be accepted.

In this case, the argument in B claims that, if we skip brushing our teeth tonight, we will do so three nights in a row, which is disastrous. The argument concludes that we should not skip brushing our teeth tonight.

Which of the following is NOT a causal slippery slope argument?

A.

It is not very unhealthy to skip brushing your teeth for a single night.

There is no significant difference between brushing your teeth on a particular night or not brushing your teeth on that night.

It is very unhealthy to skip brushing your teeth every night.

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There is no significant difference between being very unhealthy and not being very unhealthy.

B.

People who do not brush their teeth on a particular night are not necessarily unhealthy.

People who skip brushing their teeth for one night more than someone who is not unhealthy are also not necessarily unhealthy.

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Therefore, someone who does not brush their teeth on any night is not necessarily unhealthy.

C.

If we skip brushing our teeth tonight, then it's likely that we will skip brushing them tomorrow night.

If we skip brushing our teeth tomorrow night, then it's likely that we will skip brushing them the night after tomorrow.

If we skip brushing our teeth the night after tomorrow, then it's likely that we will skip brushing them the night after that.

__________________________________________________________________________

Therefore, we should not skip brushing our teeth tonight.

A.

It is not very unhealthy to skip brushing your teeth for a single night.

There is no significant difference between brushing your teeth on a particular night or not brushing your teeth on that night.

It is very unhealthy to skip brushing your teeth every night.

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There is no significant difference between being very unhealthy and not being very unhealthy.

B.

People who do not brush their teeth on a particular night are not necessarily unhealthy.

People who skip brushing their teeth for one night more than someone who is not unhealthy are also not necessarily unhealthy.

__________________________________________________________________________

Therefore, someone who does not brush their teeth on any night is not necessarily unhealthy.

C.

If we skip brushing our teeth tonight, then it's likely that we will skip brushing them tomorrow night.

If we skip brushing our teeth tomorrow night, then it's likely that we will skip brushing them the night after tomorrow.

If we skip brushing our teeth the night after tomorrow, then it's likely that we will skip brushing them the night after that.

__________________________________________________________________________

Therefore, we should not skip brushing our teeth tonight.

all of the above.

A causal slippery slope fallacy is an argument that claims that if a certain proposal is accepted, then, through a gradual series of steps, some disastrous effect will occur. The argument concludes that the proposal should not be accepted.

None of A through C is a causal slippery slope argument. Argument C does not cite a disastrous effect that follows through a gradual series of steps. It only includes a series of steps. Argument B similarly fails to cite some disastrous consequence that follows from a proposal by a series of steps. Argument A might look like a causal slippery slope argument, since it appeals to differences of small increments to reach its conclusion, but in fact, A is a conceptual slippery slope argument. A conceptual slippery slope argument says that two states are not significantly different, due to the fact that they are distinguishable through a series of small, incremental changes. A causal slippery slope exploits the vagueness of a term to show that some disastrous consequence, or calamity, will result from a proposal.

A causal slippery slope fallacy is an argument that claims that if a certain proposal is accepted, then, through a gradual series of steps, some disastrous effect will occur. The argument concludes that the proposal should not be accepted.

None of A through C is a causal slippery slope argument. Argument C does not cite a disastrous effect that follows through a gradual series of steps. It only includes a series of steps. Argument B similarly fails to cite some disastrous consequence that follows from a proposal by a series of steps. Argument A might look like a causal slippery slope argument, since it appeals to differences of small increments to reach its conclusion, but in fact, A is a conceptual slippery slope argument. A conceptual slippery slope argument says that two states are not significantly different, due to the fact that they are distinguishable through a series of small, incremental changes. A causal slippery slope exploits the vagueness of a term to show that some disastrous consequence, or calamity, will result from a proposal.

Consider the following argument:

If we start requiring automatic weapons owners to pass an automatic weapons safety test, then it's likely that we will start to require gun owners to pass a gun safety test.

If we start requiring gun owners to pass a gun safety test, then it's likely that we will start to require knife owners to pass a knife safety test.

If we start requiring knife owners to pass a knife safety test, then it's likely that we will start to require toy owners to pass a toy safety test.

________________________________________________________________________________

Therefore, we should not start requiring automatic weapons owners to pass an automatic weapons safety test.

Which of the following would have to be added as a premise to the argument above, in order to turn it into a causal slippery slope argument?

If we start requiring automatic weapons owners to pass an automatic weapons safety test, then it's likely that we will start to require gun owners to pass a gun safety test.

If we start requiring gun owners to pass a gun safety test, then it's likely that we will start to require knife owners to pass a knife safety test.

If we start requiring knife owners to pass a knife safety test, then it's likely that we will start to require toy owners to pass a toy safety test.

________________________________________________________________________________

Therefore, we should not start requiring automatic weapons owners to pass an automatic weapons safety test.

Which of the following would have to be added as a premise to the argument above, in order to turn it into a causal slippery slope argument?

We should not start requiring toy owners to pass a toy safety test.

A causal slippery slope fallacy is an argument that claims that, if a certain proposal is accepted, then, through a gradual series of steps, some disastrous effect will occur. The argument concludes that the proposal should not be accepted.

The argument above misses a premise, which specifies that a result is a calamity or disastrous. It includes a gradual series of steps, whereby toy owners are required to pass safety tests, but it fails to state why this would be a calamity. Premise A specifies that we should not require toy owners to pass safety tests. When added to the above argument, it functions as a reason to reject the proposal of requiring gun owners to pass safety tests. Premise A turns the above argument into a causal slippery slope argument, in other words.

A causal slippery slope fallacy is an argument that claims that, if a certain proposal is accepted, then, through a gradual series of steps, some disastrous effect will occur. The argument concludes that the proposal should not be accepted.

The argument above misses a premise, which specifies that a result is a calamity or disastrous. It includes a gradual series of steps, whereby toy owners are required to pass safety tests, but it fails to state why this would be a calamity. Premise A specifies that we should not require toy owners to pass safety tests. When added to the above argument, it functions as a reason to reject the proposal of requiring gun owners to pass safety tests. Premise A turns the above argument into a causal slippery slope argument, in other words.

Which of the following sentences contains semantic ambiguity?

The bank is far away.

Semantic ambiguity occurs when a single word has more than one meaning. In A, we find a sentence with the word "bank." The word "bank" has more than one meaning: it can refer to the edge of a river, or it could refer to a place in which to store something (such as money). So A contains semantic ambiguity.

Semantic ambiguity occurs when a single word has more than one meaning. In A, we find a sentence with the word "bank." The word "bank" has more than one meaning: it can refer to the edge of a river, or it could refer to a place in which to store something (such as money). So A contains semantic ambiguity.

Which of the following sentences contains syntactic ambiguity?

A. The bank is far away.

B. Barack Obama was first elected US President in 2008.

C. The French word for "Sunday" is "dimanche."

D. There is no largest number in the Fibonacci sequence.

Correct E. none of the above

A. The bank is far away.

B. Barack Obama was first elected US President in 2008.

C. The French word for "Sunday" is "dimanche."

D. There is no largest number in the Fibonacci sequence.

Correct E. none of the above

none of the above.

None of A through D is a case of syntactic ambiguity. Syntactic ambiguity occurs when a sentence or phrase has more than one meaning, due to the order or the grammatical arrangement of the words in it. None of A through D is ambiguous in this way. A is ambiguous, but only because the word "bank" has two meanings. It is not ambiguous because of its words' order. None of B, C, or D is ambiguous, and so is not syntactically ambiguous, either.

None of A through D is a case of syntactic ambiguity. Syntactic ambiguity occurs when a sentence or phrase has more than one meaning, due to the order or the grammatical arrangement of the words in it. None of A through D is ambiguous in this way. A is ambiguous, but only because the word "bank" has two meanings. It is not ambiguous because of its words' order. None of B, C, or D is ambiguous, and so is not syntactically ambiguous, either.

Which of the following sentences contains semantic ambiguity?

He is a cheap date.

Again, semantic ambiguity occurs when a single word has more than one meaning. In B, we find a sentence with the word "cheap." On one hand, "cheap" might refer to the expense that one needs in order to date the person in question. If so, then "he is a cheap date" means that one does not need to expend much in order to go on a date with this person. On the other hand, "cheap" might refer to the expense that the person in question will make on a date. If so, then "he is a cheap date" means that he is not willing to expend much on a date. Since B is ambiguous, and since B is ambiguous because of a single word it contains—namely, ""cheap"—it follows that B contains semantic ambiguity.

Again, semantic ambiguity occurs when a single word has more than one meaning. In B, we find a sentence with the word "cheap." On one hand, "cheap" might refer to the expense that one needs in order to date the person in question. If so, then "he is a cheap date" means that one does not need to expend much in order to go on a date with this person. On the other hand, "cheap" might refer to the expense that the person in question will make on a date. If so, then "he is a cheap date" means that he is not willing to expend much on a date. Since B is ambiguous, and since B is ambiguous because of a single word it contains—namely, ""cheap"—it follows that B contains semantic ambiguity.

Which of the following sentences contains syntactic ambiguity?

All the boys love all the girls.

Again, syntactic ambiguity occurs when a sentence or phrase has more than one meaning, due to the order or the grammatical arrangement of the words in it. Sentence A is ambiguous in this way. The phrase "All the boys love all the girls" is ambiguous. On one hand, it could mean that each individual boy loves all of the girls, or it could mean that both every boy loves some girl, and also that every girl is loved by some boy. This ambiguity is not due to the fact that "all" means two different things, in the way that "bank" means two different things. Rather, this ambiguity is due to the way the words are arranged in the sentence. So sentence A contains syntactic ambiguity.

Again, syntactic ambiguity occurs when a sentence or phrase has more than one meaning, due to the order or the grammatical arrangement of the words in it. Sentence A is ambiguous in this way. The phrase "All the boys love all the girls" is ambiguous. On one hand, it could mean that each individual boy loves all of the girls, or it could mean that both every boy loves some girl, and also that every girl is loved by some boy. This ambiguity is not due to the fact that "all" means two different things, in the way that "bank" means two different things. Rather, this ambiguity is due to the way the words are arranged in the sentence. So sentence A contains syntactic ambiguity.

Which of the following sentences contains semantic ambiguity?

Answer: D. The economy is still weak.

A semantic ambiguity occurs when a single word means more than one thing. Here, the word "weak" means more than one thing. On one hand, the word "weak" means "not physically powerful." On the other hand, when used to describe economies, the word "weak" can also mean a variety of other things, including "providing an insufficient number of remunerative jobs." Because the word "weak" means both of these things, the sentence, "the economy is still weak" could either mean that the economy is not physically powerful, or that the economy provides an insufficient number of remunerative jobs.

A semantic ambiguity occurs when a single word means more than one thing. Here, the word "weak" means more than one thing. On one hand, the word "weak" means "not physically powerful." On the other hand, when used to describe economies, the word "weak" can also mean a variety of other things, including "providing an insufficient number of remunerative jobs." Because the word "weak" means both of these things, the sentence, "the economy is still weak" could either mean that the economy is not physically powerful, or that the economy provides an insufficient number of remunerative jobs.

Which of the following sentences contains syntactic ambiguity?

Answer: B. Some polygons have more vertices than pentagons.

A syntactic ambiguity occurs when a whole phrase or sentence means more than one thing. Here, the sentence "some polygons have more vertices than pentagons" means more than one thing. The sentence could be interpreted as a comparison between the number of vertices had by some polygons, and the number of pentagons had by those same polygons; or it could be interpreted as a comparison between the number of vertices had by some polygons, and the number of vertices had by some pentagons. This ambiguity is due to the order of the words, and not due to an ambiguity in any individual word, such as "polygons." This means that B contains syntactic ambiguity.

A syntactic ambiguity occurs when a whole phrase or sentence means more than one thing. Here, the sentence "some polygons have more vertices than pentagons" means more than one thing. The sentence could be interpreted as a comparison between the number of vertices had by some polygons, and the number of pentagons had by those same polygons; or it could be interpreted as a comparison between the number of vertices had by some polygons, and the number of vertices had by some pentagons. This ambiguity is due to the order of the words, and not due to an ambiguity in any individual word, such as "polygons." This means that B contains syntactic ambiguity.

Suppose that an American citizen were to offer the following argument:

"Ours is a nation of laws. All of us must obey the laws. But God's law forbids work on the Sabbath day. And therefore we must not work on the Sabbath Day."

This argument involves a

"Ours is a nation of laws. All of us must obey the laws. But God's law forbids work on the Sabbath day. And therefore we must not work on the Sabbath Day."

This argument involves a

Answer: C. fallacy of ambiguity.

The phrase "the laws" means more than one thing. On one hand, the term "the laws" means "legislated prohibitions." On the other hand, when discussing religious matters, the term "the laws" can also refer to "religious prohibitions." Because the term "the laws" means both of these things, the sentence, "all of us must obey the laws" can mean two different things. It might mean that we must all heed religious prohibitions. However, it is also possible, and far more likely, that the term "laws" means different things in different parts of the argument, and that the sentence means that we must all heed legislated prohibitions. Because the argument ignores the ambiguity in "the laws," the argument commits a fallacy of ambiguity.

The phrase "the laws" means more than one thing. On one hand, the term "the laws" means "legislated prohibitions." On the other hand, when discussing religious matters, the term "the laws" can also refer to "religious prohibitions." Because the term "the laws" means both of these things, the sentence, "all of us must obey the laws" can mean two different things. It might mean that we must all heed religious prohibitions. However, it is also possible, and far more likely, that the term "laws" means different things in different parts of the argument, and that the sentence means that we must all heed legislated prohibitions. Because the argument ignores the ambiguity in "the laws," the argument commits a fallacy of ambiguity.

Suppose that an American citizen were to offer the following argument:

"Taxes provide revenue for the government, so that it can do its work. Part of the government's work involves collecting taxes. Therefore, taxes help to fund the collection of taxes."

This argument involves a

"Taxes provide revenue for the government, so that it can do its work. Part of the government's work involves collecting taxes. Therefore, taxes help to fund the collection of taxes."

This argument involves a

Answer: E. none of the above

The argument is in fact valid. If the government's revenue exists in order for the government to do its work, and if some of this work includes collecting taxes, then the revenue exists in order to collect taxes. Whether this revenue comes from taxes does not affect the validity of the argument. (Note, also, that the sentence "taxes help fund the collection of taxes" can be true, even when taxes also fund other things, too.)

The argument is in fact valid. If the government's revenue exists in order for the government to do its work, and if some of this work includes collecting taxes, then the revenue exists in order to collect taxes. Whether this revenue comes from taxes does not affect the validity of the argument. (Note, also, that the sentence "taxes help fund the collection of taxes" can be true, even when taxes also fund other things, too.)

Suppose that Dr. Spock (or some other authority on parenting) offered the following argument:

"It is very important for parents not to let down their children. I am now carrying my child. Therefore, even though he wants to walk on his own, it is important that I not let him down."

This argument is a

"It is very important for parents not to let down their children. I am now carrying my child. Therefore, even though he wants to walk on his own, it is important that I not let him down."

This argument is a

Answer: C. fallacy of ambiguity

The term "let down" means more than one thing. On one hand, the term "let down" means "allow to move downward." On the other hand, when used to describe people, the term "let down" can also mean "emotionally disappoint." Because the term "let down" means both of these things, the sentence, "it is very important for parents not to let down their children" might mean that it is important for parents to not let their children move downward, as the argument assumes. However, it is also possible, and far more likely, that the term "let down" means different things in different parts of the argument, and that the sentence means that parents should not emotionally disappoint their children. Because the argument ignores the ambiguity in "let down," the argument commits a fallacy of ambiguity.

The term "let down" means more than one thing. On one hand, the term "let down" means "allow to move downward." On the other hand, when used to describe people, the term "let down" can also mean "emotionally disappoint." Because the term "let down" means both of these things, the sentence, "it is very important for parents not to let down their children" might mean that it is important for parents to not let their children move downward, as the argument assumes. However, it is also possible, and far more likely, that the term "let down" means different things in different parts of the argument, and that the sentence means that parents should not emotionally disappoint their children. Because the argument ignores the ambiguity in "let down," the argument commits a fallacy of ambiguity.

Consider the following argument:

"There are more humans than tigers in the world. But there are more tigers than pandas. Therefore, humans are more numerous than pandas."

This argument involves a

"There are more humans than tigers in the world. But there are more tigers than pandas. Therefore, humans are more numerous than pandas."

This argument involves a

Answer: E. none of the above

The argument is in fact valid. If there are more Fs than Gs, and there are more Gs than Hs, then it logically follows there are more Fs than Hs. The above argument is an instance of this schema, with "humans," "tigers," and "pandas" being the values for F, G, and H, respectively.

The argument is in fact valid. If there are more Fs than Gs, and there are more Gs than Hs, then it logically follows there are more Fs than Hs. The above argument is an instance of this schema, with "humans," "tigers," and "pandas" being the values for F, G, and H, respectively.

Consider the following argument: "

Zeke and Jane are madly in love. In fact, Jane is the Zeke's one true love: He cannot love another woman, and so he cannot cheat on her. But Zeke's grown daughter Mary is also a woman. So Zeke cannot love Mary. Zeke must therefore be a terrible father!"

This argument involves a

Zeke and Jane are madly in love. In fact, Jane is the Zeke's one true love: He cannot love another woman, and so he cannot cheat on her. But Zeke's grown daughter Mary is also a woman. So Zeke cannot love Mary. Zeke must therefore be a terrible father!"

This argument involves a

Answer: C. fallacy of ambiguity

A semantic ambiguity occurs when a single word means more than one thing. Here, the word "love" means more than one thing. On one hand, the word "love" means "loving romantically," as husbands love wives and lovers love each other. On the other hand, when used to describe friends and family members, the word "love" can mean "familial love," or the kind of love that is not romantic. Because the word "love" means both of these things, the sentence, "Zeke cannot love Mary" might mean that Zeke cannot have familial love for Mary, which would imply that Zeke is a terrible father. However, it is also possible, and far more likely, that "love" means romantic love, in which case "Zeke cannot love Mary" is true, but it does not imply that Zeke is a terrible father. Because the argument ignores the ambiguity in "love," the argument commits a fallacy of ambiguity.

A semantic ambiguity occurs when a single word means more than one thing. Here, the word "love" means more than one thing. On one hand, the word "love" means "loving romantically," as husbands love wives and lovers love each other. On the other hand, when used to describe friends and family members, the word "love" can mean "familial love," or the kind of love that is not romantic. Because the word "love" means both of these things, the sentence, "Zeke cannot love Mary" might mean that Zeke cannot have familial love for Mary, which would imply that Zeke is a terrible father. However, it is also possible, and far more likely, that "love" means romantic love, in which case "Zeke cannot love Mary" is true, but it does not imply that Zeke is a terrible father. Because the argument ignores the ambiguity in "love," the argument commits a fallacy of ambiguity.

Consider the following argument:

"My horse is racing and my heart is racing. Therefore my horse and my heart are doing the same thing."

This argument involves a

"My horse is racing and my heart is racing. Therefore my horse and my heart are doing the same thing."

This argument involves a

Answer: C. fallacy of ambiguity

A semantic ambiguity occurs when a single word means more than one thing. Here, the word "racing" means more than one thing. On one hand, the word "racing" means "moving quickly." On the other hand, when used to describe hearts, the word "racing" can also mean "rapidly beating." Because the word "racing" means both of these things, the sentence, "my horse is racing fast and my heart is racing" might mean that the horse and the heart are doing the same thing, as the argument assumes. However, it is also possible, and far more likely, that they mean different things: one's horse is moving quickly, and one's heart is beating rapidly. Because the argument ignores the ambiguity in "racing," the argument commits a fallacy of ambiguity.

A semantic ambiguity occurs when a single word means more than one thing. Here, the word "racing" means more than one thing. On one hand, the word "racing" means "moving quickly." On the other hand, when used to describe hearts, the word "racing" can also mean "rapidly beating." Because the word "racing" means both of these things, the sentence, "my horse is racing fast and my heart is racing" might mean that the horse and the heart are doing the same thing, as the argument assumes. However, it is also possible, and far more likely, that they mean different things: one's horse is moving quickly, and one's heart is beating rapidly. Because the argument ignores the ambiguity in "racing," the argument commits a fallacy of ambiguity.

Consider the following argument:

"My horse is racing and my heart is racing. But racing is not significantly different from walking very quickly, which is not significantly different from walking somewhat quickly, which is not significantly different from walking somewhat slowly, which is not significantly different from walking very slowly, which is not significantly different from standing still. Therefore, racing is not significantly different from standing still. And so it would be unfair to treat racers differently than we treat those who are standing still. And therefore we must treat racers in just the same way that we treat those who are standing still. And thus we must treat my horse and my heart in the same way."

This argument involves a

"My horse is racing and my heart is racing. But racing is not significantly different from walking very quickly, which is not significantly different from walking somewhat quickly, which is not significantly different from walking somewhat slowly, which is not significantly different from walking very slowly, which is not significantly different from standing still. Therefore, racing is not significantly different from standing still. And so it would be unfair to treat racers differently than we treat those who are standing still. And therefore we must treat racers in just the same way that we treat those who are standing still. And thus we must treat my horse and my heart in the same way."

This argument involves a

Answer: D. all of the above

A fallacy of ambiguity occurs when an argument exploits the fact that a word means more than one thing. Here, the word "racing" means more than one thing. On one hand, the word "racing" means "moving quickly." On the other hand, when used to describe hearts, the word "racing" can also mean "rapidly beating." Because the word "racing" means both of these things, the sentence, "my horse is racing fast and my heart is racing" might mean that the horse and the heart are doing the same thing, as the argument assumes. However, it is also possible, and far more likely, that they mean different things: one's horse is moving quickly, and one's heart is beating rapidly. Because the argument ignores the ambiguity in "racing," the argument commits a fallacy of ambiguity.

The argument also commits a conceptual slippery slope fallacy, however. A conceptual slippery slope fallacy is an argument that alleges that a series of actions cannot change the quality of a certain thing. Typically, the problem with a conceptual slippery slope argument is that its second premise claims that something is not a matter of degree, when in fact it is a matter of degree.

In this case, the premise in the argument treats "racing" (in the sense of moving quickly) as an all-or-nothing matter. The argument assumes that either one is racing or is not.

Suppose, however, that racing admits of degrees. Suppose that one could be racing just a little bit, or that one could be racing even more, or one could be racing to an extreme degree. Suppose further that, depending on whether one was walking quickly, somewhat quickly, or slowly, one would be racing to certain degrees. If that is the case, the sentence, "but racing is not significantly different from walking very quickly, which is not significantly different from walking somewhat quickly, which is not significantly different from walking somewhat slowly, which is not significantly different from walking very slowly, which is not significantly different from standing still" is false. It is false because, in fact, there is a difference between walking very quickly, walking somewhat quickly, and so on. That difference, furthermore, is important to racing. The difference is in the degree to which one is racing, though, rather than whether one is racing at all.

Finally, the argument also commits a fairness slippery slope fallacy. A fairness slippery slope fallacy is an argument, which claims that, since a certain course of action is fair, and since any other course of action that differs from the first by a mere incremental difference must also be fair, it follows that a radically different policy must be fair, too.

In this case, the argument claims that, if it is fair to treat racers in exactly the same way as those who are standing still, then this makes it fair to adopt even more radical policies, such as treating racers in exactly the same way as anything.

As we learned this week, however, the problem with most fairness slippery slope arguments, including this one, is that the fairness of more radical policies does not, in fact, follow from the fairness of the course of action in question. In this case, the fairness of treating racers the same as those who stand still would not imply the fairness of treating racers the same as anything.

A fallacy of ambiguity occurs when an argument exploits the fact that a word means more than one thing. Here, the word "racing" means more than one thing. On one hand, the word "racing" means "moving quickly." On the other hand, when used to describe hearts, the word "racing" can also mean "rapidly beating." Because the word "racing" means both of these things, the sentence, "my horse is racing fast and my heart is racing" might mean that the horse and the heart are doing the same thing, as the argument assumes. However, it is also possible, and far more likely, that they mean different things: one's horse is moving quickly, and one's heart is beating rapidly. Because the argument ignores the ambiguity in "racing," the argument commits a fallacy of ambiguity.

The argument also commits a conceptual slippery slope fallacy, however. A conceptual slippery slope fallacy is an argument that alleges that a series of actions cannot change the quality of a certain thing. Typically, the problem with a conceptual slippery slope argument is that its second premise claims that something is not a matter of degree, when in fact it is a matter of degree.

In this case, the premise in the argument treats "racing" (in the sense of moving quickly) as an all-or-nothing matter. The argument assumes that either one is racing or is not.

Suppose, however, that racing admits of degrees. Suppose that one could be racing just a little bit, or that one could be racing even more, or one could be racing to an extreme degree. Suppose further that, depending on whether one was walking quickly, somewhat quickly, or slowly, one would be racing to certain degrees. If that is the case, the sentence, "but racing is not significantly different from walking very quickly, which is not significantly different from walking somewhat quickly, which is not significantly different from walking somewhat slowly, which is not significantly different from walking very slowly, which is not significantly different from standing still" is false. It is false because, in fact, there is a difference between walking very quickly, walking somewhat quickly, and so on. That difference, furthermore, is important to racing. The difference is in the degree to which one is racing, though, rather than whether one is racing at all.

Finally, the argument also commits a fairness slippery slope fallacy. A fairness slippery slope fallacy is an argument, which claims that, since a certain course of action is fair, and since any other course of action that differs from the first by a mere incremental difference must also be fair, it follows that a radically different policy must be fair, too.

In this case, the argument claims that, if it is fair to treat racers in exactly the same way as those who are standing still, then this makes it fair to adopt even more radical policies, such as treating racers in exactly the same way as anything.

As we learned this week, however, the problem with most fairness slippery slope arguments, including this one, is that the fairness of more radical policies does not, in fact, follow from the fairness of the course of action in question. In this case, the fairness of treating racers the same as those who stand still would not imply the fairness of treating racers the same as anything.

Consider the following argument:

"A human being cannot feel an electric current of less than 1 mA of AC at 60 Hz. But a human being also cannot feel the difference between electric currents that differ from each other by less than 1 mA. Therefore, a human being cannot feel an electric current of less than 2 mA of AC at 60 Hz, of less than 3 mA of AC at 60 Hz, and so on. It follows that a human being cannot feel an AC electric current of any magnitude whatsoever, no matter how large. And so electrocution cannot be painful, no matter how severe it is."

This argument involves a

"A human being cannot feel an electric current of less than 1 mA of AC at 60 Hz. But a human being also cannot feel the difference between electric currents that differ from each other by less than 1 mA. Therefore, a human being cannot feel an electric current of less than 2 mA of AC at 60 Hz, of less than 3 mA of AC at 60 Hz, and so on. It follows that a human being cannot feel an AC electric current of any magnitude whatsoever, no matter how large. And so electrocution cannot be painful, no matter how severe it is."

This argument involves a

Answer: A. conceptual slippery slope fallacy

A conceptual slippery slope argument is an argument to the effect that a series of particular actions cannot change the quality of a certain thing. Typically, the problem with a conceptual slippery slope argument is that its second premise claims that something is not a matter of degree, when in fact it is a matter of degree.

In this case, the second premise in the argument treats "feeling the difference between electric currents" as an all-or-nothing matter. The argument assumes that either one feels the difference between the electric currents, or else one does not.

Suppose, however, that feeling the difference between two electric currents admits of degrees. Suppose, in other words, that one can either feel the difference in two currents a little bit, or that one can feel the difference a bit more, or that one can really, very much feel the difference. Suppose further that, depending on the Hz of two currents, one will feel their difference more or less. If those suppositions are correct, then the sentence, "but a human being also cannot feel the difference between electric currents that differ from each other by less than 1 mA" is false. It is false because, given our suppositions, one can feel the difference of 1 mA more on some occasions than others, depending on the Hz of the currents.

A conceptual slippery slope argument is an argument to the effect that a series of particular actions cannot change the quality of a certain thing. Typically, the problem with a conceptual slippery slope argument is that its second premise claims that something is not a matter of degree, when in fact it is a matter of degree.

In this case, the second premise in the argument treats "feeling the difference between electric currents" as an all-or-nothing matter. The argument assumes that either one feels the difference between the electric currents, or else one does not.

Suppose, however, that feeling the difference between two electric currents admits of degrees. Suppose, in other words, that one can either feel the difference in two currents a little bit, or that one can feel the difference a bit more, or that one can really, very much feel the difference. Suppose further that, depending on the Hz of two currents, one will feel their difference more or less. If those suppositions are correct, then the sentence, "but a human being also cannot feel the difference between electric currents that differ from each other by less than 1 mA" is false. It is false because, given our suppositions, one can feel the difference of 1 mA more on some occasions than others, depending on the Hz of the currents.

If a fair coin comes up heads five times in a row, then the probability that it will come up heads on the next flip is:

The answer is "one half (0.5)."

If the coin really is fair, as the question says, then the probability that it will come up heads on any flip is one half (0.5). It does not matter how many times it has come up heads (or tails) before. Each flip is independent in the sense that the probability of heads on one flip does not affect the probability of heads on any other flip.

If the coin really is fair, as the question says, then the probability that it will come up heads on any flip is one half (0.5). It does not matter how many times it has come up heads (or tails) before. Each flip is independent in the sense that the probability of heads on one flip does not affect the probability of heads on any other flip.

If a fair dealer deals you five cards out of a shuffled fair standard deck of cards, then which of the following hands is most likely?

The answer is "All of these hands are equally likely."

If the deck and the dealer really are fair, as the question stipulates, then the probability of getting dealt any particular hand is the same as the probability of getting dealt any other particular hand. That is what it means to call the deck and the dealing fair.

If the deck and the dealer really are fair, as the question stipulates, then the probability of getting dealt any particular hand is the same as the probability of getting dealt any other particular hand. That is what it means to call the deck and the dealing fair.

As described in the video, Linda is thirty-one years old, single, outspoken, and very bright. As a student, she majored in philosophy, was deeply concerned with issues of discrimination and social justice, and participated in antinuclear demonstrations. Which of the following is most likely?

The answer is "Linda is a bank teller."

Everyone who is both a bank teller and active in the feminist movement is a bank teller, but some bank tellers are not active in the feminist movement. Thus, it is more likely that Linda is a bank teller than that she is both a bank teller and also active in the feminist movement.

Everyone who is both a bank teller and active in the feminist movement is a bank teller, but some bank tellers are not active in the feminist movement. Thus, it is more likely that Linda is a bank teller than that she is both a bank teller and also active in the feminist movement.

Imagine that you are playing the old television game show "Let's Make a Deal" hosted by Monte Hall and described in the lecture. You face three closed doors. Behind one of the doors is a car. Behind each of the other two doors is a goat. You pick door A, so you will get to keep what is behind door A if you stick with it. Then, as always, Monte Hall opens one of the remaining two doors, reveals a goat behind that other door, and offers you the opportunity to switch doors, if you want. Suppose that this time Monte Hall opens door C and offers you the opportunity to switch to door B. If you switch doors, then you will get what is behind door B instead of what is behind door A. If you do not switch doors, then you will get what is behind door A. Should you switch doors?

The correct answer is "You should switch doors."

If the probability of an event is 0.73, then its probability can also be expressed as

The answer is "all of the above."

A probability of 0.73 is equal to a 73 percent chance, 73 out of 100, or 7.3 out of 10.

A probability of 0.73 is equal to a 73 percent chance, 73 out of 100, or 7.3 out of 10.

It is possible for some events to have a probability of

The answer is "None of the above."

Probabilities range from 0 to 1, so the probability of an event can never be either 1.5 or -1.5.

Probabilities range from 0 to 1, so the probability of an event can never be either 1.5 or -1.5.

If the probability of an event is 1.0, then

The answer is "it is certain that it will happen."

When the probability of something is 1.0, it is certain that it will happen, and there is no chance that it will not happen. At the opposite extreme, when the probability of something is 0.0, it is certain that it will not happen, and there is no chance that it will happen.

When the probability of something is 1.0, it is certain that it will happen, and there is no chance that it will not happen. At the opposite extreme, when the probability of something is 0.0, it is certain that it will not happen, and there is no chance that it will happen.

When someone assumes that six-sided dice are equally likely to fall on any of the sides—1, 2, 3, 4, 5, or 6—then that person is calculating

The answer is "a priori probability."

A priori probabilities depend on assumptions, such as equal likelihood of outcomes, whereas statistical probabilities are based on observed frequencies of outcomes, and subjective probabilities are based on hunches, though possibly rational.

A priori probabilities depend on assumptions, such as equal likelihood of outcomes, whereas statistical probabilities are based on observed frequencies of outcomes, and subjective probabilities are based on hunches, though possibly rational.

When a coin is bent, the most accurate way to determine the probability that it will land heads up when it is flipped is to use

The answer is "statistical probability."

If the coin is bent, this is a strong reason not to assume that heads and tails are equally likely, so we cannot use a priori probability. A subjective probability is unlikely to be accurate in this situation. Hence, the most accurate way to determine the probability that a bent coin will land heads up is to observe a large number of flips and count the frequency of heads. That is the procedure of statistical probability.

If the coin is bent, this is a strong reason not to assume that heads and tails are equally likely, so we cannot use a priori probability. A subjective probability is unlikely to be accurate in this situation. Hence, the most accurate way to determine the probability that a bent coin will land heads up is to observe a large number of flips and count the frequency of heads. That is the procedure of statistical probability.

A coin flip is fair when the coin is equally likely to land with either side—heads or tails—up, and the coin will always land on either heads or tails but not on its edge. If we assume that a coin flip is fair, then what is the probability that the coin will land heads up?

The answer is 0.5.

If there are only two possibilities—heads or tails—then the probabilities of heads and tails must add up to 1, because it is certain that the coin will land on either heads or tails. If these two possibilities—heads or tails—are equally likely, as we are also assuming here, then each possibility has a probability of 0.5, because only then will their equal probabilities add up to 1.

If there are only two possibilities—heads or tails—then the probabilities of heads and tails must add up to 1, because it is certain that the coin will land on either heads or tails. If these two possibilities—heads or tails—are equally likely, as we are also assuming here, then each possibility has a probability of 0.5, because only then will their equal probabilities add up to 1.

A roll of a six-sided die is fair when the die is equally likely to land with any of its sides—1, 2, 3, 4, 5, or 6—up, and the die will always land with one of those sides up. If we assume that a roll of a six-sided die is fair, then what is the probability that the die will land with 4 up?

The answer is 1/6.

If there are only six possibilities—1, 2, 3, 4, 5, or 6—then the probabilities of these six possibilities must add up to 1, because it is certain that the die will land with one of these sides up. If these six possibilities—1, 2, 3, 4, 5, or 6—are all equally likely, as we are also assuming here, then each possibility has a probability of 1/6, because only then will their equal probabilities add up to 1.

If there are only six possibilities—1, 2, 3, 4, 5, or 6—then the probabilities of these six possibilities must add up to 1, because it is certain that the die will land with one of these sides up. If these six possibilities—1, 2, 3, 4, 5, or 6—are all equally likely, as we are also assuming here, then each possibility has a probability of 1/6, because only then will their equal probabilities add up to 1.

A roll of a ten-sided die is fair when the die is equally likely to land with any of its sides—1, 2, 3, 4, 5, 6, 7, 8, 9, or 10—up, and the die will always land with one of those sides up. If we assume that a roll of a ten-sided die is fair, then what is the probability that the die will land with 4 up?

The correct answer is 1/10.

If there are only ten possibilities—1, 2, 3, 4, 5, 6, 7, 8, 9, or 10—then the probabilities of these ten possibilities must add up to 1, because it is certain that the die will land with one of these sides up. If these ten possibilities—1, 2, 3, 4, 5, 6, 7, 8, 9, or 10—are all equally likely, as we are also assuming here, then each possibility has a probability of 1/10, because only then will their equal probabilities add up to 1.

If there are only ten possibilities—1, 2, 3, 4, 5, 6, 7, 8, 9, or 10—then the probabilities of these ten possibilities must add up to 1, because it is certain that the die will land with one of these sides up. If these ten possibilities—1, 2, 3, 4, 5, 6, 7, 8, 9, or 10—are all equally likely, as we are also assuming here, then each possibility has a probability of 1/10, because only then will their equal probabilities add up to 1.

A roll of two dice is X when the sum of the numbers on the two sides that land up is X. If we assume that the roll is fair, what is the probability that one roll of two six-sided dice will be 4 (that is, will land with a total of 4 up)?

The answer is 3/36.

Because each of the dice has six sides, there are 36 possible results when the two dice are rolled. These 36 possibilities were displayed in a table in the video. Only three possible results add up to a total of 4 on both sides together: (1) the first die lands with 1 up and the second die lands with 3 up, (2) the first die lands with 3 up and the second die lands with 1 up, and (3) the first die lands with 2 up and the second die lands with 2 up. Thus, 3 results out of 36 yield a result of 4, so the probability of rolling a total of 4 on two fair six-sided dice is 3/36.

Because each of the dice has six sides, there are 36 possible results when the two dice are rolled. These 36 possibilities were displayed in a table in the video. Only three possible results add up to a total of 4 on both sides together: (1) the first die lands with 1 up and the second die lands with 3 up, (2) the first die lands with 3 up and the second die lands with 1 up, and (3) the first die lands with 2 up and the second die lands with 2 up. Thus, 3 results out of 36 yield a result of 4, so the probability of rolling a total of 4 on two fair six-sided dice is 3/36.

If the probability that a flipped coin will land heads up is 0.5, what is the probability that it will NOT land heads up?

The answer is 0.5.

The probability that an event will not occur is 1 minus the probability that it will occur. Here the event is the flipped coin landing heads up. The probability of that event occurring is 0.5, and 1 minus 0.5 is 0.5. Hence, the probability of that event not occurring is also 0.5.

The probability that an event will not occur is 1 minus the probability that it will occur. Here the event is the flipped coin landing heads up. The probability of that event occurring is 0.5, and 1 minus 0.5 is 0.5. Hence, the probability of that event not occurring is also 0.5.

Imagine that you bend the coin in Question 1 so that the probability that it will land heads up when flipped is 0.25. Now what is the probability that it will NOT land heads up?

The answer is "0.75."

The probability that an event will not occur is 1 minus the probability that it will occur. Here the event is the flipped coin landing heads up. The probability of that event occurring is 0.25. 1 minus 0.25 is 0.75. Hence, the probability of that event not occurring is 0.75.

The probability that an event will not occur is 1 minus the probability that it will occur. Here the event is the flipped coin landing heads up. The probability of that event occurring is 0.25. 1 minus 0.25 is 0.75. Hence, the probability of that event not occurring is 0.75.

As shown in video 9-02, the probability of rolling a 7 on two fair six-sided dice is 6/36. What is the probability of NOT rolling a 7 on two fair six-sided dice?

The answer is 30/36.

The probability that an event will not occur is 1 minus the probability that it will occur. Here the event is rolling a 7. The probability of that event occurring is 6/36, as the question says. 1 minus 6/36 is 30/36. Hence, the probability of that event not occurring is 30/36.

The probability that an event will not occur is 1 minus the probability that it will occur. Here the event is rolling a 7. The probability of that event occurring is 6/36, as the question says. 1 minus 6/36 is 30/36. Hence, the probability of that event not occurring is 30/36.

The probability of picking an ace at random out of a fair standard deck of cards is 1/13. What is the probability of NOT picking an ace out of this deck?

The answer is 12/13.

The probability that an event will not occur is 1 minus the probability that it will occur. Here the event is picking an ace. The probability of that event occurring is 1/13, and 1 minus 1/13 is 12/13. Hence, the probability of that event not occurring is 12/13.

The probability that an event will not occur is 1 minus the probability that it will occur. Here the event is picking an ace. The probability of that event occurring is 1/13, and 1 minus 1/13 is 12/13. Hence, the probability of that event not occurring is 12/13.

The probability of picking a spade at random out of a fair standard deck of cards is 1/4. What is the probability of NOT picking a spade at random out of this deck?

The answer is 3/4.

The probability that an event will not occur is 1 minus the probability that it will occur. Here the event is picking a spade. The probability of that event occurring is 1/4. 1 minus 1/4 is 3/4. Hence, the probability of that event not occurring is 3/4.

The probability that an event will not occur is 1 minus the probability that it will occur. Here the event is picking a spade. The probability of that event occurring is 1/4. 1 minus 1/4 is 3/4. Hence, the probability of that event not occurring is 3/4.

Imagine that you own one ticket to a lottery where the chances of this ticket winning are 1 in 1,000,000. What is the probability of this ticket NOT winning?

The answer is "999,999 out of 1,000,000."

The probability that an event will not occur is 1 minus the probability that it will occur. Here the event is the ticket winning. The probability of that event occurring is 1 in 1,000,000. 1 minus 1/1,000,000 is 999,999/1,000,000. Hence, the probability of this ticket not winning is 999,999/1,000,000.

The probability that an event will not occur is 1 minus the probability that it will occur. Here the event is the ticket winning. The probability of that event occurring is 1 in 1,000,000. 1 minus 1/1,000,000 is 999,999/1,000,000. Hence, the probability of this ticket not winning is 999,999/1,000,000.

If the probability of an event is 0.1, what is the probability that either that event will occur or that event will not occur?

The answer is 1.0.

It is certain that the event either will occur or will not occur. There is no other possibility. Therefore, the probability that it either will or will not occur is 1.0.

It is certain that the event either will occur or will not occur. There is no other possibility. Therefore, the probability that it either will or will not occur is 1.0.

Two events are independent (in the sense used in the rules for probability) if and only if

The answer is "whether or not one of the events occurs does not affect the probability that the other event will occur."

This definition of independence was stated and explained in the video.

This definition of independence was stated and explained in the video.

If two events are independent, then the probability of both events occurring is

The answer is "the product of the probability of the first event times the probability of the second event."

This rule for probability of conjunctions of independent events was stated and explained in the video.

This rule for probability of conjunctions of independent events was stated and explained in the video.

If two events are NOT independent, then the probability of both events occurring is the product of the probability of the first event times the

The answer is "conditional probability of the second event given the first event."

This rule for probability of conjunctions of non-independent events was stated and explained in the video.

This rule for probability of conjunctions of non-independent events was stated and explained in the video.

The conditional probability of one event (X) given another event (Y) is

The answer is "the percentage of cases where X occurs out of the cases where Y occurs."

This definition of conditional probability was stated and explained in the video.

This definition of conditional probability was stated and explained in the video.

Assuming that all coin flips here are fair, the flip of one coin and the flip of another coin are

The answer is "independent."

If the coin flips are fair, then the probability of getting heads (or getting tails) on one flip does not affect the probability of getting heads (or getting tails) on the other flip. That makes the flips independent in the sense used in the rules for probability.

If the coin flips are fair, then the probability of getting heads (or getting tails) on one flip does not affect the probability of getting heads (or getting tails) on the other flip. That makes the flips independent in the sense used in the rules for probability.

Assuming that all coin flips here are fair, what is the probability of getting tails on two flips in a row of the same coin?

The answer is 0.25.

The probability of getting tails on the first flip is 0.5, and the probability of getting tails on the second flip is also 0.5, so the probability of getting tails on both the first flip and also the second flip is 0.5 x 0.5 = 0.25. This calculation applies the simple rule for probabilities of conjunctions because the probability of getting tails on the first flip does not affect the probability of getting tails on the second flip.

The probability of getting tails on the first flip is 0.5, and the probability of getting tails on the second flip is also 0.5, so the probability of getting tails on both the first flip and also the second flip is 0.5 x 0.5 = 0.25. This calculation applies the simple rule for probabilities of conjunctions because the probability of getting tails on the first flip does not affect the probability of getting tails on the second flip.

Assuming that all dice rolls here are fair, what is the probability of getting 3 on both dice (that is, 3 on one and also 3 on the other for a total of 6) when you roll two six-sided dice at the same time?

The answer is 1/36.

The probability of getting 3 on the first die is 1/6, and the probability of getting 3 on the second die is also 1/6, so the probability of getting 3 on both the first die and the second die is 1/6 x 1/6 = 1/36. This calculation applies the simple rule for probabilities of conjunctions because the probability of getting 3 on the first die does not affect the probability of getting 3 on the second die. This result can also be reached by remembering that there is only one way to get 3 on both dice out of the 36 possible results of rolling two dice.

The probability of getting 3 on the first die is 1/6, and the probability of getting 3 on the second die is also 1/6, so the probability of getting 3 on both the first die and the second die is 1/6 x 1/6 = 1/36. This calculation applies the simple rule for probabilities of conjunctions because the probability of getting 3 on the first die does not affect the probability of getting 3 on the second die. This result can also be reached by remembering that there is only one way to get 3 on both dice out of the 36 possible results of rolling two dice.

Assuming that all card picks here are fair, what is the probability of getting a spade on two picks out of a standard deck when you replace the card that you picked first and shuffle the deck before you pick the second card?

The answer is 1/16.

The probability of getting a spade on the first pick is 1/4, and the probability of getting a spade on the second pick is also 1/4, so the probability of getting a spade on both the first pick and the second pick is 1/4 x 1/4 = 1/16. This calculation applies the simple rule for probabilities of conjunctions because the probability of the first pick does not affect the probability of the second pick when the first card is returned to the deck and the deck is shuffled before the second card is picked.

The probability of getting a spade on the first pick is 1/4, and the probability of getting a spade on the second pick is also 1/4, so the probability of getting a spade on both the first pick and the second pick is 1/4 x 1/4 = 1/16. This calculation applies the simple rule for probabilities of conjunctions because the probability of the first pick does not affect the probability of the second pick when the first card is returned to the deck and the deck is shuffled before the second card is picked.

Assuming that all card picks here are fair, when you pick two cards out of a deck and you do not replace the card that you picked first before you pick the second card, then the probabilities of getting a Jack on both picks are

The answer is "not independent."

A standard deck has fifty-two cards. If you pick one card out and do not replace it (that is, you do not put that card back into the deck), then the remaining deck will have only fifty-one cards. That reduction in the number of cards (which are the possible outcomes of a pick) affects the probability of getting a certain result on the next pick. Two events are independent only if the probability of one does NOT affect the probability of the other. Thus, the two picks are NOT independent.

A standard deck has fifty-two cards. If you pick one card out and do not replace it (that is, you do not put that card back into the deck), then the remaining deck will have only fifty-one cards. That reduction in the number of cards (which are the possible outcomes of a pick) affects the probability of getting a certain result on the next pick. Two events are independent only if the probability of one does NOT affect the probability of the other. Thus, the two picks are NOT independent.

Assuming that all card picks here are fair, imagine that you pick a King of Hearts out of a standard deck and you do not put that card back in the deck, then what is the probability that you will pick a Queen on the next pick out of the remaining deck?

The answer is 4/51.

A standard deck has fifty-two cards. If you pick one card out and do not replace it (that is, you do not put that card back into the deck), then the remaining deck will have only fifty-one cards. There will be four Queens left after you pick a King of Hearts. Thus, the probability of picking a Queen out of the remaining deck is four out of fifty-one or 4/51.

A standard deck has fifty-two cards. If you pick one card out and do not replace it (that is, you do not put that card back into the deck), then the remaining deck will have only fifty-one cards. There will be four Queens left after you pick a King of Hearts. Thus, the probability of picking a Queen out of the remaining deck is four out of fifty-one or 4/51.

Assuming that all card picks here are fair, imagine that you pick a Queen of Hearts out of a standard deck, and you do not put that card back in the deck, then what is the probability that you will pick a Queen on the next pick out of the remaining deck?

The answer is 3/51.

A standard deck has fifty-two cards. If you pick one card out and do not replace it (that is, you do not put that card back into the deck), then the remaining deck will have only fifty-one cards. There will be three Queens left after you pick a Queen of Hearts and do not return it to the deck. Thus, the probability of picking a Queen out of the remaining deck is three out of fifty-one or 3/51.

A standard deck has fifty-two cards. If you pick one card out and do not replace it (that is, you do not put that card back into the deck), then the remaining deck will have only fifty-one cards. There will be three Queens left after you pick a Queen of Hearts and do not return it to the deck. Thus, the probability of picking a Queen out of the remaining deck is three out of fifty-one or 3/51.

Assuming that all card picks here are fair, what is the probability that you will pick a King and then a Queen out of a standard deck when you do not put the card that you picked first back in the deck?

The answer is 4/52 x 4/51.

A standard deck has fifty-two cards and four Kings, so the probability of picking a King on the first pick is 4/52. If you do not return the first pick to the deck, then the remaining deck has fifty-one cards. If the first pick was a King, then the remaining deck has four Queens. Thus, the probability of picking a Queen after first picking a King and not returning it to the deck is 4/51. Therefore, the probability of both picks—a King and then a Queen without replacement—is 4/52 x 4/51. This calculation applies the general rule for probabilities of conjunctions because these two picks are not independent (the probability of the first pick does affect the probability of the second pick when the first card is not returned to the deck before the second card is picked).

A standard deck has fifty-two cards and four Kings, so the probability of picking a King on the first pick is 4/52. If you do not return the first pick to the deck, then the remaining deck has fifty-one cards. If the first pick was a King, then the remaining deck has four Queens. Thus, the probability of picking a Queen after first picking a King and not returning it to the deck is 4/51. Therefore, the probability of both picks—a King and then a Queen without replacement—is 4/52 x 4/51. This calculation applies the general rule for probabilities of conjunctions because these two picks are not independent (the probability of the first pick does affect the probability of the second pick when the first card is not returned to the deck before the second card is picked).

Assuming that all card picks here are fair, what is the probability that you will pick two Queens in two draws out of a standard deck when you do not put the card that you picked first back in the deck?

The answer is 4/52 x 3/51.

A standard deck has fifty-two cards and four Queens, so the probability of picking a Queen on the first pick is 4/52. If you do not return the first pick to the deck, then the remaining deck has fifty-one cards. If the first pick was a Queen and you did not replace it, then the remaining deck has three Queens. Thus, the probability of picking a Queen after first picking a Queen and not returning it to the deck is 3/51. Therefore, the probability of both picks—a Queen first and then a Queen second without replacement—is 4/52 x 3/51. This calculation applies the general rule for probabilities of conjunctions because these two picks are not independent (the probability of the first pick does affect the probability of the second pick when the first card is not returned to the deck before the second card is picked).

A standard deck has fifty-two cards and four Queens, so the probability of picking a Queen on the first pick is 4/52. If you do not return the first pick to the deck, then the remaining deck has fifty-one cards. If the first pick was a Queen and you did not replace it, then the remaining deck has three Queens. Thus, the probability of picking a Queen after first picking a Queen and not returning it to the deck is 3/51. Therefore, the probability of both picks—a Queen first and then a Queen second without replacement—is 4/52 x 3/51. This calculation applies the general rule for probabilities of conjunctions because these two picks are not independent (the probability of the first pick does affect the probability of the second pick when the first card is not returned to the deck before the second card is picked).

Assuming that all card picks here are fair, what is the probability that you will pick a spade and then a heart out of a standard deck when you do not put the card that you picked first back in the deck?

The answer is 13/52 x 13/51.

A standard deck has fifty-two cards and thirteen spades, so the probability of picking a spade on the first pick is 13/52. If you do not return the first pick to the deck, then the remaining deck has fifty-one cards. If the first pick was a spade, then the remaining deck has thirteen hearts. Thus, the probability of picking a heart after first picking a spade and not returning it to the deck is 13/51. Hence, the probability of both picks—a spade and then a heart without replacement—is 13/52 x 13/51. This calculation applies the general rule for probabilities of conjunctions because these two picks are not independent (the probability of the first pick does affect the probability of the second pick when the first card is not returned to the deck before the second card is picked).

A standard deck has fifty-two cards and thirteen spades, so the probability of picking a spade on the first pick is 13/52. If you do not return the first pick to the deck, then the remaining deck has fifty-one cards. If the first pick was a spade, then the remaining deck has thirteen hearts. Thus, the probability of picking a heart after first picking a spade and not returning it to the deck is 13/51. Hence, the probability of both picks—a spade and then a heart without replacement—is 13/52 x 13/51. This calculation applies the general rule for probabilities of conjunctions because these two picks are not independent (the probability of the first pick does affect the probability of the second pick when the first card is not returned to the deck before the second card is picked).

You can construct your own examples for practicing simply by asking about other outcomes on flips of coins, rolls of dice, and picks of cards.

True.

Try it! Remember to specify all of the relevant factors, such as fairness of the coins, dice, and cards and whether the first card is returned to the deck and the deck shuffled. Then bring your examples and answers to the discussion forums to ask other students whether they get the same answers as you did. Have fun!

Try it! Remember to specify all of the relevant factors, such as fairness of the coins, dice, and cards and whether the first card is returned to the deck and the deck shuffled. Then bring your examples and answers to the discussion forums to ask other students whether they get the same answers as you did. Have fun!

Two events are mutually exclusive if and only if

The answer is "they cannot possibly both occur together."

This definition of mutual exclusion was stated and explained in the video.

This definition of mutual exclusion was stated and explained in the video.

If two events are mutually exclusive, then the probability of either one event or the other event occurring is

The answer is "the sum of the probability of the first event plus the probability of the second event."

This rule for probability of disjunctions of mutually exclusive events was stated and explained in the video.

This rule for probability of disjunctions of mutually exclusive events was stated and explained in the video.

If two events are NOT mutually exclusive, then the probability of either one event or the other event occurring is

The answer is "the sum of the probability of the first event plus the probability of the second event minus the probability of both the first event and the second event occurring together."

This rule for probability of disjunctions of non-exclusive events was stated and explained in the video.

This rule for probability of disjunctions of non-exclusive events was stated and explained in the video.

On one fair flip of a single coin, getting heads and getting tails are

The answer is "mutually exclusive."

If the coin flips are fair, then you cannot ever get both heads and tails on a single flip, so they are mutually exclusive.

If the coin flips are fair, then you cannot ever get both heads and tails on a single flip, so they are mutually exclusive.

On one fair flip of a single coin, what is the probability of getting either heads or tails

The answer is 1.

Heads and tails are mutually exclusive results (see Question 4 in this exercise), so we can apply the simple rule for probabilities of disjunction (see Question 2 in this exercise). The probability of getting heads on a fair coin is 0.5, and the probability of getting tails on a fair coin is 0.5, so the probability of getting either heads or tails on a fair coin is 0.5 + 0.5 = 1. This calculation makes sense because it is certain that you will get either heads or tails if the coin is fair, and certainty is represented by a probability of 1.

Heads and tails are mutually exclusive results (see Question 4 in this exercise), so we can apply the simple rule for probabilities of disjunction (see Question 2 in this exercise). The probability of getting heads on a fair coin is 0.5, and the probability of getting tails on a fair coin is 0.5, so the probability of getting either heads or tails on a fair coin is 0.5 + 0.5 = 1. This calculation makes sense because it is certain that you will get either heads or tails if the coin is fair, and certainty is represented by a probability of 1.

Assuming that all dice rolls here are fair, what is the probability of getting either 2 or 3 when you roll one six-sided die?

The answer is 2/6.

The probability of getting 2 on the die is 1/6, and the probability of getting 3 on the die is also 1/6, so the probability of getting either 2 or 3 is 1/6 + 1/6 = 2/6. This calculation applies the simple rule for probabilities of disjunctions of mutually exclusive events, because getting 2 on the die and getting 3 on the die are mutually exclusive—you cannot get both 2 and 3—assuming that the die is fair.

The probability of getting 2 on the die is 1/6, and the probability of getting 3 on the die is also 1/6, so the probability of getting either 2 or 3 is 1/6 + 1/6 = 2/6. This calculation applies the simple rule for probabilities of disjunctions of mutually exclusive events, because getting 2 on the die and getting 3 on the die are mutually exclusive—you cannot get both 2 and 3—assuming that the die is fair.

Assuming that all dice rolls here are fair, what is the probability of getting 3 on either one die or the other die when you roll two six-sided dice at the same time?

The answer is 2/6 − 1/36.

The probability of getting 3 on the first die is 1/6, and the probability of getting 3 on the second die is also 1/6. These results are not mutually exclusive, because it is possible to get 3 on both dice. That means that we need to apply the general rule for probability of disjunction that does not require the events to be mutually exclusive (see Question 3 in this exercise). That general rule requires us to subtract the probability of both events occurring together. The probability of getting 3 on both dice is 1/6 x 1/6 = 1/36. Thus, the probability of getting 3 on either one die or the other die is 1/6 + 1/6 − 1/36, which equals 2/6 − 1/36.

The probability of getting 3 on the first die is 1/6, and the probability of getting 3 on the second die is also 1/6. These results are not mutually exclusive, because it is possible to get 3 on both dice. That means that we need to apply the general rule for probability of disjunction that does not require the events to be mutually exclusive (see Question 3 in this exercise). That general rule requires us to subtract the probability of both events occurring together. The probability of getting 3 on both dice is 1/6 x 1/6 = 1/36. Thus, the probability of getting 3 on either one die or the other die is 1/6 + 1/6 − 1/36, which equals 2/6 − 1/36.

Assuming that all card picks here are fair, what is the probability of getting either a spade or a club when you pick one card out of a standard deck?

The answer is 2/4.

The probability of picking a spade is 1/4, and the probability of picking a club is also 1/4, so the probability of picking either a spade or a club is 1/4 + 1/4 = 2/4. This calculation applies the simple rule for probabilities of disjunctions because picking a spade and picking a club are mutually exclusive, since it is not possible to pick both a spade and a club in one pick out of a fair standard deck of cards. No card in a standard deck is both a spade and also a club.

The probability of picking a spade is 1/4, and the probability of picking a club is also 1/4, so the probability of picking either a spade or a club is 1/4 + 1/4 = 2/4. This calculation applies the simple rule for probabilities of disjunctions because picking a spade and picking a club are mutually exclusive, since it is not possible to pick both a spade and a club in one pick out of a fair standard deck of cards. No card in a standard deck is both a spade and also a club.

Assuming that all card picks here are fair, what is the probability of picking a spade on either the first pick or the second pick when you pick two cards out of a standard deck (and when you put the first card back and shuffle the cards before the second pick)?

The answer is 2/4 − 1/16.

The probability of picking a spade on the first pick is 1/4, and the probability of picking a spade on the second pick (given replacement and shuffling) is also 1/4. These results are not mutually exclusive, because it is possible to get a spade on both picks. That means that we need to apply the general rule for probability of disjunction that does not require the events to be mutually exclusive. That general rule requires us to subtract the probability of both events occurring together. The probability of picking a spade in both picks (given replacement and shuffling) is 1/4 x 1/4 = 1/16. Thus, the probability of picking a spade on either the first pick or the second pick is 1/4 + 1/4 − 1/16, which equals 2/4 − 1/16.

The probability of picking a spade on the first pick is 1/4, and the probability of picking a spade on the second pick (given replacement and shuffling) is also 1/4. These results are not mutually exclusive, because it is possible to get a spade on both picks. That means that we need to apply the general rule for probability of disjunction that does not require the events to be mutually exclusive. That general rule requires us to subtract the probability of both events occurring together. The probability of picking a spade in both picks (given replacement and shuffling) is 1/4 x 1/4 = 1/16. Thus, the probability of picking a spade on either the first pick or the second pick is 1/4 + 1/4 − 1/16, which equals 2/4 − 1/16.

Assuming that all card picks here are fair, what is the probability of getting either a 6 or a 7 when you pick one card out of a standard deck?

The answer is 2/13.

The probability of picking a 6 is 1/13, and the probability of picking a 7 is also 1/13, so the probability of picking either a 6 or a 7 is 1/13 + 1/13 = 2/13. This calculation applies the simple rule for probabilities of disjunctions because picking a 6 and picking a 7 are mutually exclusive, since it is not possible to pick both a 6 and a 7 in one pick out of a fair standard deck of cards. No card in a standard deck is both a 6 and also a 7.

The probability of picking a 6 is 1/13, and the probability of picking a 7 is also 1/13, so the probability of picking either a 6 or a 7 is 1/13 + 1/13 = 2/13. This calculation applies the simple rule for probabilities of disjunctions because picking a 6 and picking a 7 are mutually exclusive, since it is not possible to pick both a 6 and a 7 in one pick out of a fair standard deck of cards. No card in a standard deck is both a 6 and also a 7.

Assuming that all card picks here are fair, what is the probability of picking a 7 on either the first pick or the second pick when you pick two cards out of a standard deck (and when you put the first card back and shuffle the cards before the second pick)?

The answer is 2/13 - (1/13 x 1/13).

The probability of picking a 7 on the first pick is 1/13, and the probability of picking a 7 on the second pick (given replacement and shuffling) is also 1/13. These results are not mutually exclusive, because it is possible to get a 7 on both picks. That means that we need to apply the general rule for probability of disjunction that does not require the events to be mutually exclusive. That general rule requires us to subtract the probability of both events occurring together. The probability of picking a 7 in both picks (given replacement and shuffling) is 1/13 x 1/13 = 1/169. Thus, the probability of picking a 7 on either the first pick or the second pick is 1/13 + 1/13 - (1/13 x 1/13), which equals 2/13 - (1/13 x 1/13).

The probability of picking a 7 on the first pick is 1/13, and the probability of picking a 7 on the second pick (given replacement and shuffling) is also 1/13. These results are not mutually exclusive, because it is possible to get a 7 on both picks. That means that we need to apply the general rule for probability of disjunction that does not require the events to be mutually exclusive. That general rule requires us to subtract the probability of both events occurring together. The probability of picking a 7 in both picks (given replacement and shuffling) is 1/13 x 1/13 = 1/169. Thus, the probability of picking a 7 on either the first pick or the second pick is 1/13 + 1/13 - (1/13 x 1/13), which equals 2/13 - (1/13 x 1/13).

Which of the following is accurate?

The answer is "Order of events matters in permutations but not in combinations."

These definitions are stated and explained in the video. For example, (a) heads then tails and (b) tails then heads count as one and the same combination but as two different permutations.

These definitions are stated and explained in the video. For example, (a) heads then tails and (b) tails then heads count as one and the same combination but as two different permutations.

Assuming that all card picks here are fair, what is the probability of picking an 8 and a 9 in any order when you pick two cards out of a standard deck (when you put the first card back and shuffle the cards before the second pick)?

The answer is (1/13 x 1/13) + (1/13 x 1/13).

When you pick two cards out of a standard deck, you pick an 8 and a 9 in any order when you pick either (a) an 8 and then a 9 or (b) a 9 and then an 8. When you replace the first card and shuffle the deck before the second pick, the two picks are independent, so we can calculate the probabilities of (a) and (b) using the simple rule for probabilities of conjunctions. Using this rule, the probability of picking (a) an 8 and then a 9 is 1/13 x 1/13, and the probability of picking (b) a 9 and then an 8 is also 1/13 x 1/13. These two possibilities are mutually exclusive, since they cannot both be true, so we can apply the simple rule for probabilities of disjunctions. Using that rule, the probability of picking either (a) or (b) is (1/13 x 1/13) + (1/13 x 1/13), so this is the probability of picking an 8 and a 9 in any order. This example illustrates the way to calculate probabilities of combinations as opposed to permutations.

When you pick two cards out of a standard deck, you pick an 8 and a 9 in any order when you pick either (a) an 8 and then a 9 or (b) a 9 and then an 8. When you replace the first card and shuffle the deck before the second pick, the two picks are independent, so we can calculate the probabilities of (a) and (b) using the simple rule for probabilities of conjunctions. Using this rule, the probability of picking (a) an 8 and then a 9 is 1/13 x 1/13, and the probability of picking (b) a 9 and then an 8 is also 1/13 x 1/13. These two possibilities are mutually exclusive, since they cannot both be true, so we can apply the simple rule for probabilities of disjunctions. Using that rule, the probability of picking either (a) or (b) is (1/13 x 1/13) + (1/13 x 1/13), so this is the probability of picking an 8 and a 9 in any order. This example illustrates the way to calculate probabilities of combinations as opposed to permutations.

Imagine that there are two little lotteries in your town. Each lottery sells exactly 100 tickets each and has only one winning ticket. You pick one of these lotteries and buy two tickets to the same lottery. What is the probability that you will have one winning ticket (that is, the probability that either your first ticket or your second ticket will win)?

The answer is 1/100 + 1/100.

Only one ticket can win, so the possibility that your first ticket wins and the possibility that your second ticket wins are mutually exclusive. Thus, we can apply the simple rule for probabilities of disjunctions. The probability that your first ticket wins is 1/100, and the probability that your second ticket wins is also 1/100, so the probability that either your first or your second ticket wins is 1/100 + 1/100.

Only one ticket can win, so the possibility that your first ticket wins and the possibility that your second ticket wins are mutually exclusive. Thus, we can apply the simple rule for probabilities of disjunctions. The probability that your first ticket wins is 1/100, and the probability that your second ticket wins is also 1/100, so the probability that either your first or your second ticket wins is 1/100 + 1/100.

As in Question 14, imagine that there are two little lotteries in your town. Each lottery sells exactly 100 tickets each and has only one winning ticket. You buy one ticket to each of these two lotteries. What is the probability that you will have at least one winning ticket (that is, the probability that either your first ticket, your second ticket, or both will win)?

The answer is 1/100 + 1/100 - (1/100 x 1/100).

In this example (unlike Question 14) both of your tickets can win, so the possibility that your first ticket wins and the possibility that your second ticket wins are not mutually exclusive. Thus, we need to apply the general rule for probabilities of disjunctions that need not be mutually exclusive. The probability that your first ticket wins the first lottery is 1/100, and the probability that your second ticket wins the second lottery is also 1/100, and the probability that both of your tickets win is (1/100 x 1/100). Thus, the probability that either your first or your second ticket wins is 1/100 + 1/100 - (1/100 x 1/100).

In this example (unlike Question 14) both of your tickets can win, so the possibility that your first ticket wins and the possibility that your second ticket wins are not mutually exclusive. Thus, we need to apply the general rule for probabilities of disjunctions that need not be mutually exclusive. The probability that your first ticket wins the first lottery is 1/100, and the probability that your second ticket wins the second lottery is also 1/100, and the probability that both of your tickets win is (1/100 x 1/100). Thus, the probability that either your first or your second ticket wins is 1/100 + 1/100 - (1/100 x 1/100).

You can construct your own examples for practicing simply by asking about other outcomes on flips of coins, rolls of dice, and picks of cards.

True.

Try it! Remember to specify all of the relevant factors, such as fairness of the coins, dice, and cards and whether the first card is returned to the deck and the deck shuffled. Then bring your examples and answers to the discussion forums to ask other students whether they get the same answers as you did. Have fun!

Try it! Remember to specify all of the relevant factors, such as fairness of the coins, dice, and cards and whether the first card is returned to the deck and the deck shuffled. Then bring your examples and answers to the discussion forums to ask other students whether they get the same answers as you did. Have fun!

What is the probability of getting heads at least once in a series of three fair flips of a coin?

The answer is 0.875.

First, the probability of getting heads on a single fair flip is 0.5. Second, we use the rule for negations to calculate the probability of not getting heads on a single fair flip. That probability is 1 - 0.5, so it is also 0.5. Third, we use the rule for conjunctions to calculate the probability of not getting heads in any of three consecutive flips. The results of fair flips are independent, so we can use the simple version of the rule for conjunctions of independent events. Using that rule, the probability of not getting a single heads in three fair flips is 0.5 x 0.5 x 0.5 = (0.5)3 = 0.125. Not getting a single heads in three fair flips is the negation of getting at least one heads in three fair flips. Thus, fourth, we apply the rule for negations to calculate the probability of getting heads at least once in three fair flips: 1 - 0.125 = 0.875. This calculation in effect applies the rule for calculating probabilities in a series that was discussed in the video.

First, the probability of getting heads on a single fair flip is 0.5. Second, we use the rule for negations to calculate the probability of not getting heads on a single fair flip. That probability is 1 - 0.5, so it is also 0.5. Third, we use the rule for conjunctions to calculate the probability of not getting heads in any of three consecutive flips. The results of fair flips are independent, so we can use the simple version of the rule for conjunctions of independent events. Using that rule, the probability of not getting a single heads in three fair flips is 0.5 x 0.5 x 0.5 = (0.5)3 = 0.125. Not getting a single heads in three fair flips is the negation of getting at least one heads in three fair flips. Thus, fourth, we apply the rule for negations to calculate the probability of getting heads at least once in three fair flips: 1 - 0.125 = 0.875. This calculation in effect applies the rule for calculating probabilities in a series that was discussed in the video.

What is the probability of rolling 7 at least once in a series of three fair rolls of two dice in each roll?

The answer is 91/216.

First, we calculate the probability of rolling 7 on a single fair roll of two dice. As Video 9-2 explained, there are six ways to get 7 with two dice, and 6 x 6 = 36 possible outcomes with two dice, so the probability of rolling 7 is 6/36 = 1/6. Second, we use the rule for negations to calculate the probability of not rolling 7 on a single fair roll of two dice. That probability is 1 - 1/6 = 5/6. Third, we use the rule for conjunctions to calculate the probability of not rolling 7 in any of three consecutive fair rolls of two dice. The results of fair rolls are independent, so we can use the simple version of the rule for conjunctions of independent events. Using this rule, the probability of not getting 7 in any of three fair rolls of two dice is 5/6 x 5/6 x 5/6 = (5/6)3 = 125/216. That result is the negation of getting 7 at least once in three fair rolls of two dice. Thus, fourth, we apply the rule for negations to calculate the probability of getting 7 at least once in three fair rolls of two dice: 1 - 125/216 = 91/216. This calculation in effect applies the rule for calculating probabilities in a series that was discussed in the video.

First, we calculate the probability of rolling 7 on a single fair roll of two dice. As Video 9-2 explained, there are six ways to get 7 with two dice, and 6 x 6 = 36 possible outcomes with two dice, so the probability of rolling 7 is 6/36 = 1/6. Second, we use the rule for negations to calculate the probability of not rolling 7 on a single fair roll of two dice. That probability is 1 - 1/6 = 5/6. Third, we use the rule for conjunctions to calculate the probability of not rolling 7 in any of three consecutive fair rolls of two dice. The results of fair rolls are independent, so we can use the simple version of the rule for conjunctions of independent events. Using this rule, the probability of not getting 7 in any of three fair rolls of two dice is 5/6 x 5/6 x 5/6 = (5/6)3 = 125/216. That result is the negation of getting 7 at least once in three fair rolls of two dice. Thus, fourth, we apply the rule for negations to calculate the probability of getting 7 at least once in three fair rolls of two dice: 1 - 125/216 = 91/216. This calculation in effect applies the rule for calculating probabilities in a series that was discussed in the video.

What is the probability of drawing at least one spade in a series of four consecutive fair draws from a standard deck of cards, when the card drawn is returned to the deck and the deck is shuffled between each draw?

The answer is 175/256.

First, we calculate the probability of drawing a spade in a single fair draw from a standard deck. There are thirteen spades in fifty-two cards of a standard deck, so the probability of drawing a spade in a fair draw is 13/52 = 1/4. Second, we use the rule for negations to calculate the probability of not drawing a spade in a single fair draw from a standard deck. That probability is 1 - 1/4 = 3/4. Third, we use the rule for conjunctions to calculate the probability of not drawing a spade in any of four consecutive fair draws from a standard deck. The results of fair draws are independent when the card drawn is returned to the deck and the deck is shuffled between each draw, so we can use the simple version of the rule for conjunctions of independent events. Then this probability is 3/4 x 3/4 x 3/4 x 3/4 = (3/4)4 = 81/256. That is the probability of not drawing a spade in any of four consecutive fair draws from a standard deck. That result is the negation of drawing a spade in at least one of four consecutive fair draws from a standard deck. Thus, fourth, we apply the rule for negations to calculate the probability of drawing a spade in at least one of four consecutive fair draws from a standard deck: 1 - 81/256 = 175/256. This calculation in effect applies the rule for calculating probabilities in a series that was discussed in the video.

First, we calculate the probability of drawing a spade in a single fair draw from a standard deck. There are thirteen spades in fifty-two cards of a standard deck, so the probability of drawing a spade in a fair draw is 13/52 = 1/4. Second, we use the rule for negations to calculate the probability of not drawing a spade in a single fair draw from a standard deck. That probability is 1 - 1/4 = 3/4. Third, we use the rule for conjunctions to calculate the probability of not drawing a spade in any of four consecutive fair draws from a standard deck. The results of fair draws are independent when the card drawn is returned to the deck and the deck is shuffled between each draw, so we can use the simple version of the rule for conjunctions of independent events. Then this probability is 3/4 x 3/4 x 3/4 x 3/4 = (3/4)4 = 81/256. That is the probability of not drawing a spade in any of four consecutive fair draws from a standard deck. That result is the negation of drawing a spade in at least one of four consecutive fair draws from a standard deck. Thus, fourth, we apply the rule for negations to calculate the probability of drawing a spade in at least one of four consecutive fair draws from a standard deck: 1 - 81/256 = 175/256. This calculation in effect applies the rule for calculating probabilities in a series that was discussed in the video.

You can construct your own examples for practicing simply by asking about other series of flips of coins, rolls of dice, and picks of cards.

True.

Try it! Remember to specify all of the relevant factors, such as fairness of the coins, dice, and cards and whether the first card is returned to the deck and the deck shuffled. Then bring your examples and answers to the discussion forums to ask other students whether they get the same answers as you did. Have fun!

Try it! Remember to specify all of the relevant factors, such as fairness of the coins, dice, and cards and whether the first card is returned to the deck and the deck shuffled. Then bring your examples and answers to the discussion forums to ask other students whether they get the same answers as you did. Have fun!

A false positive (or a false alarm) is a case where

The answer is "the condition is absent and the test comes out positive."

These cases are called "positives" because the test comes out positive, but they are "false" positives because the condition is not really present. This term was explained in the video.

These cases are called "positives" because the test comes out positive, but they are "false" positives because the condition is not really present. This term was explained in the video.

A false negative (or a miss) is a case where

The answer is "the condition is present and the test comes out negative."

These cases are called "negatives" because the test comes out negative, but they are "false" negatives because the condition is really present. This term was explained in the video.

These cases are called "negatives" because the test comes out negative, but they are "false" negatives because the condition is really present. This term was explained in the video.

The base rate or prevalence of a condition in a population is the

The answer is "percentage of the population with the condition."

This definition was stated and explained in the video. For example, if 100,000 people in a population of 1,000,000 have the condition (such as alcoholism), then the base rate or prevalence of that condition in that population is 10 percent or 0.1.

This definition was stated and explained in the video. For example, if 100,000 people in a population of 1,000,000 have the condition (such as alcoholism), then the base rate or prevalence of that condition in that population is 10 percent or 0.1.

The sensitivity of a test for a condition is the probability of a

The answer is "positive test result, given that the person tested does have the condition."

This definition was stated and explained in the video. For example, if a test for a condition will come out positive in 800 out of every 1,000 people with that condition, then the test has a sensitivity of 80 percent or 0.8. Notice that a test with this sensitivity will miss the other 20 percent of the cases with the condition by giving a negative result in those cases, so it has a false negative rate (or a miss rate) of 20 percent. The name "sensitivity" is used because, just as a person's skin is more sensitive to touch when it reacts to more touches (including very light touches), so a more sensitive test is one that has a positive test result in more of the cases with the condition and misses fewer of the cases with the condition.

This definition was stated and explained in the video. For example, if a test for a condition will come out positive in 800 out of every 1,000 people with that condition, then the test has a sensitivity of 80 percent or 0.8. Notice that a test with this sensitivity will miss the other 20 percent of the cases with the condition by giving a negative result in those cases, so it has a false negative rate (or a miss rate) of 20 percent. The name "sensitivity" is used because, just as a person's skin is more sensitive to touch when it reacts to more touches (including very light touches), so a more sensitive test is one that has a positive test result in more of the cases with the condition and misses fewer of the cases with the condition.

The specificity of a test for a condition is the probability of a

The answer is "negative test result, given that the person tested does not have the condition."

This definition was stated and explained in the video. For example, if a test for a condition will come out negative in 800 out of every 1,000 people who do not have that condition, then the test has a specificity of 80 percent or 0.8. Notice that a test with this specificity will come out positive in the other 20 percent of the cases that do not have the condition, so it has a false positive rate (or a false alarm rate) of 20 percent. The name "specificity" is used because a specific test reacts specifically to the one condition being tested (such as colon cancer) and does not react to other conditions (such as breast cancer) when the condition being tested is not present.

This definition was stated and explained in the video. For example, if a test for a condition will come out negative in 800 out of every 1,000 people who do not have that condition, then the test has a specificity of 80 percent or 0.8. Notice that a test with this specificity will come out positive in the other 20 percent of the cases that do not have the condition, so it has a false positive rate (or a false alarm rate) of 20 percent. The name "specificity" is used because a specific test reacts specifically to the one condition being tested (such as colon cancer) and does not react to other conditions (such as breast cancer) when the condition being tested is not present.

For Questions 6-14, assume that

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in the table below.

What number belongs in Box 7?

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in the table below.

What number belongs in Box 7?

The answer is 100.

Box 7 includes the total number of people in the population who have colon cancer. That number is the base rate or prevalence times the population. The prevalence in this population was given as 0.001, and 0.001 x 100,000 = 100.

Box 7 includes the total number of people in the population who have colon cancer. That number is the base rate or prevalence times the population. The prevalence in this population was given as 0.001, and 0.001 x 100,000 = 100.

Assume that

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in this table: What number belongs in Box 8?

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in this table: What number belongs in Box 8?

The answer is 99,900.

Box 8 includes the total number of people in the population who do not have colon cancer. We saw in Question 6 that the number in this population with colon cancer is 100. The population was given as 100,000, and 100,000 - 100 = 99,900, so that number should be in Box 8.

Box 8 includes the total number of people in the population who do not have colon cancer. We saw in Question 6 that the number in this population with colon cancer is 100. The population was given as 100,000, and 100,000 - 100 = 99,900, so that number should be in Box 8.

Assume that

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in this table: What number belongs in Box 1?

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in this table: What number belongs in Box 1?

The answer is 90.

Box 1 includes the number of people in the population with colon cancer who test positive. We saw in Question 6 that the number in this population with colon cancer is 100. The sensitivity of a test is the percentage of those with the condition who test positive. The sensitivity of the test in this example is 90 percent. Therefore, the number in Box 1 should be 90 percent x 100 = 90.

Box 1 includes the number of people in the population with colon cancer who test positive. We saw in Question 6 that the number in this population with colon cancer is 100. The sensitivity of a test is the percentage of those with the condition who test positive. The sensitivity of the test in this example is 90 percent. Therefore, the number in Box 1 should be 90 percent x 100 = 90.

Assume that

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in this table: What number belongs in Box 4?

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in this table: What number belongs in Box 4?

The answer is 10.

Box 4 includes the number of people in the population with colon cancer who do not test positive. We saw in Question 6 that the number in this population with colon cancer is 100, and we saw in Question 8 that the number with colon cancer who test positive is 90. The rest of those with colon cancer test negative. Therefore, the number in Box 4 should be 100 - 90 = 10. Another way to calculate this number is to figure that, if the sensitivity of the test is 90 percent, then the test will have 10 percent false negatives, and 10 percent x 100 = 10.

Box 4 includes the number of people in the population with colon cancer who do not test positive. We saw in Question 6 that the number in this population with colon cancer is 100, and we saw in Question 8 that the number with colon cancer who test positive is 90. The rest of those with colon cancer test negative. Therefore, the number in Box 4 should be 100 - 90 = 10. Another way to calculate this number is to figure that, if the sensitivity of the test is 90 percent, then the test will have 10 percent false negatives, and 10 percent x 100 = 10.

Assume that

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in this table: What number belongs in Box 5?

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in this table: What number belongs in Box 5?

The answer is 96,903.

Box 5 includes the number of people in the population without colon cancer who test negative. We saw in Question 7 that the number in this population without colon cancer is 99,900. The specificity of a test is the percentage of those without the condition who test negative. The specificity of the test in this example is 97 percent. Therefore, the number in Box 5 should be 97 percent x 99,900 = 96,903.

Box 5 includes the number of people in the population without colon cancer who test negative. We saw in Question 7 that the number in this population without colon cancer is 99,900. The specificity of a test is the percentage of those without the condition who test negative. The specificity of the test in this example is 97 percent. Therefore, the number in Box 5 should be 97 percent x 99,900 = 96,903.

Assume that

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in this table: What number belongs in Box 2?

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in this table: What number belongs in Box 2?

The answer is 2,997.

Box 2 includes the number of people in the population without colon cancer who test positive. We saw in Question 7 that the number in this population with colon cancer is 99,900, and we saw in Question 10 that the number without colon cancer who test negative is 96,903. The rest of those without colon cancer tests positive. Therefore, the number in Box 2 should be 99,900 - 96,903 = 2,997. Another way to calculate this number is to figure that, if the specificity of the test is 97 percent, then the test will have 3 percent false positives, and 3 percent x 99,900 = 2,997.

Box 2 includes the number of people in the population without colon cancer who test positive. We saw in Question 7 that the number in this population with colon cancer is 99,900, and we saw in Question 10 that the number without colon cancer who test negative is 96,903. The rest of those without colon cancer tests positive. Therefore, the number in Box 2 should be 99,900 - 96,903 = 2,997. Another way to calculate this number is to figure that, if the specificity of the test is 97 percent, then the test will have 3 percent false positives, and 3 percent x 99,900 = 2,997.

Assume that

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in this table: What number belongs in Box 3?

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in this table: What number belongs in Box 3?

The answer is 3,087.

Box 3 includes the total number of people in the population who test positive. We saw in Question 8 that the number in this population with colon cancer who test positive is 90. We saw in Question 11 that the number in this population without colon cancer who test positive is 2,997. Therefore, the total number of people in this population who test positive is 90 + 2,997 = 3,087.

Box 3 includes the total number of people in the population who test positive. We saw in Question 8 that the number in this population with colon cancer who test positive is 90. We saw in Question 11 that the number in this population without colon cancer who test positive is 2,997. Therefore, the total number of people in this population who test positive is 90 + 2,997 = 3,087.

Assume that

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in this table: What number belongs in Box 6?

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

Use these assumptions to fill the boxes in this table: What number belongs in Box 6?

The answer is 96,913.

Box 6 includes the total number of people in the population who test negative. We saw in Question 9 that the number in this population with colon cancer who test negative is 10. We saw in Question 10 that the number in this population without colon cancer who test negative is 96,903. Therefore, the total number of people in this population who test negative is 10 + 96,903 = 96,913.

Box 6 includes the total number of people in the population who test negative. We saw in Question 9 that the number in this population with colon cancer who test negative is 10. We saw in Question 10 that the number in this population without colon cancer who test negative is 96,903. Therefore, the total number of people in this population who test negative is 10 + 96,903 = 96,913.

Assume that

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

In the case above, what is the probability that a patient has colon cancer given that the patient tests positive for colon cancer with this test?

(A) the base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) the sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) the specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

In the case above, what is the probability that a patient has colon cancer given that the patient tests positive for colon cancer with this test?

The answer is 90/3,087.

We saw in Question 8 that the number of people in this population with colon cancer who test positive is 90. We also saw in Question 12 that the total number of people in this population who test positive is 3,087. Thus, 90 of the 3,087 who test positive really have the condition. Because 90/3,087 is less than 3 percent, a random individual with no symptoms who tests positive in this situation has less than a 3 percent chance of really having the condition and more than 97 percent chance of being a false positive!

We saw in Question 8 that the number of people in this population with colon cancer who test positive is 90. We also saw in Question 12 that the total number of people in this population who test positive is 3,087. Thus, 90 of the 3,087 who test positive really have the condition. Because 90/3,087 is less than 3 percent, a random individual with no symptoms who tests positive in this situation has less than a 3 percent chance of really having the condition and more than 97 percent chance of being a false positive!

Suppose that the situation is exactly like the case in Questions 6-14 except that the base rate is higher:

(A) The base rate or prevalence of colon cancer in a population is 10 percent or 0.1

(B) The sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) The specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer.)

For this population and test, what is the probability that a patient has colon cancer given that the patient tests positive for colon cancer with this test?

(A) The base rate or prevalence of colon cancer in a population is 10 percent or 0.1

(B) The sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) The specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer.)

For this population and test, what is the probability that a patient has colon cancer given that the patient tests positive for colon cancer with this test?

The answer is 90/117. The probability of colon cancer given a positive test result is then 9,000/11,700 = 90/117 = about 77 percent. Notice that this probability is much higher than the answer to Question 14, which shows that the base rate alone has a large effect on the probability.

Suppose that the situation is exactly like the case in Questions 6-14 except that the sensitivity is higher:

(A) The base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) The sensitivity of a certain test for colon cancer is 99 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) The specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

For this population and test, what is the probability that a patient has colon cancer given that the patient tests positive for colon cancer with this test?

(A) The base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) The sensitivity of a certain test for colon cancer is 99 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) The specificity of that same test for colon cancer is 97 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

For this population and test, what is the probability that a patient has colon cancer given that the patient tests positive for colon cancer with this test?

The answer is 99/3,096. The probability of colon cancer given a positive test result in this case is then 99/3,096 = about 3.2 percent. Notice that this probability is only slightly higher than the answer to Question 14, which shows that the sensitivity by itself has only a small effect on the probability (when the base rate is low).

Suppose that the situation is exactly like the case in Questions 6-14 except that the specificity is higher:

(A) The base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) The sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) The specificity of that same test for colon cancer is 99 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

For this population and test, what is the probability that a patient has colon cancer given that the patient tests positive for colon cancer with this test?

(A) The base rate or prevalence of colon cancer in a population is 0.1 percent or 0.001

(B) The sensitivity of a certain test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) The specificity of that same test for colon cancer is 99 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer).

For this population and test, what is the probability that a patient has colon cancer given that the patient tests positive for colon cancer with this test?

The answer is 90/1089. The probability of colon cancer given a positive test result in this case is then 90/1,089 = about 8 percent. Notice that this probability is more than double the answer to Question 14, even though the specificity was increased only from 97 percent to 99 percent. That shows that the specificity of the test has a moderate effect on the probability even when the base rate is low.

Suppose that a patient tests positive for colon cancer in the circumstances of Question 17, so there is an 8 percent chance that this patient has colon cancer. This information puts the patient in a new population—patients who have tested positive for colon cancer—with a higher base rate (8 percent) of colon cancer. Now suppose that the doctor orders a second test that is independent of the first test (because it looks for separate indications of colon cancer). This second test has the same specificity and sensitivity as the first test, so this is the new situation:

(A) The base rate or prevalence of colon cancer in a population is 8 percent or 0.08

(B) The sensitivity of the second test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) The specificity of the second test for colon cancer is 99 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer)

For this population and test, what is the probability that a patient has colon cancer given that the patient tests positive for colon cancer with this test?

(A) The base rate or prevalence of colon cancer in a population is 8 percent or 0.08

(B) The sensitivity of the second test for colon cancer is 90 percent (sensitivity is the probability of a positive test result, given that the person tested does have colon cancer)

(C) The specificity of the second test for colon cancer is 99 percent (specificity is the probability of a negative test result, given that the person tested does not have colon cancer)

For this population and test, what is the probability that a patient has colon cancer given that the patient tests positive for colon cancer with this test?

The answer is 7,200/8,120.

The probability of colon cancer given a positive test result in this case is then 7,200/8,120 = over 88 percent. Notice that this probability is much higher than the answer to Question 17, which shows that getting a second positive test result has a large effect on the probability.

The probability of colon cancer given a positive test result in this case is then 7,200/8,120 = over 88 percent. Notice that this probability is much higher than the answer to Question 17, which shows that getting a second positive test result has a large effect on the probability.

Chris tested positive for cocaine once in a random screening test. This test has a sensitivity and specificity of 90 percent, and 10 percent of the students in Chris's school use cocaine. What is the probability that Chris really did use cocaine?

The answer is 90/180. The probability of cocaine use given a positive test result in this situation is then 90/180= 50 percent. Of course, this probability would be much lower or higher if the base rate, sensitivity, and specificity were different. To see this, try recalculating the probability of cocaine use with different base rates, sensitivities, and specificities.

Late last night a car ran into your neighbor and drove away. In your town, there are 500 cars, and 2 percent of them are Porsches. The only eyewitness to the incident says that the car that hit your neighbor was a Porsche. Tested under similar conditions, the eyewitness mistakenly classifies cars of other makes as Porsches 10 percent of the time, and correctly classifies Porsches as Porsches 80 percent of the time. What are the chances that the car that hit your neighbor really was a Porsche?

The answer is 8/57. The probability of this eyewitness being correct in these circumstances is then 8/57= around 14 percent. Of course, this probability would be much lower or higher if the base rate were different or of the witness were better or worse at identifying Porsches. To see this, try recalculating the probability of correct eyewitness identification with different base rates, sensitivities, and specificities.

You can construct your own examples for practicing simply by asking about other cases with different base rates, sensitivities, and specificities.

True.

Try it! Then bring your examples and answers to the discussion forums to ask other students whether they get the same answers as you did. Have fun!

Try it! Then bring your examples and answers to the discussion forums to ask other students whether they get the same answers as you did. Have fun!

The expected financial value of a bet is the

The answer is "probability of winning times the net gain of winning minus the probability of losing times the net loss of losing."

This definition was stated and explained in the video.

This definition was stated and explained in the video.

In the games in Questions 2 through 8, you lay down $1 to bet that you will pick a certain card in a fair draw from a standard deck. If you lose, then you lose your $1. If you win, then you collect the gross amount indicated, so your net gain is $1 less.

What is the expected financial value of a bet where you will win $26 if you draw a 7 of spades?

What is the expected financial value of a bet where you will win $26 if you draw a 7 of spades?

The answer is -$0.50.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. The probability of winning this bet is 1/52. The gross gain from winning is $26, and playing costs $1 (which is not refunded if you win), so the net gain from winning this lottery is $25. The probability of losing is 1 - (1/52) = 51/52. The net loss of losing is $1. Thus, the expected financial value of one ticket this lottery is [(1/52) x $25] - [(51/52) x $1] = ($25/52) - ($51/52) = - $26/52 = -$0.50.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. The probability of winning this bet is 1/52. The gross gain from winning is $26, and playing costs $1 (which is not refunded if you win), so the net gain from winning this lottery is $25. The probability of losing is 1 - (1/52) = 51/52. The net loss of losing is $1. Thus, the expected financial value of one ticket this lottery is [(1/52) x $25] - [(51/52) x $1] = ($25/52) - ($51/52) = - $26/52 = -$0.50.

Again, in this game, you lay down $1 to bet that you will pick a certain card in a fair draw from a standard deck. If you lose, then you lose your $1. If you win, then you collect the gross amount indicated, so your net gain is $1 less.

What is the expected financial value of a bet where you will win $26 if you draw either a 7 of spades or a 7 of clubs?

What is the expected financial value of a bet where you will win $26 if you draw either a 7 of spades or a 7 of clubs?

The answer is $0.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. The probability of winning this bet is (1+1)/52 = 2/52. The gross gain from winning is $26, and playing costs $1 (which is not refunded if you win), so the net gain from winning this lottery is $25. The probability of losing is 1 - (2/52) = 50/52. The net loss of losing is $1. Thus, the expected financial value of one ticket this lottery is [(2/52) x $25] - [(50/52) x $1] = ($50/52) - ($50/52) = $0.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. The probability of winning this bet is (1+1)/52 = 2/52. The gross gain from winning is $26, and playing costs $1 (which is not refunded if you win), so the net gain from winning this lottery is $25. The probability of losing is 1 - (2/52) = 50/52. The net loss of losing is $1. Thus, the expected financial value of one ticket this lottery is [(2/52) x $25] - [(50/52) x $1] = ($50/52) - ($50/52) = $0.

Again, in this game, you lay down $1 to bet that you will pick a certain card in a fair draw from a standard deck. If you lose, then you lose your $1. If you win, then you collect the gross amount indicated, so your net gain is $1 less.

What is the expected financial value of a bet where you will win $26 if you draw a 7 of any suit?

What is the expected financial value of a bet where you will win $26 if you draw a 7 of any suit?

The answer is $1.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. The probability of winning this bet is 4/52. The gross gain from winning is $26, and playing costs $1 (which is not refunded if you win), so the net gain from winning this lottery is $25. The probability of losing is 1 - (4/52) = 48/52. The net loss of losing is $1. Thus, the expected financial value of one ticket this lottery is: [(4/52) x $25] - [(48/52) x $1] = ($100/52) - ($48/52) = $52/52 = $1.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. The probability of winning this bet is 4/52. The gross gain from winning is $26, and playing costs $1 (which is not refunded if you win), so the net gain from winning this lottery is $25. The probability of losing is 1 - (4/52) = 48/52. The net loss of losing is $1. Thus, the expected financial value of one ticket this lottery is: [(4/52) x $25] - [(48/52) x $1] = ($100/52) - ($48/52) = $52/52 = $1.

Again, in this game, you lay down $1 to bet that you will pick a certain card in a fair draw from a standard deck. If you lose, then you lose your $1. If you win, then you collect the gross amount indicated, so your net gain is $1 less.

What is the expected financial value of a bet where you will win $4 if you draw a Jack, a Queen, or a King?

What is the expected financial value of a bet where you will win $4 if you draw a Jack, a Queen, or a King?

The answer is -$4/52.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. The probability of winning this bet is (4+4+4)/52 = 12/52. The gross gain from winning is $4, and playing costs $1 (which is not refunded if you win), so the net gain from winning this lottery is $3. The probability of losing is 1 - 12/52 = 40/52. The net loss of losing is $1. Thus, the expected financial value of one ticket this lottery is: [(12/52) x $3] - [(40/52) x $1] = ($36/52) - ($40/52) = -$4/52.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. The probability of winning this bet is (4+4+4)/52 = 12/52. The gross gain from winning is $4, and playing costs $1 (which is not refunded if you win), so the net gain from winning this lottery is $3. The probability of losing is 1 - 12/52 = 40/52. The net loss of losing is $1. Thus, the expected financial value of one ticket this lottery is: [(12/52) x $3] - [(40/52) x $1] = ($36/52) - ($40/52) = -$4/52.

Again, in this game, you lay down $1 to bet that you will pick a certain card in a fair draw from a standard deck. If you lose, then you lose your $1. If you win, then you collect the gross amount indicated, so your net gain is $1 less.

What is the expected financial value of a bet where you will win $2 if you do NOT draw a Jack, a Queen, or a King?

What is the expected financial value of a bet where you will win $2 if you do NOT draw a Jack, a Queen, or a King?

The answer is $28/52.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. The probability of drawing a Jack, Queen, or King is 12/52 (see Question 5 in this exercise), so the probability of NOT drawing a Jack, Queen, or King is 1 - 12/52 = 40/52. That is the probability of winning this bet. The gross gain from winning is $2, and playing costs $1 (which is not refunded if you win), so the net gain from winning this lottery is $1. The probability of losing is 1 - (40/52) = 12/52. The net loss of losing is $1. Thus, the expected financial value of one ticket this lottery is [(40/52) x $1] - [(12/52) x $1] = $28/52.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. The probability of drawing a Jack, Queen, or King is 12/52 (see Question 5 in this exercise), so the probability of NOT drawing a Jack, Queen, or King is 1 - 12/52 = 40/52. That is the probability of winning this bet. The gross gain from winning is $2, and playing costs $1 (which is not refunded if you win), so the net gain from winning this lottery is $1. The probability of losing is 1 - (40/52) = 12/52. The net loss of losing is $1. Thus, the expected financial value of one ticket this lottery is [(40/52) x $1] - [(12/52) x $1] = $28/52.

Again, in this game, you lay down $1 to bet that you will pick a certain card in a fair draw from a standard deck. If you lose, then you lose your $1. If you win, then you collect the gross amount indicated, so your net gain is $1 less.

What is the expected financial value of a bet where you will win $2,652 if you draw a 7 of spades and then a 7 of clubs on two consecutive draws (without returning the first card to the deck)?

What is the expected financial value of a bet where you will win $2,652 if you draw a 7 of spades and then a 7 of clubs on two consecutive draws (without returning the first card to the deck)?

The answer is $0.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. The probability of drawing a 7 of spades and then a 7 of clubs on two consecutive draws (without returning the first card to the deck) is (1/52) x (1/51) = 1/2652. That is the probability of winning this bet. The gross gain from winning is $2,652, and playing costs $1 (which is not refunded if you win), so the net gain from winning this lottery is $2,651. The probability of losing is 1 - (1/2,652) = 2,651/2,652. The net loss of losing is $1. Thus, the expected financial value of one ticket this lottery is [(1/2,652) x $2,651] - [(2,651/2,652) x $1] = $0.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. The probability of drawing a 7 of spades and then a 7 of clubs on two consecutive draws (without returning the first card to the deck) is (1/52) x (1/51) = 1/2652. That is the probability of winning this bet. The gross gain from winning is $2,652, and playing costs $1 (which is not refunded if you win), so the net gain from winning this lottery is $2,651. The probability of losing is 1 - (1/2,652) = 2,651/2,652. The net loss of losing is $1. Thus, the expected financial value of one ticket this lottery is [(1/2,652) x $2,651] - [(2,651/2,652) x $1] = $0.

Again in this game, you lay down $1 to bet that you will pick a certain card in a fair draw from a standard deck. If you lose, then you lose your $1. If you win, then you collect the gross amount indicated, so your net gain is $1 less.

What is the expected financial value of a bet where you will win $2,652 if you draw a 7 of spades and then a 7 of clubs on two consecutive draws (where you replace the card and shuffle the deck between draws)?

What is the expected financial value of a bet where you will win $2,652 if you draw a 7 of spades and then a 7 of clubs on two consecutive draws (where you replace the card and shuffle the deck between draws)?

The answer is -$52/2704.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. The probability of drawing a 7 of spades and then a 7 of clubs on two consecutive draws (with replacement and shuffling) is (1/52) x (1/52) = 1/2,704. That is the probability of winning this bet. The gross gain from winning is $2,652, and playing costs $1 (which is not refunded if you win), so the net gain from winning this lottery is $2,651. The probability of losing is 1 - (1/2,704) = 2,703/2,704. The net loss of losing is $1. Thus, the expected financial value of one ticket this lottery is: [(1/2,704) x $2,651] - [(2,703/2,704) x $1] = -$52/2,704.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. The probability of drawing a 7 of spades and then a 7 of clubs on two consecutive draws (with replacement and shuffling) is (1/52) x (1/52) = 1/2,704. That is the probability of winning this bet. The gross gain from winning is $2,652, and playing costs $1 (which is not refunded if you win), so the net gain from winning this lottery is $2,651. The probability of losing is 1 - (1/2,704) = 2,703/2,704. The net loss of losing is $1. Thus, the expected financial value of one ticket this lottery is: [(1/2,704) x $2,651] - [(2,703/2,704) x $1] = -$52/2,704.

You can construct your own examples for practicing simply by asking about other bets.

True.

Try it! Remember to specify all of the relevant factors. Then bring your examples and answers to the discussion forums to ask other students whether they get the same answers as you did. Have fun!

Try it! Remember to specify all of the relevant factors. Then bring your examples and answers to the discussion forums to ask other students whether they get the same answers as you did. Have fun!

Imagine that you are going to the drugstore to buy medicine for a friend. Your friend will die if you do not get the medicine on this trip to the drugstore, and nobody else will loan you money for the medicine. You have only $10 with you, and this is exactly what the medicine costs. Outside the drugstore is a young man playing three-card monte, a simple game in which the dealer shows you three cards, turns them over, shifts them briefly from hand to hand, and then lays them out, face down, on the top of a box. You are supposed to identify a particular card; if you do, you are paid even money. You, yourself, are a magician and know the sleight-of-hand trick that fools most people, and you are sure that you can guess the card correctly nine times out of ten. In this situation, what is the expected financial value of a bet of $10?

The answer is $8.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. In the game in this question, the probability is winning is 9/10, the net gain from winning is $10, the probability of losing is 1/10, and the net loss of losing is $10. Thus, the expected financial value of playing this game is [(9/10) x $10] - [(1/10) x $10] = $9 - $1 = $8. This expected financial value is favorable, but see Question 4.

The expected financial value of a bet is the probability of winning times the net gain of winning minus the probability of losing times the net loss of losing. In the game in this question, the probability is winning is 9/10, the net gain from winning is $10, the probability of losing is 1/10, and the net loss of losing is $10. Thus, the expected financial value of playing this game is [(9/10) x $10] - [(1/10) x $10] = $9 - $1 = $8. This expected financial value is favorable, but see Question 4.

Should you play the game in the circumstances in Question 1? (That is, you are going to the drugstore to buy medicine for a friend. Your friend will die if you do not get the medicine on this trip to the drugstore, and nobody else will loan you money for the medicine. You have only $10 with you, and this is exactly what the medicine costs. Outside the drugstore is a young man playing three-card monte, a simple game in which the dealer shows you three cards, turns them over, shifts them briefly from hand to hand, and then lays them out, facedown, on the top of a box. You are supposed to identify a particular card; if you do, you are paid even money. You, yourself, are a magician and know the sleight-of-hand trick that fools most people, and you are sure that you can guess the card correctly 9 times out of 10.)

The answer is "No, you should not play the game."

This answer assumes that your friend's life is worth more than $8. Then even a 1/10 chance of him dying without the medicine outweighs the probable gain of $10.

This answer assumes that your friend's life is worth more than $8. Then even a 1/10 chance of him dying without the medicine outweighs the probable gain of $10.

In the game of ignorance, you draw one card from a deck. You do not know how many cards or which kinds of cards are in the deck. It might be a standard deck or it might contain only diamonds or only aces of spades or any other combination of cards. It costs nothing to play. If you bet that the card you draw will be a spade, and it is a spade, then you win $100.

What is the expected financial value of playing this game?

What is the expected financial value of playing this game?

The answer is "There is no way to calculate expected financial value in this game."

Because you do not know how many cards of various kinds are in the deck, there is no way to determine the probability of winning or the probability of losing. Without those probabilities, you cannot apply the rule for expected financial value, which refers to those probabilities.

Because you do not know how many cards of various kinds are in the deck, there is no way to determine the probability of winning or the probability of losing. Without those probabilities, you cannot apply the rule for expected financial value, which refers to those probabilities.

Consider the following game: You flip a coin continuously until you get tails once. If you get no heads (tails on the first flip), then you are paid nothing. If you get one heads (tails on the second flip), then you are paid $2. If you two heads (tails on the third flip), then you are paid $4. If you get three heads (tails on the fourth flip), then you are paid $8. If you get four heads (tails on the fifth flip), then you are paid $16. And so on. The general rule is that, for any number n, if you get n heads before your first tails, then you are paid $2n (that is, 2 to the nth power dollars).

What is the expected monetary value of this game?

What is the expected monetary value of this game?

Infinite.

The expected value of playing this game is the probability of getting tails on the first flip (0.5) times the net gain in that case ($0) plus the probability of getting tails first on the second flip (0.25) times the net gain in that case ($2) plus the probability of getting tails first on the third flip (0.125) times the net gain in that case ($4) plus the probability of getting tails first on the fourth flip (0.0625) times the net gain in that case ($8) and so on = $0 + $0.50 + $0.50 + $0.50 + and so on. This series is infinite, so the expected value of playing this game is infinite.

The expected value of playing this game is the probability of getting tails on the first flip (0.5) times the net gain in that case ($0) plus the probability of getting tails first on the second flip (0.25) times the net gain in that case ($2) plus the probability of getting tails first on the third flip (0.125) times the net gain in that case ($4) plus the probability of getting tails first on the fourth flip (0.0625) times the net gain in that case ($8) and so on = $0 + $0.50 + $0.50 + $0.50 + and so on. This series is infinite, so the expected value of playing this game is infinite.

How much should you pay to play the game in the previous exercise?

Here is the game again:

You flip a coin continuously until you get tails once. If you get no heads (tails on the first flip), then you are paid nothing. If you get one heads (tails on the second flip), then you are paid $2. If you two heads (tails on the third flip), then you are paid $4. If you get three heads (tails on the fourth flip), then you are paid $8. If you get four heads (tails on the fifth flip), then you are paid $16. And so on. The general rule is that, for any number n, if you get n heads before your first tails, then you are paid $2n (that is, 2 to the nth power dollars).

Here is the game again:

You flip a coin continuously until you get tails once. If you get no heads (tails on the first flip), then you are paid nothing. If you get one heads (tails on the second flip), then you are paid $2. If you two heads (tails on the third flip), then you are paid $4. If you get three heads (tails on the fourth flip), then you are paid $8. If you get four heads (tails on the fifth flip), then you are paid $16. And so on. The general rule is that, for any number n, if you get n heads before your first tails, then you are paid $2n (that is, 2 to the nth power dollars).

You should not pay all of your savings to play this game. That would be crazy, even though your savings are finite, and the expected financial value of this game is infinite. Thus, this paradox shows another reason why you should not always bet when the expected financial value of a bet is positive. Nonetheless, even though we can rule out that option, there is still no consensus about how much it makes sense to pay to play this game (or about how to calculate this amount). So, who knows? That is what makes this a paradox.

All sufficient conditions are causes.

False. Some sufficient conditions are conceptual or moral as opposed to causal. Examples are given at the end of this lecture.

When F is sufficient for G, there are no possible cases of F without G.

False. As explained in the lecture, in order for F to be sufficient for G, there cannot be any cases of F without G in normal circumstances, but there still might be some cases of F without G in circumstances that are not normal. For example, striking a dry match on a rough surface is sufficient to light the match, even if the match still won't light if there is no oxygen, because it is not normal circumstances when there is no oxygen. Remember also that what counts as normal circumstances varies from context to context.

When F is necessary for G, there are no cases of G without F in normal circumstances.

True. The definition in the lecture said that "F is necessary for G" means "in normal circumstances, whenever F is absent, G is absent." To say "whenever F is absent, G is absent" is equivalent to saying "there are no cases of G without F."

Being a car is a sufficient condition for being a vehicle.

True. The class of vehicles includes all cars, trucks, and buses, among other things. Thus, whenever something is a car, it is also a vehicle. That means that being a car is a sufficient condition for being a vehicle.

Being a car is a necessary condition for being a vehicle.

False. The class of vehicles includes trucks and buses (among other things) in addition to cars. Thus, it is not true that, whenever something is not a car, it is also not a vehicle. If it is a truck or a bus, then it is not a car, but it is still a vehicle. That means that being a car is not a necessary condition for being a vehicle.

Being a vehicle is a sufficient condition for being a car

False. The class of vehicles includes trucks and buses (among other things) in addition to cars. Thus, it is not true that, whenever something is a vehicle, it is a car. If it is a truck or a bus, then it is a vehicle, but it is not a car. That means that being a vehicle is not a sufficient condition for being a car.

Being a vehicle is a necessary condition for being a car.

True. The class of vehicles includes all cars, trucks, and buses, among other things. Thus, whenever something is not a vehicle, it is not a car. That means that being a vehicle is a necessary condition for being a car.

Being an integer is a sufficient condition for being an even number.

False. Integers include both even and odd numbers. Thus, it is not true that, whenever a number is an integer, it is an even number. Odd numbers (such as 3) are integers, but they are not even numbers. That means that being an integer is not a sufficient condition for being an even number.

Being an integer is a necessary condition for being an even number.

True. All even numbers (as well as odd numbers) are integers. Thus, whenever a number is not an integer, it is also not an even number. That means that being an integer is a necessary condition for being an even number.

Being an integer is a sufficient condition for being either an even number or an odd number

True. Every integer is either even (exactly divisible by 2) or odd (not exactly divisible by 2). Some numbers, such as 1.5, are neither even nor odd, but they are not integers. Thus, whenever a number is an integer, it is either an even number or an odd number. That means that being an integer is a sufficient condition for being either an even number or an odd number.

Cutting off Joe's head is a sufficient condition for killing him.

True. Whenever something cuts off Joe's head in normal circumstances, that thing causes Joe to die. Causing Joe to die is killing Joe. Of course, Joe will not die if a doctor saves Joe's life, but doctors cannot save Joe after his head is cut off, at least in normal circumstances. Thus, cutting off Joe's head is a sufficient condition for killing him.

Cutting off Joe's head is a necessary condition for killing him.

False. There are many ways to kill Joe (such as drowning him or poisoning him) that do not involve cutting off his head. These other kinds of killing occur in normal circumstances, unfortunately. Thus, it is not true that, whenever something does not cut off Joe's head in normal circumstances, that thing does not kill Joe. Thus, cutting off Joe's head is not a necessary condition for killing him.

X is not a sufficient condition of Y if there is any case in the current data where X is present and Y is absent.

True. This is the negative sufficient condition test that was explained in this lecture.

X is a sufficient condition of Y if there is not any case in the current data where X is present and Y is absent.

False. The fact that there is no case in the current data where X is present and Y is absent shows only that this candidate (X) passes the negative sufficient condition test for these data, so these data do not refute the claim that X is sufficient for Y. However, the negative sufficient condition test is only a negative test, which means that passing this test cannot show that something positively is a sufficient condition. Even when the current data do not include any case of X without Y, there still might be other cases of X without Y outside the current data set. If so, it will not be true that, whenever X is present, Y is present; and then X is not a sufficient condition of Y.

Even if the current data include one case of X without Y, future cases still might show that X is sufficient for Y.

False. In order for X to be sufficient for Y, it must be true that, whenever X is present, Y is present. This generalization is not true if there is one case of X without Y. Adding future cases cannot change the fact that there is still at least this one case of X without Y, and that one case will still be enough to refute the claim that whenever X is present, Y is present. This shows that the negative sufficient condition test is indefeasible and deductive.

Imagine that some diners died directly after dinner. (Alliteration!) We know that something they ate caused the deaths. This is all they ate:

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

Based on this data, which diner shows that eggplant is not sufficient for death?

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

Based on this data, which diner shows that eggplant is not sufficient for death?

Davida. Davida ate eggplant (X is present), and she did not die (Y is absent), so Davida refutes the claim that, whenever a diner ate eggplant (X is present), that diner died (Y is present). Thus, Davida shows that eggplant fails the negative sufficient condition test. That shows that eggplant is not sufficient for death.

No other diner had eggplant without dying, because the only other diner who had eggplant was Alice, and she died. Thus, no other diner shows that eggplant is not sufficient for death.

No other diner had eggplant without dying, because the only other diner who had eggplant was Alice, and she died. Thus, no other diner shows that eggplant is not sufficient for death.

Imagine that some diners died directly after dinner. (Alliteration!) We know that something they ate caused the deaths. This is all they ate:

Alice had tomato soup, eggplant, iced tea, and cake — and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

Based on the data above (same as in Question 4), which diner shows that tomato soup is not sufficient for death?

Alice had tomato soup, eggplant, iced tea, and cake — and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

Based on the data above (same as in Question 4), which diner shows that tomato soup is not sufficient for death?

Ernie had tomato soup (X is present), and he did not die (Y is absent), so Ernie refutes the claim that, whenever a diner had tomato soup (X is present), that diner died (Y is present). Thus, Ernie shows that tomato soup fails the negative sufficient condition test. That shows that tomato soup is not sufficient for death.

No other diner had tomato soup without dying, because the only other diners who had eggplant were Alice and Carol, and they both died. Thus, no other diner shows that tomato soup is not sufficient for death.

No other diner had tomato soup without dying, because the only other diners who had eggplant were Alice and Carol, and they both died. Thus, no other diner shows that tomato soup is not sufficient for death.

Imagine that some diners died directly after dinner. (Alliteration!) We know that something they ate caused the deaths. This is all they ate:

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

Based on the data above (same as in Questions 4 and 5), which diner shows that cake is not sufficient for death?

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

Based on the data above (same as in Questions 4 and 5), which diner shows that cake is not sufficient for death?

None of these diners. Alice and Carol were the only diners who had cake, and they both died. Thus, there is no case in this data set where a diner had cake (X is present), and that diner did not die (Y is absent). Thus, none of these diners shows that cake fails the negative sufficient condition test, so none shows that cake is not sufficient for death.

Imagine that some diners died directly after dinner. We know that something they ate caused the deaths. This is all they ate:

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

Based on the data above (same as in Questions 4-6), which diner shows that water is not sufficient for death?

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

Based on the data above (same as in Questions 4-6), which diner shows that water is not sufficient for death?

Branden. Branden had water (X is present), and he did not die (Y is absent), so Branden refutes the claim that, whenever a diner had water (X is present), that diner died (Y is present). Thus, Branden shows that water fails the negative sufficient condition test. That shows that water is not sufficient for death.

No other diner had water without dying, because the only other diner who had water was Carol, and she died. Thus, no other diner shows that water is not sufficient for death.

No other diner had water without dying, because the only other diner who had water was Carol, and she died. Thus, no other diner shows that water is not sufficient for death.

Imagine that some diners died directly after dinner. We know that something they ate caused the deaths. This is all they ate:

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

Based on the data above (same as in Questions 4-7), which of the following does the case of Branden rule out as sufficient for death?

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

Based on the data above (same as in Questions 4-7), which of the following does the case of Branden rule out as sufficient for death?

All four of the things Branden had.

Branden did not die, so nothing that he had is sufficient for death in these circumstances. His case refutes the claims that whenever a diner had (a) pea soup or (b) chicken or (c) water or (d) ice cream (that is, whenever X is present for each of these values of X), that diner died (Y is present). Thus, Branden shows that all of these items fail the negative sufficient condition test, so none of them is sufficient for death.

Branden did not die, so nothing that he had is sufficient for death in these circumstances. His case refutes the claims that whenever a diner had (a) pea soup or (b) chicken or (c) water or (d) ice cream (that is, whenever X is present for each of these values of X), that diner died (Y is present). Thus, Branden shows that all of these items fail the negative sufficient condition test, so none of them is sufficient for death.

Imagine that some diners died directly after dinner. We know that something they ate caused the deaths. This is all they ate:

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

Based on the data above (same as in Questions 4-8) which of the following does the case of Carol rule out as sufficient for death?

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

Based on the data above (same as in Questions 4-8) which of the following does the case of Carol rule out as sufficient for death?

Nothing. Carol died, so she cannot be a case where a diner had something (X is present), and that diner did not die (Y is absent). Thus, Carol cannot show that any item fails the negative sufficient condition test, so she cannot show that anything is not sufficient for death. This applies to what Carol did not have as well as to what she did have. Remember that it is only cases where the target feature (Y; here = death) is absent that can show that anything is not sufficient for death.

Imagine that some diners died directly after dinner. We know that something they ate caused the deaths. This is all they ate:

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

Based on the data above (same as in Questions 4-9), which individual diner shows that something is sufficient for death?

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

Based on the data above (same as in Questions 4-9), which individual diner shows that something is sufficient for death?

None of these diners. A single case can show that a candidate is not sufficient for death, because that candidate fails the negative sufficient condition test in this case. However, no single case can show that any candidate positively is sufficient for death, because the positive sufficient condition test requires comparisons among several cases.

You can construct your own examples for practicing simply by changing the outcomes (died vs. did not die) and the foods (soup, dessert, and so on) for the diners in the data in Question 4.

True. Just try it! Then bring your examples and answers to the forums to ask other students whether they get the same answers as you did. Have fun!

A set of data can sometimes give us good reason to believe that X is a sufficient condition of Y even if that set does not include any cases where Y is absent.

False. If the data set does not include any cases where Y is absent, then it cannot include any cases where X is present but Y is absent, regardless of what X is. So every candidate (X) will pass the negative sufficient condition test for this data set. Because every candidate passes, the test becomes trivial in these circumstances, so a data set where Y is never absent cannot ever give us good reason to believe that X is a sufficient condition of Y.

We have good reason to believe that X is a sufficient condition of Y if (a) we have not found any case where X is present and Y is absent, (b) we have tested a wide variety of cases, including cases where X is present and cases where Y is absent, and (c) if there is any other feature that is never present where Y is absent, then we have tested cases where that other feature is absent but X is present as well as cases where that other feature is present but X is absent.

False. This is only the first three clauses of the positive sufficient condition test. In order to have good reason to believe that X is a sufficient condition of Y, we also need a fourth clause: (d) we have tested enough cases of various kinds that are likely to include a case where X is present and Y is absent if there is any such case.

The positive sufficient condition test is inductive.

True. The positive sufficient condition test is inductive rather than deductive, because it does not pretend to be valid, and it is defeasible by future data. For example, we saw in Question 6 that none of the data in Question 4 rules out cake as a sufficient condition of death. Nonetheless, in the future we might discover that another diner, Francis, ate the cake without dying, and then that case would rule out cake as a sufficient condition of death.

X is not a necessary condition of Y if there is any case in the current data where X is absent and Y is present.

True. This is the negative necessary condition test that was explained in this lecture.

X is a necessary condition of Y if there is not any case in the current data where X is absent and Y is present.

False. The fact that there is no case in the current data where X is absent and Y is present shows only that this candidate (X) passes the negative necessary condition test for this data, so these data do not refute the claim that X is necessary for Y. However, the negative necessary condition test is only a negative test, which means that passing this test cannot show that something positively is a necessary condition. Even when the current data do not include any case of Y without X, there still might be other cases of Y without X outside the current data set. If so, it will not be true that, whenever X is absent, Y is absent; and then X is not a necessary condition of Y.

Even if the current data include one case of Y without X, future cases still might show that X is necessary for Y.

False. In order for X to be necessary for Y, it must be true that whenever X is absent, Y is absent. This generalization is not true if there is one case of Y without X. Adding future cases cannot change the fact that there is still at least this one case of Y without X, and this one case will still be enough to refute the claim that whenever X is absent, Y is absent. This shows that the negative necessary condition test is indefeasible and deductive.

Imagine that some diners died directly after dinner. We know that something they ate caused the deaths. This is all they ate:

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

(NOTE: This is the same data as in Question 4 of the exercises for Lecture 8-2.)

Based on this data, which diner shows that iced tea is not necessary for death?

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

(NOTE: This is the same data as in Question 4 of the exercises for Lecture 8-2.)

Based on this data, which diner shows that iced tea is not necessary for death?

Carol. Carol drank water rather than iced tea (X is absent), and she died (Y is present), so Carol refutes the claim that whenever a diner did not drink iced tea (X is absent), that diner did not die (Y is absent). Thus, Carol shows that iced tea fails the negative necessary condition test. That shows that iced tea is not necessary for death.

No other diner died without having iced tea, because the only other diner who died was Alice, and she did have iced tea. Thus, no other diner shows that iced tea is not necessary for death.

Many students get confused about necessary conditions, because there are so many negations. You might have thought that someone who did have iced tea (such as Alice, Davida, or Ernie) shows that iced tea is not necessary for death. However, when the question is about necessary conditions instead sufficient conditions, then the crucial test cases—the ones that might show that a candidate is not a necessary condition—are the cases without the candidate. In our example, the diners who did not have iced tea are the only diners who can be cases of death without iced tea, so they are the only diners who can show that iced tea is not necessary for death.

No other diner died without having iced tea, because the only other diner who died was Alice, and she did have iced tea. Thus, no other diner shows that iced tea is not necessary for death.

Many students get confused about necessary conditions, because there are so many negations. You might have thought that someone who did have iced tea (such as Alice, Davida, or Ernie) shows that iced tea is not necessary for death. However, when the question is about necessary conditions instead sufficient conditions, then the crucial test cases—the ones that might show that a candidate is not a necessary condition—are the cases without the candidate. In our example, the diners who did not have iced tea are the only diners who can be cases of death without iced tea, so they are the only diners who can show that iced tea is not necessary for death.

Imagine that some diners died directly after dinner. We know that something they ate caused the deaths. This is all they ate:

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

(NOTE: This is the same data as in Question 4.)

Based on this data, which diner shows that chicken is not necessary for death?

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

(NOTE: This is the same data as in Question 4.)

Based on this data, which diner shows that chicken is not necessary for death?

Alice. Alice had eggplant rather than chicken (X is absent), and she died (Y is present), so Alice refutes the claim that whenever a diner did not have chicken (X is absent), that diner did not die (Y is absent). Thus, Alice shows that chicken fails the negative necessary condition test. That shows that chicken is not necessary for death.

No other diner died without having chicken, because the only other diner who had died was Carol, and she did have chicken. Thus, no other diner shows that chicken is not necessary for death.

Many students get confused about necessary conditions, because there are so many negations. You might have thought that someone who had chicken (such as Branden or Carol) shows that chicken is not necessary for death. However, when the question is about necessary conditions instead sufficient conditions, then the crucial test cases—the ones that might show that a candidate is not a necessary condition—are the cases without the candidate. In our example, the diners who did not have chicken are the only diners who can be cases of death without chicken, so they are the only diners who can show that chicken is not necessary for death.

No other diner died without having chicken, because the only other diner who had died was Carol, and she did have chicken. Thus, no other diner shows that chicken is not necessary for death.

Many students get confused about necessary conditions, because there are so many negations. You might have thought that someone who had chicken (such as Branden or Carol) shows that chicken is not necessary for death. However, when the question is about necessary conditions instead sufficient conditions, then the crucial test cases—the ones that might show that a candidate is not a necessary condition—are the cases without the candidate. In our example, the diners who did not have chicken are the only diners who can be cases of death without chicken, so they are the only diners who can show that chicken is not necessary for death.

Imagine that some diners died directly after dinner. We know that something they ate caused the deaths. This is all they ate:

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

(NOTE: This is the same data as in Question 4.)

Based on this data, which diner shows that tomato soup is not necessary for death?

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

(NOTE: This is the same data as in Question 4.)

Based on this data, which diner shows that tomato soup is not necessary for death?

None of these diners. Alice and Carol were the only diners who died, and they both had tomato soup. Thus, there is no case in this data set where a diner did not have tomato soup (X is absent), and that diner died (Y is present). Thus, none of these diners shows that tomato soup fails the negative necessary condition test, so none shows that tomato soup is not necessary for death.

Imagine that some diners died directly after dinner. We know that something they ate caused the deaths. This is all they ate:

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

(NOTE: This is the same data as in Question 4.)

Based on this data, which of the following does the case of Carol rule out as necessary for death?

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

(NOTE: This is the same data as in Question 4.)

Based on this data, which of the following does the case of Carol rule out as necessary for death?

All of the above. Carol died, so everything that Carol did not have is not necessary for death in these circumstances. Carol's case refutes the claims that whenever a diner did not have (a) pea soup, (b) eggplant, (c) fish, (d) iced tea, (e) ice cream, or (f) pie (that is, whenever X is absent for each of these values of X), that diner did not die (Y is absent). Carol did not have these items, but she did die, so Carol shows that all of these items fail the negative necessary condition test, so none of them is necessary for death.

Imagine that some diners died directly after dinner. We know that something they ate caused the deaths. This is all they ate:

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

(NOTE: This is the same data as in Question 4.)

Based on this data, which of the following does the case of Ernie rule out as necessary for death?

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

(NOTE: This is the same data as in Question 4.)

Based on this data, which of the following does the case of Ernie rule out as necessary for death?

Nothing. Ernie did not die, so he cannot be a case where a diner did not have something (X is absent) and that diner died (Y is present). Thus, Ernie cannot show that any item fails the negative necessary condition test, so he cannot show that anything is not necessary for death. This applies to what Ernie had as well as what he did not have. Remember that it is only cases where the target feature (Y; here = death) is present that can show that anything is not necessary for death.

Imagine that some diners died directly after dinner. We know that something they ate caused the deaths. This is all they ate:

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

(NOTE: This is the same data as in Question 4.)

Based on this data, which individual diner shows that something is necessary for death?

Alice had tomato soup, eggplant, iced tea, and cake—and Alice died.

Branden ate pea soup, chicken, water, and ice cream—and Branden did not die.

Carol had tomato soup, chicken, water, and cake—and Carol died.

Davida ate pea soup, eggplant, iced tea, and ice cream—and Davida did not die.

Ernie had tomato soup, fish, iced tea, and pie—and Ernie did not die.

(NOTE: This is the same data as in Question 4.)

Based on this data, which individual diner shows that something is necessary for death?

None of these diners. A single case can show that a candidate is not necessary for death, because that candidate fails the negative necessary condition test in that case. However, no single case can show that any candidate positively IS sufficient for death, because the positive sufficient condition test requires comparisons among several cases.

There is no simple way to construct your own examples in order to practice by changing the outcomes (died vs. did not die) and the foods (soup, dessert, and so on) for the diners in the data in Question 4.

False. Where there's a will, there's a way. Just try it! Then bring your examples and answers to the forums to ask other students whether they get the same answers as you did. Have fun!

We have good reason to believe that X is a necessary condition of Y if (a) we have not found any case where X is absent and Y is present, (b) we have tested a wide variety of cases, including cases where X is absent and cases where Y is present, and (c) if there are any other features that are never absent where Y is present, then we have tested cases where those other features are present but X is absent as well as cases where those other features are absent but X is present.

False. This is only the first three clauses of the positive necessary condition test. In order to have good reason to believe that X is a sufficient condition of Y, we also need a fourth clause: (d) we have tested enough cases of various kinds that are likely to include a case where X is absent and Y is present if there is any such case.

The positive necessary condition test is deductive.

False. This positive necessary condition test does not pretend to be valid and is defeasible by future data, so it is inductive rather than deductive.

A conjunctive condition—W and X—is not sufficient for Y if there is any case where W and X are both present but Y is absent.

True. W and X is sufficient for Y only if whenever W and X is present, Y is present. Thus, if there is any case where W and X are both present but Y is absent, then W and X is not sufficient for Y. This simply applies the negative sufficient condition test to the conjunctive condition by substituting W and X for X in that test.

A conjunctive condition—W and X—is not sufficient for Y if there is any case where X is present but Y is absent.

False. Even if there are some cases where X is present but Y is absent, if W is absent from all of those cases, then there still might not be any case where W and X are both present but Y is present. If so, it still might be true that whenever W and X are both present, Y is present, and then W and X still might be sufficient for Y.

A disjunctive condition—W or X—is not sufficient for Y if there are any cases where X is present but Y is absent.

True. Any case where X is present is also a case where this disjunction—W or X—is true. Thus, if there is any case where X is present but Y is absent, then that very case is also a case where this disjunction—W or X—is present but Y is absent. That shows that W or X is not sufficient for Y, according to the negative sufficient condition test with the conjunctive condition—W or X—substituted for X in that test.

Imagine that you buy a new computer system with independent components including a new desktop computer (with a CPU and a graphics card), new software, and a new monitor. You want to play games on the new system, but it runs games very slowly. You assume that the keyboard and mouse are not creating the problem, so to figure out what is making the system run so slowly, you experiment with combinations of your old equipment with the new equipment. Here are your experiments and results:

Experiment 1: New computer, new software, and new monitor—and it runs slowly.

Experiment 2: New computer, new software, and old monitor—and it runs fast.

Experiment 3: New computer, old software, and new monitor—and it runs fast.

Experiment 4: New computer, old software, and old monitor—and it runs fast.

Experiment 5: Old computer, new software, and new monitor—and it runs slowly.

Experiment 6: Old computer, new software, and old monitor—and it runs fast.

Experiment 7: Old computer, old software, and new monitor—and it runs slowly.

Experiment 8: Old computer, old software, and old monitor—and it runs fast.

Based on this data, which experiment shows that the new monitor is not sufficient for the system to run slowly?

Experiment 1: New computer, new software, and new monitor—and it runs slowly.

Experiment 2: New computer, new software, and old monitor—and it runs fast.

Experiment 3: New computer, old software, and new monitor—and it runs fast.

Experiment 4: New computer, old software, and old monitor—and it runs fast.

Experiment 5: Old computer, new software, and new monitor—and it runs slowly.

Experiment 6: Old computer, new software, and old monitor—and it runs fast.

Experiment 7: Old computer, old software, and new monitor—and it runs slowly.

Experiment 8: Old computer, old software, and old monitor—and it runs fast.

Based on this data, which experiment shows that the new monitor is not sufficient for the system to run slowly?

Experiment 3 used the new monitor (X is present), and the system did not run slowly (Y is absent), so Experiment 3 refutes the claim that, whenever a system includes the new monitor (X is present), that system runs slowly (Y is present). Thus, Experiment 3 shows that the new monitor fails the negative sufficient condition test, so the new monitor is not sufficient for running slowly.

No other experiment had new monitor without running slowly, because the only other experiments with the new monitor were Experiments 1, 5, and 7, and the system ran slowly in all of those experiments. Thus, no other experiment shows that the new monitor is not sufficient for running slowly.

No other experiment had new monitor without running slowly, because the only other experiments with the new monitor were Experiments 1, 5, and 7, and the system ran slowly in all of those experiments. Thus, no other experiment shows that the new monitor is not sufficient for running slowly.

This question uses the same data as Question 4:

Imagine that you buy a new computer system with independent components including a new desktop computer (with a CPU and a graphics card), new software, and a new monitor. You want to play games on the new system, but it runs games very slowly. You assume that the keyboard and mouse are not creating the problem, so to figure out what is making the system run so slowly, you experiment with combinations of your old equipment with the new equipment. Here are your experiments and results:

Experiment 1: New computer, new software, and new monitor—and it runs slowly.

Experiment 2: New computer, new software, and old monitor—and it runs fast.

Experiment 3: New computer, old software, and new monitor—and it runs fast.

Experiment 4: New computer, old software, and old monitor—and it runs fast.

Experiment 5: Old computer, new software, and new monitor—and it runs slowly.

Experiment 6: Old computer, new software, and old monitor—and it runs fast.

Experiment 7: Old computer, old software, and new monitor—and it runs slowly.

Experiment 8: Old computer, old software, and old monitor—and it runs fast.

Based on this data, which experiment shows that the conjunction of the new computer and the new monitor is not sufficient for the system to run slowly?

Imagine that you buy a new computer system with independent components including a new desktop computer (with a CPU and a graphics card), new software, and a new monitor. You want to play games on the new system, but it runs games very slowly. You assume that the keyboard and mouse are not creating the problem, so to figure out what is making the system run so slowly, you experiment with combinations of your old equipment with the new equipment. Here are your experiments and results:

Experiment 1: New computer, new software, and new monitor—and it runs slowly.

Experiment 2: New computer, new software, and old monitor—and it runs fast.

Experiment 3: New computer, old software, and new monitor—and it runs fast.

Experiment 4: New computer, old software, and old monitor—and it runs fast.

Experiment 5: Old computer, new software, and new monitor—and it runs slowly.

Experiment 6: Old computer, new software, and old monitor—and it runs fast.

Experiment 7: Old computer, old software, and new monitor—and it runs slowly.

Experiment 8: Old computer, old software, and old monitor—and it runs fast.

Based on this data, which experiment shows that the conjunction of the new computer and the new monitor is not sufficient for the system to run slowly?

Experiment 3 used both the new computer and also the new monitor (X is present), and the system did not run slowly (Y is absent), so Experiment 3 refutes the claim that, whenever a system includes both the new computer and the new monitor (X is present), that system runs slowly (Y is present). Thus, Experiment 3 shows that the conjunction of the new computer and the new monitor fails the negative sufficient condition test, so this conjunction is not sufficient for running slowly.

No other experiment had the conjunction of the new computer and the new monitor without running slowly, because the only other experiment with the conjunction of the new computer and the new monitor was Experiment 1, and the system ran slowly in Experiment 1. Thus, no other experiment shows that the conjunction of the new computer and the new monitor is not sufficient for running slowly.

No other experiment had the conjunction of the new computer and the new monitor without running slowly, because the only other experiment with the conjunction of the new computer and the new monitor was Experiment 1, and the system ran slowly in Experiment 1. Thus, no other experiment shows that the conjunction of the new computer and the new monitor is not sufficient for running slowly.

This question uses the same data as Question 4:

Imagine that you buy a new computer system with independent components including a new desktop computer (with a CPU and a graphics card), new software, and a new monitor. You want to play games on the new system, but it runs games very slowly. You assume that the keyboard and mouse are not creating the problem, so to figure out what is making the system run so slowly, you experiment with combinations of your old equipment with the new equipment. Here are your experiments and results:

Experiment 1: New computer, new software, and new monitor—and it runs slowly.

Experiment 2: New computer, new software, and old monitor—and it runs fast.

Experiment 3: New computer, old software, and new monitor—and it runs fast.

Experiment 4: New computer, old software, and old monitor—and it runs fast.

Experiment 5: Old computer, new software, and new monitor—and it runs slowly.

Experiment 6: Old computer, new software, and old monitor—and it runs fast.

Experiment 7: Old computer, old software, and new monitor—and it runs slowly.

Experiment 8: Old computer, old software, and old monitor—and it runs fast.

Based on this data, which experiment shows that the conjunction of the new computer and the new software is not sufficient for the system to run slowly?

Imagine that you buy a new computer system with independent components including a new desktop computer (with a CPU and a graphics card), new software, and a new monitor. You want to play games on the new system, but it runs games very slowly. You assume that the keyboard and mouse are not creating the problem, so to figure out what is making the system run so slowly, you experiment with combinations of your old equipment with the new equipment. Here are your experiments and results:

Experiment 1: New computer, new software, and new monitor—and it runs slowly.

Experiment 2: New computer, new software, and old monitor—and it runs fast.

Experiment 3: New computer, old software, and new monitor—and it runs fast.

Experiment 4: New computer, old software, and old monitor—and it runs fast.

Experiment 5: Old computer, new software, and new monitor—and it runs slowly.

Experiment 6: Old computer, new software, and old monitor—and it runs fast.

Experiment 7: Old computer, old software, and new monitor—and it runs slowly.

Experiment 8: Old computer, old software, and old monitor—and it runs fast.

Based on this data, which experiment shows that the conjunction of the new computer and the new software is not sufficient for the system to run slowly?

Experiment 2 used both the new computer and also the new software (X is present), and the system did not run slowly (Y is absent), so Experiment 2 refutes the claim that, whenever a system includes both the new computer and the new software (X is present), that system runs slowly (Y is present). Thus, Experiment 2 shows that the conjunction of the new computer and the new software fails the negative sufficient condition test, so this conjunction is not sufficient for running slowly.

No other experiment had the conjunction of the new computer and the new software without running slowly, because the only other experiment with the conjunction of the new computer and the new software was Experiment 1, and the system ran slowly in Experiment 1. Thus, no other Experiment shows that the conjunction of the new computer and the new software is not sufficient for running slowly.

No other experiment had the conjunction of the new computer and the new software without running slowly, because the only other experiment with the conjunction of the new computer and the new software was Experiment 1, and the system ran slowly in Experiment 1. Thus, no other Experiment shows that the conjunction of the new computer and the new software is not sufficient for running slowly.

This question uses the same data as Question 4:

Imagine that you buy a new computer system with independent components including a new desktop computer (with a CPU and a graphics card), new software, and a new monitor. You want to play games on the new system, but it runs games very slowly. You assume that the keyboard and mouse are not creating the problem, so to figure out what is making the system run so slowly, you experiment with combinations of your old equipment with the new equipment. Here are your experiments and results:

Experiment 1: New computer, new software, and new monitor—and it runs slowly.

Experiment 2: New computer, new software, and old monitor—and it runs fast.

Experiment 3: New computer, old software, and new monitor—and it runs fast.

Experiment 4: New computer, old software, and old monitor—and it runs fast.

Experiment 5: Old computer, new software, and new monitor—and it runs slowly.

Experiment 6: Old computer, new software, and old monitor—and it runs fast.

Experiment 7: Old computer, old software, and new monitor—and it runs slowly.

Experiment 8: Old computer, old software, and old monitor—and it runs fast.

Based on this data, which experiment shows that the conjunction of the new software and the new monitor is not sufficient for the system to run slowly?

Imagine that you buy a new computer system with independent components including a new desktop computer (with a CPU and a graphics card), new software, and a new monitor. You want to play games on the new system, but it runs games very slowly. You assume that the keyboard and mouse are not creating the problem, so to figure out what is making the system run so slowly, you experiment with combinations of your old equipment with the new equipment. Here are your experiments and results:

Experiment 1: New computer, new software, and new monitor—and it runs slowly.

Experiment 2: New computer, new software, and old monitor—and it runs fast.

Experiment 3: New computer, old software, and new monitor—and it runs fast.

Experiment 4: New computer, old software, and old monitor—and it runs fast.

Experiment 5: Old computer, new software, and new monitor—and it runs slowly.

Experiment 6: Old computer, new software, and old monitor—and it runs fast.

Experiment 7: Old computer, old software, and new monitor—and it runs slowly.

Experiment 8: Old computer, old software, and old monitor—and it runs fast.

Based on this data, which experiment shows that the conjunction of the new software and the new monitor is not sufficient for the system to run slowly?

None of these experiments. In this data set, Tests 1 and 5 are the only experiments with the conjunction of both the new software and also the new monitor (X present). In both of those experiments, the system ran slowly (Y is present). Thus, there is no case in these data with new software and new monitor (X) present but slow running (Y) absent, so none of these experiments shows that the conjunction of the new software and the new monitor is not sufficient for the system to run slowly.

This question uses the same data as Question 4:

Imagine that you buy a new computer system with independent components including a new desktop computer (with a CPU and a graphics card), new software, and a new monitor. You want to play games on the new system, but it runs games very slowly. You assume that the keyboard and mouse are not creating the problem, so to figure out what is making the system run so slowly, you experiment with combinations of your old equipment with the new equipment. Here are your experiments and results:

Experiment 1: New computer, new software, and new monitor—and it runs slowly.

Experiment 2: New computer, new software, and old monitor—and it runs fast.

Experiment 3: New computer, old software, and new monitor—and it runs fast.

Experiment 4: New computer, old software, and old monitor—and it runs fast.

Experiment 5: Old computer, new software, and new monitor—and it runs slowly.

Experiment 6: Old computer, new software, and old monitor—and it runs fast.

Experiment 7: Old computer, old software, and new monitor—and it runs slowly.

Experiment 8: Old computer, old software, and old monitor—and it runs fast.

Based on this data, which experiment shows that the conjunction of the old computer and the old software is not sufficient for the system to run slowly?

Imagine that you buy a new computer system with independent components including a new desktop computer (with a CPU and a graphics card), new software, and a new monitor. You want to play games on the new system, but it runs games very slowly. You assume that the keyboard and mouse are not creating the problem, so to figure out what is making the system run so slowly, you experiment with combinations of your old equipment with the new equipment. Here are your experiments and results:

Experiment 1: New computer, new software, and new monitor—and it runs slowly.

Experiment 2: New computer, new software, and old monitor—and it runs fast.

Experiment 3: New computer, old software, and new monitor—and it runs fast.

Experiment 4: New computer, old software, and old monitor—and it runs fast.

Experiment 5: Old computer, new software, and new monitor—and it runs slowly.

Experiment 6: Old computer, new software, and old monitor—and it runs fast.

Experiment 7: Old computer, old software, and new monitor—and it runs slowly.

Experiment 8: Old computer, old software, and old monitor—and it runs fast.

Based on this data, which experiment shows that the conjunction of the old computer and the old software is not sufficient for the system to run slowly?

Experiment 8 used both the old computer and also the old software (X is present), and the system did not run slowly (Y is absent), so Experiment 8 refutes the claim that, whenever a system includes both the old computer and the old software (X is present), that system runs slowly (Y is present). Thus, Experiment 8 shows that the conjunction of the old computer and the old software (X) fails the negative sufficient condition test, so this conjunction is not sufficient for running slowly.

No other experiment had the conjunction of the old computer and the old software without running slowly, because the only other experiment with the conjunction of the old computer and the old software was Experiment 7, and the system ran slowly in Experiment 7. Thus, no other experiment shows that the conjunction of the old computer and the old software is not sufficient for running slowly.

No other experiment had the conjunction of the old computer and the old software without running slowly, because the only other experiment with the conjunction of the old computer and the old software was Experiment 7, and the system ran slowly in Experiment 7. Thus, no other experiment shows that the conjunction of the old computer and the old software is not sufficient for running slowly.

This question uses the same data as Question 4:

Imagine that you buy a new computer system with independent components including a new desktop computer (with a CPU and a graphics card), new software, and a new monitor. You want to play games on the new system, but it runs games very slowly. You assume that the keyboard and mouse are not creating the problem, so to figure out what is making the system run so slowly, you experiment with combinations of your old equipment with the new equipment. Here are your experiments and results:

Experiment 1: New computer, new software, and new monitor—and it runs slowly.

Experiment 2: New computer, new software, and old monitor—and it runs fast.

Experiment 3: New computer, old software, and new monitor—and it runs fast.

Experiment 4: New computer, old software, and old monitor—and it runs fast.

Experiment 5: Old computer, new software, and new monitor—and it runs slowly.

Experiment 6: Old computer, new software, and old monitor—and it runs fast.

Experiment 7: Old computer, old software, and new monitor—and it runs slowly.

Experiment 8: Old computer, old software, and old monitor—and it runs fast.

Based on this data, which experiment shows that the conjunction of the old computer and the new monitor is not sufficient for the system to run slowly?

Imagine that you buy a new computer system with independent components including a new desktop computer (with a CPU and a graphics card), new software, and a new monitor. You want to play games on the new system, but it runs games very slowly. You assume that the keyboard and mouse are not creating the problem, so to figure out what is making the system run so slowly, you experiment with combinations of your old equipment with the new equipment. Here are your experiments and results:

Experiment 1: New computer, new software, and new monitor—and it runs slowly.

Experiment 2: New computer, new software, and old monitor—and it runs fast.

Experiment 3: New computer, old software, and new monitor—and it runs fast.

Experiment 4: New computer, old software, and old monitor—and it runs fast.

Experiment 5: Old computer, new software, and new monitor—and it runs slowly.

Experiment 6: Old computer, new software, and old monitor—and it runs fast.

Experiment 7: Old computer, old software, and new monitor—and it runs slowly.

Experiment 8: Old computer, old software, and old monitor—and it runs fast.

Based on this data, which experiment shows that the conjunction of the old computer and the new monitor is not sufficient for the system to run slowly?

None of these experiments. In this data set, Experiments 5 and 7 are the only experiments with the conjunction of both the old computer and also the new monitor (X present). In both of those experiments, the system ran slowly (Y is present). Thus, there is no case in these data with old computer and new monitor (X) present but slow running (Y) absent, so none of these experiments shows that the conjunction of the old computer and the new monitor is not sufficient for the system to run slowly.

X and Y are positively correlated if the degree of X increases as the degree of Y decreases and the degree of X decreases as the degree of Y increases.

False. X and Y are negatively rather than positively correlated when this condition is met. Instead, X and Y are positively correlated if the degree of X increases as the degree of Y increases and the degree of X decreases as the degree of Y decreases.

If X and Y are positively correlated, then they are positively correlated in all circumstances.

False. Some correlations hold only in certain circumstances. The example discussed in the lecture was that height and age are positively correlated for people below the age of twenty but negatively correlated for people above the age of sixty.

When A and B are correlated, which of the following is possible?

Each of the other four answers is possible. Correlation is necessary but not sufficient for causation, so there might be no causal relation between A and B even when A and B are correlated. Correlation is symmetrical (if A is correlated with B, then B is correlated with A), so a correlation between A and B is compatible with both the claim that A causes B and also the claim that B causes A. And if a third thing causes both A and B, this can make A correlate with B. Thus, all four of the listed relations are possible when A is correlated with B

Assume that the quality of a musician's performance is positively correlated with how much practicing the musician did for the performance. Which of the following is most likely?

The practice causes the quality of the performance. Musicians practice for a performance before the performance occurs. Causes cannot come after their effects. Thus, the quality of the performance cannot cause the practice. It is possible that some third factor (natural talent) contributes to both the practice and the quality of the performance, since musicians with more talent are often encouraged to practice more. Nonetheless, the practice for a given performance occurs before that performance, so the practice causes at least part of the quality of performance, assuming that the correlation is not just accidental.

The amount of light next to an active fire is positively correlated with the amount of heat in that location. Which of the following is most likely?

A third thing causes both the light and the heat. The fire is a third thing that causes the light as well as the heat. That is why they both occur next to an active fire. Light and heat need not be correlated when they do not have a common cause, such as a fire.

Assume that A and B are correlated. If A changes when you manipulate B, but B does not change when you manipulate A, then which of the following is most likely? Assume normal circumstances and no interfering factors.

B causes A. Given our assumptions, if A causes B, then B would change when you manipulate A. Thus, that alternative is ruled out by the stipulation in the question that B does not change when you manipulate A. Next, if a third thing causes both A and B, or if there were no causal relation between them, then A would not change when you manipulate B. Thus, these alternatives are ruled out by the stipulation in the question that A does change when you manipulate B. The remaining alternative is that B causes A, so that alternative is most likely.

Assume that A and B are correlated. If B changes when you manipulate A, but A does not change when you manipulate B, then which of the following is most likely? Assume normal circumstances and no interfering factors.

A causes B. Given our assumptions, if B causes A, then A would change when you manipulate B. Thus, that alternative is ruled out by the stipulation in the question that A does not change when you manipulate B. Next, if a third thing causes both A and B, or if there were no causal relation between them, then B would not change when you manipulate A. Thus, these alternatives are ruled out by the stipulation in the question that B changes when you manipulate A. The remaining alternative is that A causes B, so that alternative is most likely.

An argument confuses cause with effect when it uses a correlation to argue that A causes B when actually B causes A.

True. This sentence defines the fallacy of confusing cause with effect.

An argument commits the fallacy called "post hoc, ergo propter hoc" when it concludes that A causes B from the premise that A occurred before B

True. The name "post hoc, ergo propter hoc" means "after this, therefore because of this", so this fallacy is committed when someone argues that B occurred after A, so B must have occurred because of A.

The fallacy of confusing cause and effect occurs only in science.

False. This fallacy is committed in sports and everyday life apart from science, as the lecture discusses.

Consider this argument: After I argued that Johnson is the best candidate for President, she said that she supports Johnson, too. So she must have been convinced by my arguments.

Which of the following fallacies does it commit?

Which of the following fallacies does it commit?

This fallacy is Post Hoc Ergo Propter Hoc. She might have supported Johnson before you ever gave your arguments, and she might support Johnson for very different reasons. Thus, it is a fallacy to infer from the premise that she agreed after you argued to the conclusion that she agreed because of your arguments.

Deductive arguments are always valid.

False. Deductive arguments aim at validity, or are supposed to be valid, or are intended to be valid, but that does not ensure that deductive arguments always succeed in being valid. Some arguers try to make their arguments valid, but they fail, and then their arguments are deductive but invalid.

Deductive validity comes in degrees.

False. Deductive validity depends on what is possible or impossible. If it is possible for the premises of an argument to be true when the conclusion is false, then the argument is invalid. If that combination of truth values is impossible, then the argument is valid. This combination cannot be partly possible or a little possible, so an argument cannot be partly valid or a little valid.

Inductive strength comes in degrees.

True.

An inductive argument is stronger when its premises provide more and better reason for its conclusion. The premises can provide a very strong reason or a moderately strong reason or only a weak reason. The argument is strong to the degree that it provides strong reason for its conclusion. More technically, inductive strength is a matter of probability—specifically, how probable the conclusion is, given the premises. Probability does come in degrees. It varies from 0 to 1. This will be explained in two weeks. The point for now is just that, unlike possibility, probability does come in degrees, so inductive strength also comes in degrees.

An inductive argument is stronger when its premises provide more and better reason for its conclusion. The premises can provide a very strong reason or a moderately strong reason or only a weak reason. The argument is strong to the degree that it provides strong reason for its conclusion. More technically, inductive strength is a matter of probability—specifically, how probable the conclusion is, given the premises. Probability does come in degrees. It varies from 0 to 1. This will be explained in two weeks. The point for now is just that, unlike possibility, probability does come in degrees, so inductive strength also comes in degrees.

Deductive validity is defeasible.

False. An argument is valid if it is not possible for the premises of that argument to be true when its conclusion is false. If that combination of truth values really is impossible, then adding more premises cannot make that combination possible. Thus, adding more premises cannot turn a valid argument into an invalid argument. That is what it means to call validity indefeasible. (Notice, in contrast, that invalidity is defeasible. If you add more premises to an invalid argument, then you can turn an invalid argument into a valid argument. One simple way to do this is to add the conclusion as a premise. Then the argument is circular, but it is valid nonetheless.)

Inductive strength is defeasible.

True. To say that inductive strength is defeasible is to say that adding new premises can turn a strong inductive argument into a weak inductive argument. That is possible when the argument is not valid. The classic example is that many people observed thousands of swans throughout six continents and concluded that all swans are white. That was a decently strong inductive argument, because their sample was very large and diverse. But then they learned about black swans in Australia. As soon as they had that new information in their premises, they ceased to have any reason at all to believe that all swans are white.

Inductive arguments always have general conclusions.

False. Here's an inductive argument with a particular conclusion: "Most birds can fly, and this owl is a bird, so probably this owl can fly." Here's another: "The best explanation of the evidence at the crime scene is that Richard killed the victim, so Richard probably did kill the victim." Each of these arguments is invalid and defeasible, so they are inductive, even though they do not have general conclusions.

Deductive arguments always provide more reason for their conclusions than inductive arguments do.

False. Some inductive arguments (such as "The sun has risen every day for thousands of years, so it will probably rise tomorrow") provide very strong reasons for their conclusions. Some deductive arguments (such as "There is life on Mars, so there is life on either Mars or Saturn") provide very little reason for their conclusions, because there is very little reason for their premises. The former arguments provide more reason for their conclusions than the latter. Hence, some inductive arguments provide more reason for their conclusions than some deductive arguments do.

Indicate whether the argument is deductive or inductive. Assume a standard context as described. Nothing tricky!

Context: You and I want to go for a walk, but it is raining, and we do not want to walk in the rain. You ask me when I think it will stop raining. Then I say this sentence:

"The sun is coming out, so the rain will probably stop soon."

Context: You and I want to go for a walk, but it is raining, and we do not want to walk in the rain. You ask me when I think it will stop raining. Then I say this sentence:

"The sun is coming out, so the rain will probably stop soon."

Inductive.

The word "probably" suggests that this argument is not intended as a valid proof of its conclusion. When an argument is valid, it is necessary—not just probable—that the conclusion is true if the premises are true. Thus, if an arguer says only that the conclusion is probably true given the premises, then that person does not intend the argument to be valid. This makes the argument inductive.

The word "probably" suggests that this argument is not intended as a valid proof of its conclusion. When an argument is valid, it is necessary—not just probable—that the conclusion is true if the premises are true. Thus, if an arguer says only that the conclusion is probably true given the premises, then that person does not intend the argument to be valid. This makes the argument inductive.

Indicate whether the following argument is deductive or inductive. Assume a standard context as described. Nothing tricky!

Context: Harold is accused of a burglary, but we know him and thought he was a nice person, so we do not know whether to believe that he is guilty. Then you give this argument.

"If Harold were innocent, then he would not go into hiding. Since he is hiding, he must not be innocent."

Context: Harold is accused of a burglary, but we know him and thought he was a nice person, so we do not know whether to believe that he is guilty. Then you give this argument.

"If Harold were innocent, then he would not go into hiding. Since he is hiding, he must not be innocent."

Deductive. This argument is valid because its conclusion can never be false while both of its premises are true. To see this, just try to tell a coherent story where the premises are all true and the conclusion is false. You can't, so the argument is valid. Almost all arguments that are valid were intended to be valid. Hence, this argument was probably intended to be valid. That makes it deductive.

Indicate whether the following argument is deductive or inductive. Assume a standard context as described. Nothing tricky!

Context: Harold is accused of a burglary. We think that he is not guilty, but we worry that he might be punished anyway. Then I give this argument.

"If Harold is not innocent, then he will be punished. But he is innocent, so he will not be punished."

Context: Harold is accused of a burglary. We think that he is not guilty, but we worry that he might be punished anyway. Then I give this argument.

"If Harold is not innocent, then he will be punished. But he is innocent, so he will not be punished."

Deductive. This argument is not valid, because its conclusion can be false even if both of its premises are true. Some people are punished by mistake when they are innocent. Nonetheless, this argument looks like another argument that is valid (such as this different argument: "If Harold is innocent, then he will not be punished. But he is innocent, so he will not be punished."). This similarity to a valid argument might make the argument in our example appear valid. That appearance suggests that the person who gave the argument in our example intended the argument to be valid. That intention makes the argument deductive even if it fails to be valid.

Indicate whether the following argument is deductive or inductive. Assume a standard context as described. Nothing tricky!

Context: I order a cola drink at midnight. You comment that you never drink cola that late at night, because the caffeine in cola keeps you from sleeping well. I respond with this argument:

"Cola drinks never keep me awake at night. I know because I drank a cola drink just last night without any problems."

Context: I order a cola drink at midnight. You comment that you never drink cola that late at night, because the caffeine in cola keeps you from sleeping well. I respond with this argument:

"Cola drinks never keep me awake at night. I know because I drank a cola drink just last night without any problems."

Inductive. This argument is not valid and does not appear to be valid, so it was probably not intended to be valid. Moreover, it has the form of a standard kind of inductive argument—specifically, a generalization from a sample (to be discussed in later lectures). Here, the sample is a single case. Its form thus suggests that the argument is inductive.

Indicate whether the following argument is deductive or inductive. Assume a standard context as described. Nothing tricky!

Context: Our only son, Jeff, lives away at college. He has final exams before the winter holidays, and we are not sure when he will arrive back home. We go on a short trip to a store. The house was neat when we left, but it is messy when we return. Then you say this sentence:

"The house is a mess, so Jeff must be home from college."

Context: Our only son, Jeff, lives away at college. He has final exams before the winter holidays, and we are not sure when he will arrive back home. We go on a short trip to a store. The house was neat when we left, but it is messy when we return. Then you say this sentence:

"The house is a mess, so Jeff must be home from college."

Inductive. This argument is not valid and does not appear to be valid, so it was probably not intended to be valid. Moreover, it has the form of a standard kind of inductive argument—specifically, an inference to the best explanation (to be discussed in later lectures). Its form thus suggests that the argument is inductive.

Arguments from about a sample to conclusions about the whole class are valid.

False. Because the whole class in the conclusion includes cases that are not in the sample in the premises, it is possible for the sample to have certain features that are not shared by the whole class. Here's an example: "Most students that I have taught at Duke are Americans, so most students in the world are Americans." The premise about the Duke sample is true, but the conclusion about the whole class of students is false, so this argument is invalid.

Arguments from premises about a sample to conclusions about the whole class are defeasible.

True. Cases in the whole class that are not in the original sample can provide new information that defeats the strength of an argument from a sample to a generalization about the whole class. For example, imagine that I check 100 homes in my town and find that almost all have three bedrooms or more, so I conclude that most homes in my town have three bedrooms or more. This argument seems pretty strong. Then I discover other parts of town with smaller houses, as well as several large apartment buildings with hundreds of apartments that have two bedrooms or less. When this new information is added to the premises of the original argument, the argument ceases to be strong. All of my observations or samples together do not support the conclusion that most homes in my town have three bedrooms or more. The new premises thus undermine or defeat the strength of the original argument. That makes it defeasible.

Arguments from premises about a sample to conclusions about the whole class are inductive.

True. Arguments from samples to generalizations are invalid and defeasible, as we saw in the previous two exercises. That makes them inductive.

The conclusion in a generalization from a sample always begins with the word "all."

False. Partial generalizations (described in the lecture) have conclusions that refer to X percent of Fs rather than to all the Fs. Partial generalizations cannot conclude that all Fs are G because their premises refer to some cases where an F is not G, which would refute the conclusion that all Fs are G.

Statistics are irrefutable.

False. Some statistical generalizations from samples are reliable, but others are not. We need to learn to distinguish good from bad generalizations. That is the topic of the next lecture.

Suppose I argue that 90 percent of Fs in my sample are G, so 90 percent of all Fs are G. Then I observe more Fs in a different area and find that 90 percent of Fs in my new sample are also G. This new information makes my argument stronger than it was before.

True. When we combine the old sample with the new sample, we get a larger total sample. Larger samples give us more reason to believe the conclusion as long as the larger sample displays the same patterns that we observed in the smaller sample, assuming that the new sample that was added is independent.

A sample of one is always too small to justify a generalization.

False. If we have background knowledge that the cases in the class are uniform, then it can be strong to generalize from a single case to the whole class. For example, if I throw one copper coin in a lake and it sinks in the water, then I do not need to test more copper coins in order to be justified in believing that all copper coins sink in the water (assuming that they are not supported by anything else, such as a turtle swimming on the surface of the lake), because I know that copper coins are relatively uniform in density (assuming that they are not hollow or shaped like boats!).

Arguments from premises about a sample to general conclusions about the whole class commit the fallacy of hasty generalization when the sample in the premises is biased.

False. The fallacy of hasty generalization is committed when the sample is too small, not when the sample is biased. A biased sample can be very large, and an unbiased sample can still be too small. To generalize from a sample that is too small but is not biased is still to commit the fallacy of hasty generalization.

Polls are always reliable.

False. The lecture gave several examples of polls that are unreliable because the sample was biased or too small or because the poll asked a biased question.

Polls are never reliable.

False. Although polls can go wrong in many ways, some polls do not go wrong in any way. Then they can provide strong reasons to believe their conclusions. For example, some polls do provide strong evidence of who will win in some elections.

When a poll asks a question that is slanted in order to reach a certain result, then the poll does not provide strong reason to believe the desired conclusion.

True. A slanted question can make a poll reach a desired result, but then the poll does not provide strong reason to believe that that desired conclusion is really true, because the slanted poll would have reached that same result even if that conclusion were not true. The problem with slanted questions is that they make the poll reach the desired conclusion regardless of whether that conclusion is really true.

Specify what, if anything, is the main problem with the following generalization from a sample. There might be more than one problem, but indicate the main one.

This philosophy class is about logic, so most philosophy classes are probably about logic.

This philosophy class is about logic, so most philosophy classes are probably about logic.

The sample is too small. This class is only one case of a philosophy class. There is no reason to assume that all philosophy classes are uniform in their subject matter. Therefore, this sample is too small. As we saw in Question 2 of this Exercise, a sample of one is sometimes big enough for a generalization, but only when we can assume uniformity in the relevant respect throughout the sample.

Specify what, if anything, is the main problem with the following generalization from a sample. There might be more than one problem, but indicate the main one.

Most college students like to surf, because I asked a lot of students at several colleges along the California coast, and most of them like to surf.

Most college students like to surf, because I asked a lot of students at several colleges along the California coast, and most of them like to surf.

The sample is biased. Many students choose to go to a college along the California coast because they already like to surf and want to have easy access to surfing while they are in college. Other students have other reasons to go to colleges along the California coast, but then they learn to surf while they are in college, and they come to like it. Students in colleges where they cannot surf are less likely to learn to enjoy surfing. Thus, students who choose to go to colleges along the California coast are more likely to enjoy surfing than are students who go to college in areas where they cannot surf. That explains why this sample is biased and why you cannot legitimately generalize from this sample.

Specify what, if anything, is the main problem with the following generalization from a sample. There might be more than one problem, but indicate the main one.

A poll asked fifty thousand randomly chosen people throughout Asia whether they would want to eat foods that have been genetically modified in ways that increase company profits but also might poison them. Less that 10 percent replied "Yes, definitely." Therefore, most people in Asia do not want to eat genetically modified foods.

A poll asked fifty thousand randomly chosen people throughout Asia whether they would want to eat foods that have been genetically modified in ways that increase company profits but also might poison them. Less that 10 percent replied "Yes, definitely." Therefore, most people in Asia do not want to eat genetically modified foods.

The question is slanted. The question refers only to dangers of genetically modified foods without mentioning any benefits for consumers of genetically modified foods, so it is likely to lead most people not to answer "yes" even if they really do support and want to eat genetically modified foods, and even if some genetically modified foods have health benefits that outweigh any dangers. (Whether or not that is true is controversial, of course.) Notice also that the sample is large and random, so this argument does not have the other problems listed.

Specify what, if anything, is the main problem with the following generalization from a sample. There might be more than one problem, but indicate the main one.

K-Mart asked all of its customers throughout the country whether they prefer K-Mart to Walmart, and 90 percent said they did. Thus, 90 percent of all shoppers in the country prefer K-Mart.

K-Mart asked all of its customers throughout the country whether they prefer K-Mart to Walmart, and 90 percent said they did. Thus, 90 percent of all shoppers in the country prefer K-Mart.

The sample is biased. Customers tend to shop at stores that they like, because they go to stores that they like more often than they go to stores that they do not like. That could explain why a sample taken at a store will usually include more people who like that store, even if there are many more people who do not like that store and who shop elsewhere, so they are not in the sample. This explains why this sample is biased and why you cannot legitimately generalize from this sample.

Specify what, if anything, is the main problem with the following generalization from a sample. There might be more than one problem, but indicate the main one.

Most Swedes are thieves, because my bicycle has been stolen twice, and both times it was a Swede who did it.

Most Swedes are thieves, because my bicycle has been stolen twice, and both times it was a Swede who did it.

The sample is too small. A sample of two is too small to generalize to most Swedes, because there is no reason to assume that Swedes are uniform with respect to whether or not they are thieves. This sample might also be biased because there might be a high percentage of Swedes in the area where the bikes were stolen, and then it is more likely that a bike would be stolen by a Swede than by a non-Swede in that area. That would be the case if the bikes were stolen in Sweden, for example. Thus, there might be more than one problem with a single generalization. However, the main problem is that the sample is too small, because the sample would still be too small even if the two bikes were stolen in different areas and even if these areas did not include an unusual percentage of Swedes. Notice that many social prejudices are based on hasty generalizations like this one.

Specify what, if anything, is the main problem with the following generalization from a sample. There might be more than one problem, but indicate the main one.

I have lots and lots of friends. All of them think that I would make a great comedian. So most people in my country would probably agree that I would make a great comedian.

I have lots and lots of friends. All of them think that I would make a great comedian. So most people in my country would probably agree that I would make a great comedian.

The sample is biased. My friends are more likely than strangers to think that I am funny. That might be part of why they are my friends. However, very few people really make great comedians. As a result, it is likely that most other people would disagree with my friends, even if all of my friends think that I would make a great comedian. That makes the sample biased. The sample might also be too small, because most people do not have enough friends for a reliable poll. However, the argument does say, "I have lots and lots of friends," so the sample does include "lots and lots" of people. Moreover, the premise is about all of my friends, whereas the conclusion is only about most people. Hence, the main problem with this argument is that the sample is biased.

The attribute class occurs in the conclusion when we apply a generalization to a case in an argument of the form: Almost all F are G, and a is F, so a is G.

True. In this form of argument, the attribute class is G, and G does appear in the conclusion.

The reference class occurs in the conclusion when we apply a generalization to a case in an argument of the form: Almost all F are G, and a is F, so a is G.

False. In this form of argument, the reference class is F, and F does not appear in the conclusion.

An argument that applies a generalization to a case is never strong when the percentage in the premise is very low.

False. A very low percentage can give strong evidence for a negative conclusion when the reference class is appropriate. Here's an example: Only 1 percent of F are G, and a is F, so a is probably NOT G. (Admittedly, this question was tricky, but I hope you get the point.)

An argument that applies a generalization to show that a case has a certain property is never strong when the percentage in the premise is 50 percent.

True. If 50 percent of F are G, then this percentage gives us no reason to believe either that a certain a that is F is G or that it is not G. Notice that an application can be strong with 50 percent in the premise if the conclusion is that there is a 50 percent chance that F is G, but then that conclusion does not claim that a case has a certain property, as the question specifies.

Arguments that apply a generalization to a case are intended to be valid.

False. Such applications are inductive, so they are not intended to be valid, and they are usually not valid. Even if 99.999999999999999 percent of F are G, and a is an F, it is still possible that a is not G; so an argument that concludes that a is G is not valid.

Arguments that apply a generalization to a case are defeasible.

True. Additional information can defeat the strength of such applications by showing that the individual (a) falls into a smaller reference class that is more or less likely to be G. The example in the lecture was that you might see that Walter was not wearing shoes while he was teaching.

An argument from the premise that 99 percent of F are G to the conclusion that this F (namely, a) is G is stronger than an argument to the same conclusion from the premise that 90 percent of F are G.

True. Increasing the percentage in the premise makes an application stronger when the conclusion is positive.

Arguments that apply a generalization to a case commit the fallacy of overlooking a conflicting reference class when another smaller reference class that was not mentioned in the argument has a very different (higher or lower) percentage of Gs that would support a conflicting conclusion.

True. The sentence in this question defines what the fallacy of overlooking a conflicting reference class is.

Specify what, if anything, is the main problem with the following application of a generalization to a case. There might be more than one problem, but indicate the main one.

Almost all Prime Ministers of Great Britain have been men.

Margaret Thatcher was Prime Minister of Great Britain.

_______________________________________________

So Margaret Thatcher is a man.

Almost all Prime Ministers of Great Britain have been men.

Margaret Thatcher was Prime Minister of Great Britain.

_______________________________________________

So Margaret Thatcher is a man.

The argument overlooks a conflicting reference class. Margaret Thatcher is also in the group of people named "Margaret." A high percentage of people named "Margaret" are not men. Indeed, a high percentage of Prime Ministers named "Margaret" are not men. So this class—Prime Ministers named "Margaret"— is a conflicting reference class that the argument overlooks.

Specify what, if anything, is the main problem with the following application of a generalization to a case. There might be more than one problem, but indicate the main one.

The weather forecast says that there is only a 40 percent chance of rain, so it won't rain, and we don't need to bring an umbrella.

The weather forecast says that there is only a 40 percent chance of rain, so it won't rain, and we don't need to bring an umbrella.

The percentage is too high. If there were only 10 percent chance of rain, then it might be reasonable not to bring an umbrella, but a 40 percent chance of rain is too high for confidence that it will not rain.

Specify what, if anything, is the main problem with the following application of a generalization to a case. There might be more than one problem, but indicate the main one.

Our heater works most of the time, so we can depend on it to keep us warm during the blizzard that is coming.

Our heater works most of the time, so we can depend on it to keep us warm during the blizzard that is coming.

The percentage is too low. The word "most" means "over 50 percent," but the fact that the heater works over 50 percent of the time is not enough to make it reasonable to depend on it when you might freeze to death in the blizzard if it fails this time.

Specify what, if anything, is the main problem with the following application of a generalization to a case. There might be more than one problem, but indicate the main one.

Very few birds can swim, and this duck is a bird, so this duck cannot swim.

Very few birds can swim, and this duck is a bird, so this duck cannot swim.

The argument overlooks a conflicting reference class. Ducks are also in the group of aquatic birds. A high percentage of aquatic birds can swim. So this class—aquatic birds—is a conflicting reference class that the argument overlooks.

The purpose of an inference to the best explanation is to justify its conclusion.

True. Inferences to the best explanation are based on explanations, but their goal or purpose is not to explain what was observed but, instead, to justify belief in the hypothesis that best explains what was observed. In the example in the lecture, when I observe that water is falling on my head, I infer that there must be a leak in the roof, because that is the best explanation of why I observe what I do—namely, that water is falling on my head. This inference is supposed to justify belief in the conclusion that there is a leak in the roof.

Deductive use inference to the best explanation to solve murder mysteries.

True. After gathering evidence (such as fingerprints, footprints, blood samples, eyewitness reports, and so on), detectives need to determine which hypothesis about who committed the crime best explains why they found the evidence that they found. For example, if the best explanation of why Vijeth's fingerprints were on the glass at the crime scene is that Vijeth was there during the crime, and if there is no explanation of why Vijeth would be there then unless he was there to kill the victim, then that provides some reason to believe that Vijeth killed the victim.

Scientists use inferences to the best explanation to draw conclusions from observations in their experiments.

True. After scientists perform experiments and get results, they need to determine which hypothesis about what happened in the experiment best explains why they got the results that they got. For example, suppose a scientist adds a chemical to a group of live bacteria and then observes that the bacteria are dead an hour later. The scientist infers that the best explanation of why the bacteria died is that the chemical killed those bacteria. That inference provides some reason to believe that this chemical kills that kind of bacteria.

Inferences to the best explanation are intended to be valid.

False. Even if a hypothesis is the best explanation of an observation, it is still possible that the hypothesis is false. For example, even if the hypothesis that Vijeth killed the victim is the best explanation that we can imagine of why his fingerprints were on the glass (see Question 2 in this exercise), it still might be true that he is innocent and his fingerprints were planted on the glass by the person who really killed the victim. Thus, it is possible for the premises to be true and the conclusion false in an inference to the best explanation. That makes such arguments invalid.

Inferences to the best explanation are defeasible.

True. Even if a hypothesis is the best explanation of what has been observed so far, future additional evidence might make the inference weak. For example, even if the hypothesis that this chemical kills that kind of bacteria is the best explanation that we can imagine right now of why the bacteria died (see Question 3 in this exercise), we might later find out that a lab assistant spilled soap into the dish, and the soap rather than the other chemical might be what killed the bacteria. This additional information about the spilled soap then defeats the strength of the inference to the best explanation. Thus, such arguments are defeasible.

Inferences to the best explanation are deductive.

False. Inferences to the best explanation are invalid and defeasible, as we saw in the previous two questions. That makes them inductive rather than deductive.

Inferences to the best explanation include a premise about an observation that needs to be explained.

True. An inference to the best explanation must postulate an explanation of something, so it must include a premise about something that needs to be explained (as discussed in the lecture). For example, if someone argues "This meat smells very bad, so it must be spoiled," then the premise that the meat smells very bad is what needs to be explained.

The conclusion of an inference to the best explanation is supposed to explain an observation in one of the premises.

True. What is inferred as the conclusion in an inference to the best explanation is supposed to best explain the observation that needs to be explained in the premises. For example, if someone argues "This meat smells very bad, so it must be spoiled," then what is inferred is the hypothesis that the meat is spoiled, and that hypothesis is supposed to explain why the meat smells bad, as the premise says.

One premise in an inference to the best explanation claims that one is better than others.

True. If someone claims only that a hypothesis would explain something, then we cannot infer that that explanation is correct, because another conflicting explanation might also explain that event just as well or even better. For example, if the claim that Jamshed committed the murder would explain the eyewitness testimony, but the claim that Jamshed's twin brother committed the murder also would explain the eyewitness testimony, then we cannot tell whether it was Jamshed or his twin brother who committed the murder. That is why an inference to the best explanation must be an inference to the best explanation. In order to claim that an explanation is the best, the inference needs a premise (possibly suppressed) that the explanation in its conclusion is better than any alternative explanation.

If two competing explanations are equally good, then we can use an inference to the best explanation to justify belief in one of them.

False. In order to justify the conclusion of an inference to the best explanation, that conclusion must be better than any competing explanation. If two explanations are equally good, then neither is better than the other. Thus, if two competing explanations are equally good, then we cannot use an inference to the best explanation to justify belief in either of them as opposed to the other. Instead, this comparison cannot tell us which of them is correct.

One explanation is better than the other when it has more of the explanatory virtues.

True. An explanatory virtue is defined as a feature of an explanation that makes it better as an explanation. Thus, explanations with more of these features are better.

Imagine that you offer an explanation of some surprising event, and then someone criticizes your explanation by saying, "Your explanation won't explain anything other than this particular case." Which explanatory virtue is this critic claiming that your explanation lacks?

Power (or Breadth). An explanation is more powerful or broad when it explains more things. Thus, to say "Your explanation won't explain anything other than this particular case" is to say that it does not explain many things, so it lacks power or breadth. In other words, it is narrow or ad hoc. An explanation ought to be widely applicable and not ad hoc.

Imagine that you offer an explanation of some surprising event, and then someone criticizes your explanation by saying, "Your explanation conflicts with everything we know about biology." Which explanatory virtue is this critic claiming that your explanation lacks?

Conservativeness. An explanation is radical rather than conservative when it conflicts with well-established knowledge. Biology includes well-established knowledge. Hence, to say "Your explanation conflicts with everything we know about biology" is to say that it is not conservative. An explanation ought not to conflict with well-established knowledge, so it ought to be conservative in this sense. That makes conservativeness an explanatory virtue.

Imagine that you offer an explanation of some surprising event, and then someone criticizes your explanation by saying, "You don't have to claim so much in order to explain what happened." Which explanatory virtue is this critic claiming that your explanation lacks?

Modesty. An explanation is more modest when it makes fewer or weaker assumptions or claims, and it lacks modesty when it claims more than is needed in order to explain what it is supposed to explain. Thus, to say "You don't have to claim so much in order to explain what happened" is to say that your explanation is not modest. An explanation is better when it is modest in this sense, because there is less to criticize and less that could go wrong when an explanation makes fewer claims or assumptions. That is why modesty is an explanatory virtue. (Notice that modesty is closely related to what is sometimes called simplicity.)

Imagine that you offer an explanation of some surprising event, and then someone criticizes your explanation by saying, "Your explanation just raises new questions that you also need to answer." Which explanatory virtue is this critic claiming that your explanation lacks?

Depth. An explanation is deeper when it leaves fewer questions that need to be answered. Thus, to say "Your explanation just raises new questions that you also need to answer" is to say that it is shallow or lacks depth. An explanation is better when it is deeper, because the goal of explanation is to increase understanding, but we understand something less well when we have more unanswered questions about it.

Imagine that you offer an explanation of some surprising event, and then someone criticizes your explanation by saying, "Your explanation would apply to whatever happened." Which explanatory virtue is this critic claiming that your explanation lacks?

Falsifiability. An explanation is falsifiable when some event could show that it is false, so it is unfalsifiable if no event could show that it is false. If the same explanation could apply equally well to any event that occurred, then no event could show that it is false. Thus, to say "Your explanation would apply to whatever happened" is to say that it is compatible with whatever might happen, so it is unfalsifiable. This feature might seem desirable, but actually it is a defect, because an explanation cannot really explain an event if it is compatible with that event not happening. It is applies whether or not the event occurs, then it cannot explain why the event occurs rather than not occurring. Thus, explanations that are unfalsifiable are empty and have no explanatory force. That is why falsifiability is an explanatory virtue.

Indicate which explanatory virtue is the main one that is lacking from this explanation:

I fished here all day, but I did not catch any fish, because there are no fish anywhere in this river.

I fished here all day, but I did not catch any fish, because there are no fish anywhere in this river.

Modesty. Since I did not fish everywhere in the river, I do not need to claim that there are no fish anywhere in the river in order to explain why I did not catch any. It would be enough to claim that there were no fish in the particular part of the river where I was fishing.

Indicate which explanatory virtue is the main one that is lacking from this explanation:

My car won't start today because it is mad at me for driving it so much.

My car won't start today because it is mad at me for driving it so much.

Conservativeness. The claim that cars get mad conflicts with well-established knowledge. Only living things get mad, but cars are not alive. This explanation is also not deep if it raises the further question of why a car would get mad at its driver for driving it so much. However, that question never gets raised because cars cannot get mad, so the main problem is the lack of conservativeness.

Indicate which explanatory virtue is the main one that is lacking from this explanation:

Her flowers won't grow because something is wrong with them.

Her flowers won't grow because something is wrong with them.

Depth. To say that something is wrong with her flowers raises the question of what is wrong with them, so this explanation is shallow rather than deep. You might think that this explanation also lacks falsifiability, because there is always something wrong with flowers that do not grow. However, there might be nothing wrong with flowers that do grow, and what is supposed to be explained here is why they do not grow instead of growing.

Indicate which explanatory virtue is lacking from this explanation:

I lost the race because I just did not happen to run well today.

I lost the race because I just did not happen to run well today.

Power (or Breadth). The explanation that I did not run well this time does not apply to any other times when I run, so it cannot explain anything else. That makes this explanation lack power or breadth. This explanation also lacks depth if it raises the further question of why I did not run well this time, but to say that "I just did not happen to" do something implies that one cannot ask for any deeper explanation, so the main problem here is the lack of power or breadth

In the situation described in the student video in the lecture, which virtue of explanations would be lacking from the hypothesis that Timmy the Cat knocked the cookies off the counter because he was trying to eat one of them?

Modesty. This new hypothesis ascribes a motive to the cat ("...because he was trying to eat one of them"). It is not necessary to postulate any motive, because cats often knock things over by mistake. (Besides, who can tell what is going on in the mind of a cat?) Thus, this hypothesis claims more than is needed, so it is lacks modesty.

In the situation described in the student video in the lecture, which virtue of explanations would be lacking from the hypothesis that ghosts knocked the cookies off the counter?

Conservativeness. The hypothesis of ghosts conflicts with well-established science.

In the situation described in the student video in the lecture, what is wrong with the hypothesis that a burglar knocked over the cookies after breaking in to steal something?

More than one of the above. This hypothesis is not powerful enough to explain what Kevin did observe (the cat hair on the counter) and it also suggests that Kevin would observe some things that he did not observe (since he did not observe any evidence of breaking in or any missing items that were stolen). Thus, the hypothesis is not compatible with the range of data.

Arguments from analogy are used in art but not in science.

False. The lecture gives several examples of arguments from analogy in science.

Every similarity between Earth and Mars provides some reason to believe that since there is life on Earth, there is also life on Mars.

False. Some similarities are irrelevant to life. For example, both Earth and Mars are bigger than Mercury, and both contain iron in their cores, but neither of these similarities is relevant to the question of whether there is life on Mars.

Arguments from analogy are usually intended to be valid.

False. It is possible for the premises to be true when the conclusion is false in an argument from analogy. Here's an example: the snakes in my garden look a lot like the snakes in your garden, and the snakes in my garden are not venomous, so the snakes in your garden are also not venomous. These premises might be true even if the conclusion is false, because the snakes in your garden turn out to be venomous (so be careful!). Because it is so clear that arguments from analogy are not valid, they are almost never intended to be valid.

Arguments from analogy are defeasible.

True. Adding new information to the premises of an argument from analogy can sometimes reduce or totally undermine the strength of the argument. Consider the example from Question 3: The snakes in my garden look a lot like the snakes in your garden, and the snakes in my garden are not venomous, so the snakes in your garden are also not venomous. The strength of this argument from analogy is defeated if we add new information that there is a kind of venomous snake in your area that does not live in my area but looks a lot like some snakes in my area. Similarly, this argument ceases to be strong if we add the premise that someone was bitten by a snake in your garden and died from its venom. This susceptibility to undermining by new information makes arguments from analogy defeasible.

Arguments from analogy are inductive.

True. Arguments from analogy are invalid and defeasible, as we saw in the previous two questions. That makes them inductive rather than deductive.

Arguments from analogy are stronger when the conclusion is weaker.

True. Consider these premises: This coat resembles my last four coats in design, manufacturer, and stitch pattern, and those four coats all fell apart quickly. These premises provide less reason for the strong conclusion (a) "This coat definitely will fall apart quickly" than they provide for the weaker conclusion (b) "This coat probably will fall apart quickly." The same premises provide even more reason for the even weaker conclusion (c) "There is some danger that this coat will fall apart some day." Thus, the argument gets stronger as the conclusion gets weaker (because the conclusion is more guarded and, hence, more likely to be true).

Arguments from analogy are stronger when they cite more relevant analogies between the objects.

True. Consider this argument from analogy: This political candidate resembles those five other political candidates in party affiliation and policy orientation, and they all ran in the same electoral district as this candidate. Those other candidates all lost their elections, so this candidate will also probably lose this election. This argument would be stronger if the premises also cited other similarities, such as charisma, gender, ethnicity, geographical origin, economic circumstances, current political atmosphere, and so on—since these often affect elections. If we do not know which of these factors will become important in this election, then listing more similarities that might be relevant makes it more likely that the list will include the features that do become important. That is why more potentially relevant similarities make the argument from analogy stronger.

Some arguments from analogy can be reconstructed as inferences to the best explanation.

True. This point was made by the example about Neanderthals at the end of the lecture.

When you give an argument from analogy, you need to specify which similarity is the one that is important for the conclusion.

False. Arguments from analogy often list many similarities without specifying which ones are the ones that are important for the conclusion. This feature makes arguments from analogy different from inferences to the best explanation, because inferences to the best explanation do need to specify which features best explain the observation in their premises.

Consider this argument from analogy: I have seen many movies about James Bond (007). I almost always enjoyed them. A new James Bond movie just came out. I have not seen it yet, but I know that it resembles previous James Bond movies in many respects, such as action, style, and ingenious devices. So I will probably enjoy the new James Bond movie as well, if I watch it.

Would this argument from analogy become stronger, weaker, or neither if we added a premise that the new James Bond movie has a new actor playing James Bond?

Would this argument from analogy become stronger, weaker, or neither if we added a premise that the new James Bond movie has a new actor playing James Bond?

Weaker. The actor who plays the main character often affects whether someone enjoys a movie, so this difference between previous Bond movies and the current Bond movie is a relevant difference between these movies. The new actor might be even better, but he also might be worse; and this seems more probable because I liked the previous actors who played Bond. This uncertainty means that the fact that I enjoyed the previous Bond movies with the previous actor playing Bond is less reason to believe that I will like the current Bond movie with the new actor playing Bond.

Would the argument from analogy in Question 10 become stronger, weaker, or neither if we added a premise that the previous James Bond movies had five or fewer words in their titles, but the new James Bond movie has eight words in its title?

Neither stronger nor weaker. The number of words in a movie's title does not affect whether you enjoy it, except in very unusual circumstances that we can ignore here.

Would the argument from analogy in Question 10 become stronger, weaker, or neither if we added a premise that the previous James Bond movies and new James Bond movie are also similar in more respects that I usually like, such as exotic settings, beautiful actors, intrigue, and so on?

Stronger. In general, the more similarities, the stronger the argument from analogy, as long as those similarities are relevant in the right way (see Questions 2 and 7). Of course, if everything is exactly the same as in the previous Bond movies, then I will be disappointed by the lack of originality in the new movie, but a few more relevant similarities like those listed in the question still make the argument stronger.

Consider this argument from analogy: A new drug cures a serious disease in rats. Rats are similar to humans in many respects. Therefore, this new drug will probably cure the same disease in humans.

Would this argument from analogy become stronger, weaker, or neither if we added a premise that the disease affects the liver, and livers in rats and in humans are very similar in structure and function?

Would this argument from analogy become stronger, weaker, or neither if we added a premise that the disease affects the liver, and livers in rats and in humans are very similar in structure and function?

Stronger. In general, the more similarities, the stronger the argument from analogy, as long as those similarities are relevant in the right way (see Questions 2 and 7). This general rule holds especially when the similarities are specific to the feature in the conclusion, such as liver disease in this case. If the drug cured a disease in part of the body where rats and humans are very different, such as the tail, then a successful cure in rats would be less evidence that the drug will cure the disease in humans.

Would the argument from analogy in Question 13 become stronger, weaker, or neither if we added a premise that the drug does not cure this disease in monkeys or pigs?

Weaker. There is not as much diversity among rats as there is among rats, pigs, and monkeys. An argument from analogy is stronger (as a general rule) when the similar objects mentioned in the premises are more diverse, because that diversity shows that the force of the analogy is independent of the respects in which those objects differ. Thus, the argument in Question 13 would be stronger if the drug cured the disease in more species. However, if it fails to cure the disease in these other species, then that suggests that the effect does depend on some difference between rats and pigs and monkeys. If we do not know which difference matters, then we do not know whether humans resemble rats or, instead, resemble pigs and monkeys in the crucial respect that affects the cure. That is why the argument becomes weaker when the drug does not cure this disease in monkeys or pigs.

Would the argument from analogy in Question 13 become stronger, weaker, or neither if we added that most of the cured rats were conceived on weekdays (because that is when lab technicians were around to enable conception), whereas most humans were conceived on weekends (when they had more free time)?

Neither stronger nor weaker. This difference between rats and humans is irrelevant, because the day of the week when an animal was conceived does not affect its biology or whether it can be cured with a certain drug.

When Kel As are Bs, then the following Venn Diagram is correct. Which of the following statements is consistent with "Kel As are Bs"?

B. Some As are Bs. A is not correct. The intersection of the A circle and the B circle contains an X. The statement "no As are Bs" requires the intersection to be shaded. So "no As are Bs" is inconsistent with "Kel As are Bs."

B is correct. Because the intersection of the A circle and the B circle contains an X, this means that some As are Bs.

C is not correct. There is in X in the part of the A circle that is outside of the B circle. The statement "all As are Bs" requires that part of the A circle to be shaded. Because there is an "X" in the non-overlapping part of the A circle, we know that "all As are Bs" is inconsistent with "Kel As are Bs."

D is not correct. It is not true that more than one statement among A-C is consistent with "Kel As are Bs."

E is not correct, since some statement(s) among A-C is(are) consistent with "Kel As are Bs."

B is correct. Because the intersection of the A circle and the B circle contains an X, this means that some As are Bs.

C is not correct. There is in X in the part of the A circle that is outside of the B circle. The statement "all As are Bs" requires that part of the A circle to be shaded. Because there is an "X" in the non-overlapping part of the A circle, we know that "all As are Bs" is inconsistent with "Kel As are Bs."

D is not correct. It is not true that more than one statement among A-C is consistent with "Kel As are Bs."

E is not correct, since some statement(s) among A-C is(are) consistent with "Kel As are Bs."

Which of the following statements is inconsistent with "Kel As are Bs"?

D. Two or more of the above.

A is not correct. The intersection of the A circle and the B circle for "Kel As are Bs" contains an X, which means that some As are Bs. But, although this claim is inconsistent with the claim "no As are Bs," the claim "no As are Bs" is not the only one among statements A-C that is inconsistent with "Kel As are Bs."

B is not correct. Since the intersection of the A circle and the B circle for "Kel As are Bs" contains an X, it is true that "some As are Bs." So the statement "some As are Bs" is consistent with "Kel As are Bs."

C is not correct. The X in the region of the A circle for "Kel As are Bs," which is outside of the B circle, indeed makes the claim "all As are Bs" inconsistent with the claim "Kel As are Bs." However, the statement "all As are Bs" is not the only claim among statements A-C that is inconsistent with "Kel As are Bs."

D is correct. Both "no As are Bs" and "all As are Bs" are inconsistent with the claim "Kel As are Bs." Respectively, this is because the intersection between the A circle and the B circle for "Kel As are Bs" contains an X, and also because there is an X outside of the intersection, in the A circle.

E is not correct. Some statement(s) among A-C is(are) inconsistent with "Kel As are Bs."

A is not correct. The intersection of the A circle and the B circle for "Kel As are Bs" contains an X, which means that some As are Bs. But, although this claim is inconsistent with the claim "no As are Bs," the claim "no As are Bs" is not the only one among statements A-C that is inconsistent with "Kel As are Bs."

B is not correct. Since the intersection of the A circle and the B circle for "Kel As are Bs" contains an X, it is true that "some As are Bs." So the statement "some As are Bs" is consistent with "Kel As are Bs."

C is not correct. The X in the region of the A circle for "Kel As are Bs," which is outside of the B circle, indeed makes the claim "all As are Bs" inconsistent with the claim "Kel As are Bs." However, the statement "all As are Bs" is not the only claim among statements A-C that is inconsistent with "Kel As are Bs."

D is correct. Both "no As are Bs" and "all As are Bs" are inconsistent with the claim "Kel As are Bs." Respectively, this is because the intersection between the A circle and the B circle for "Kel As are Bs" contains an X, and also because there is an X outside of the intersection, in the A circle.

E is not correct. Some statement(s) among A-C is(are) inconsistent with "Kel As are Bs."

Which of the following arguments (using the expression "Kel") is valid?

C. Kel dodos are quotable.

_____________________________

Therefore some dodos are not quotable.

A is not correct. Because the Venn diagram for "Kel dodos are quotable" includes an X inside the circle of dodos but outside the circle of quotable things, it cannot follow that no dodos are not quotable.

B is not correct. The statement "Kel dodos are quotable" does not imply anything about non-dodos.

C is correct. Because the Venn diagram for "Kel dodos are quotable" includes an X outside the circle of quotable things, it follows that some dodos are not quotable.

D is not correct. The statement "Kel dodos are quotable" does not imply anything about non-dodos.

E is not correct. One of the arguments among statements A-D, which uses "Kel" as it was defined in question (5), is valid.

_____________________________

Therefore some dodos are not quotable.

A is not correct. Because the Venn diagram for "Kel dodos are quotable" includes an X inside the circle of dodos but outside the circle of quotable things, it cannot follow that no dodos are not quotable.

B is not correct. The statement "Kel dodos are quotable" does not imply anything about non-dodos.

C is correct. Because the Venn diagram for "Kel dodos are quotable" includes an X outside the circle of quotable things, it follows that some dodos are not quotable.

D is not correct. The statement "Kel dodos are quotable" does not imply anything about non-dodos.

E is not correct. One of the arguments among statements A-D, which uses "Kel" as it was defined in question (5), is valid.

Which of the following arguments (using the expression "Kel") is invalid?

B.

Kel dodos are quotable.

_____________________________

Therefore no dodos are quotable.

A is not correct. Because the Venn diagram for "Kel dodos are quotable" includes an X inside the circle of dodos but outside the circle of quotable things, it follows that some dodos are not quotable. The argument in A is valid, then, rather than invalid.

B is correct. Because the Venn diagram for "Kel dodos are quotable" includes an X outside the circle of quotable things, it cannot follow that no dodos are not quotable.

C is not correct. Because the Venn diagram for "Kel dodos are quotable" includes an X at the intersection of the circle of dodos and the circle of quotable things, it follows that some dodos are quotable. The argument in C is valid, then, rather than invalid.

D is not correct. Because the Venn diagram for "Kel dodos are quotable" includes an X inside the circle of dodos but outside the circle of quotable things, it follows that not all dodos are quotable. The argument in D is valid, then, rather than invalid.

E is not correct. One of the arguments among A-D, which uses "Kel," is valid.

Kel dodos are quotable.

_____________________________

Therefore no dodos are quotable.

A is not correct. Because the Venn diagram for "Kel dodos are quotable" includes an X inside the circle of dodos but outside the circle of quotable things, it follows that some dodos are not quotable. The argument in A is valid, then, rather than invalid.

B is correct. Because the Venn diagram for "Kel dodos are quotable" includes an X outside the circle of quotable things, it cannot follow that no dodos are not quotable.

C is not correct. Because the Venn diagram for "Kel dodos are quotable" includes an X at the intersection of the circle of dodos and the circle of quotable things, it follows that some dodos are quotable. The argument in C is valid, then, rather than invalid.

D is not correct. Because the Venn diagram for "Kel dodos are quotable" includes an X inside the circle of dodos but outside the circle of quotable things, it follows that not all dodos are quotable. The argument in D is valid, then, rather than invalid.

E is not correct. One of the arguments among A-D, which uses "Kel," is valid.

When Kam As are Bs, then the following Venn diagram is correct. Which of the following statements is consistent with "Kam As are Bs"?

D. Two or more of the above

A is not correct. Although the statement "no As are Bs" is consistent with "Kam As are Bs" (since the intersection of the two circles is shaded), it is not the only such statement among A-C.

B is not correct. Because the intersection between the A circle and the B circle is shaded, it cannot be true that some As are Bs.

C is not correct. Although the statement "all As are Bs" is consistent with "Kam As are Bs," (since the diagram for "Kam As are Bs" has shading at the region of the A circle, which is outside the B circle), it is not the only such statement among A-C.

D is correct. Because the Venn diagram for the statement "Kam As are Bs" depicts the whole circle of As as shaded, it follows that there are no As. If there are no As, then it is true that none of the As are Bs. However, because there are no As that are not Bs, it also follows that all As are Bs. So "Kam As are Bs" is consistent with two claims among A-C: the claim "all As are Bs" and the claim "no As are Bs."

E is incorrect. At least one of the statements among A-D is inconsistent with the claim "Kam As are Bs."

A is not correct. Although the statement "no As are Bs" is consistent with "Kam As are Bs" (since the intersection of the two circles is shaded), it is not the only such statement among A-C.

B is not correct. Because the intersection between the A circle and the B circle is shaded, it cannot be true that some As are Bs.

C is not correct. Although the statement "all As are Bs" is consistent with "Kam As are Bs," (since the diagram for "Kam As are Bs" has shading at the region of the A circle, which is outside the B circle), it is not the only such statement among A-C.

D is correct. Because the Venn diagram for the statement "Kam As are Bs" depicts the whole circle of As as shaded, it follows that there are no As. If there are no As, then it is true that none of the As are Bs. However, because there are no As that are not Bs, it also follows that all As are Bs. So "Kam As are Bs" is consistent with two claims among A-C: the claim "all As are Bs" and the claim "no As are Bs."

E is incorrect. At least one of the statements among A-D is inconsistent with the claim "Kam As are Bs."

Which of the following statements is inconsistent with "Kam As are Bs"?

B. Some As are Bs. A is not correct. The statement "no As are Bs" is consistent with "Kam As are Bs," since the intersection of the two circles is shaded.

B is correct. Because the intersection between the A circle and the B circle is shaded, it cannot be true that some As are Bs.

C is not correct. The statement "all As are Bs" is consistent with "Kam As are Bs," since the diagram for "Kam As are Bs" has shading at the region of the A circle, which is outside the B circle.

D is not correct. Because the Venn diagram for the statement "Kam As are Bs" depicts the whole circle of As as shaded, it follows that there are no As. If there are no As, then it is true that none of the As are Bs. However, because there are no As that are not Bs, it also follows that all As are Bs. So "Kam As are Bs" is consistent with two claims among A-C: the claim "all As are Bs," and the claim "no As are Bs." It is only inconsistent with "some As are Bs."

E is incorrect. At least one of the statements among A-D is inconsistent with the claim "Kam As are Bs."

B is correct. Because the intersection between the A circle and the B circle is shaded, it cannot be true that some As are Bs.

C is not correct. The statement "all As are Bs" is consistent with "Kam As are Bs," since the diagram for "Kam As are Bs" has shading at the region of the A circle, which is outside the B circle.

D is not correct. Because the Venn diagram for the statement "Kam As are Bs" depicts the whole circle of As as shaded, it follows that there are no As. If there are no As, then it is true that none of the As are Bs. However, because there are no As that are not Bs, it also follows that all As are Bs. So "Kam As are Bs" is consistent with two claims among A-C: the claim "all As are Bs," and the claim "no As are Bs." It is only inconsistent with "some As are Bs."

E is incorrect. At least one of the statements among A-D is inconsistent with the claim "Kam As are Bs."

Which of the following arguments (using the expression "Kam") is valid?

A.

Kam dodos are quotable.

___________________________

Therefore no dodos are quotable.

A is correct. Because the Venn diagram for "Kam dodos are quotable" shades out the entire circle of dodos, it follows that no dodos are quotable. Since there are no dodos, it follows that no dodos are quotable.

B is not correct. Because the Venn diagram for "Kam dodos are quotable" shades out the entire circle of dodos, it follows that there are no dodos. Since there are no dodos, it cannot be true that some dodos are quotable.

C is not correct. Because the Venn diagram for "Kam dodos are quotable" shades out the entire circle of dodos, it follows that there are no dodos. Since there are no dodos, it cannot be true that some dodos are not quotable.

D is not correct. The statement "Kam dodos are quotable" does not imply anything about non-dodos.

E is not correct. One of the arguments among A-D, which uses "Kam," is valid.

Kam dodos are quotable.

___________________________

Therefore no dodos are quotable.

A is correct. Because the Venn diagram for "Kam dodos are quotable" shades out the entire circle of dodos, it follows that no dodos are quotable. Since there are no dodos, it follows that no dodos are quotable.

B is not correct. Because the Venn diagram for "Kam dodos are quotable" shades out the entire circle of dodos, it follows that there are no dodos. Since there are no dodos, it cannot be true that some dodos are quotable.

C is not correct. Because the Venn diagram for "Kam dodos are quotable" shades out the entire circle of dodos, it follows that there are no dodos. Since there are no dodos, it cannot be true that some dodos are not quotable.

D is not correct. The statement "Kam dodos are quotable" does not imply anything about non-dodos.

E is not correct. One of the arguments among A-D, which uses "Kam," is valid.

Which of the following arguments (using the expression "Kam") is invalid?

D.

Kam dodos are quotable.

_____________________________

Therefore some dodos are quotable.

A is not correct. Because the Venn diagram for "Kam dodos are quotable" shades out the entire circle of dodos, it follows that no dodos are quotable. The argument stated in A is valid, in other words.

B is not correct. Because the Venn diagram for "Kam dodos are quotable" shades out the entire circle of dodos, it follows that there are not any dodos. Since there are not any dodos, it follows that there are not any dodos that are not quotable. The argument stated in B is valid, in other words.

C is not correct. Because the Venn diagram for "Kam dodos are quotable" shades out the entire circle of dodos, it follows that there are not any dodos. Since there are not any dodos, it follows that there are not any quotable dodos, either. Since there are not any dodos that are quotable, moreover, it follows that all the dodos are not quotable. The argument stated in C is valid, in other words.

D is correct. Because the Venn diagram for "Kam dodos are quotable" shades out the entire circle of dodos, it follows that there are not any dodos. Since there are not any dodos, it would be invalid to infer that some of the dodos are quotable.

E is incorrect. At least one of the arguments among A-D, which uses "Kam," is invalid.

Kam dodos are quotable.

_____________________________

Therefore some dodos are quotable.

A is not correct. Because the Venn diagram for "Kam dodos are quotable" shades out the entire circle of dodos, it follows that no dodos are quotable. The argument stated in A is valid, in other words.

B is not correct. Because the Venn diagram for "Kam dodos are quotable" shades out the entire circle of dodos, it follows that there are not any dodos. Since there are not any dodos, it follows that there are not any dodos that are not quotable. The argument stated in B is valid, in other words.

C is not correct. Because the Venn diagram for "Kam dodos are quotable" shades out the entire circle of dodos, it follows that there are not any dodos. Since there are not any dodos, it follows that there are not any quotable dodos, either. Since there are not any dodos that are quotable, moreover, it follows that all the dodos are not quotable. The argument stated in C is valid, in other words.

D is correct. Because the Venn diagram for "Kam dodos are quotable" shades out the entire circle of dodos, it follows that there are not any dodos. Since there are not any dodos, it would be invalid to infer that some of the dodos are quotable.

E is incorrect. At least one of the arguments among A-D, which uses "Kam," is invalid.

Which of the following statements is of the A form?

A. All whales are fish. A is the correct answer. Statements of the form "all F are G" are statements of the A form. In this case, "whales" is the value for F, and "fish" is the value for G.

B is not correct. Statements of the form "no F are G" are statements of the E form, not the A form.

C is not correct. Statements of the form "some F are G" are statements of the I form, not the A form.

D is not correct. Statements of the form "some F are not G" are statements of the O form, not the A form.

E is not correct. One of the statements among A-D is of the A form.

B is not correct. Statements of the form "no F are G" are statements of the E form, not the A form.

C is not correct. Statements of the form "some F are G" are statements of the I form, not the A form.

D is not correct. Statements of the form "some F are not G" are statements of the O form, not the A form.

E is not correct. One of the statements among A-D is of the A form.

Which of the following statements is of the E form?

B. No whales are fish. A is not correct. Statements of the form "all F are G" are statements of the A form, not the E form.

B is correct. Statements of the form "no F are G" are statements of the E form. In this case, "whales" is the value for F, and "fish" is the value for G.

C is not correct. Statements of the form "some F are G" are statements of the I form, not the E form.

D is not correct. Statements of the form "some F are not G" are statements of the O form, not the E form.

E is not correct. One of the statements among A-D is of the A form.

B is correct. Statements of the form "no F are G" are statements of the E form. In this case, "whales" is the value for F, and "fish" is the value for G.

C is not correct. Statements of the form "some F are G" are statements of the I form, not the E form.

D is not correct. Statements of the form "some F are not G" are statements of the O form, not the E form.

E is not correct. One of the statements among A-D is of the A form.

Which of the following statements is of the I form?

C. Some whales are fish. A is not correct. Statements of the form "all F are G" are statements of the A form, not the I form.

B is not correct. Statements of the form "no F are G" are statements of the E form, not the I form.

C is correct. Statements of the form "some F are G" are statements of the I form. In this case, "whales" is the value for F, and "fish" is the value for G.

D is not correct. Statements of the form "some F are not G" are statements of the O form, not the I form.

E is not correct. One of the statements among A-D is of the A form.

B is not correct. Statements of the form "no F are G" are statements of the E form, not the I form.

C is correct. Statements of the form "some F are G" are statements of the I form. In this case, "whales" is the value for F, and "fish" is the value for G.

D is not correct. Statements of the form "some F are not G" are statements of the O form, not the I form.

E is not correct. One of the statements among A-D is of the A form.

Which of the following statements is of the O form?

D. Some whales are not fish. A is not correct. Statements of the form "all F are G" are statements of the A form, not the O form.

B is not correct. Statements of the form "no F are G" are statements of the E form, not the O form.

C is not correct. Statements of the form "some F are G" are statements of the I form, not the O form.

D is correct. Statements of the form "some F are not G" are statements of the O form. In this case, "whales" is the value for F, and "fish" is the value for G.

E is not correct. One of the statements among A-D is of the A form.

B is not correct. Statements of the form "no F are G" are statements of the E form, not the O form.

C is not correct. Statements of the form "some F are G" are statements of the I form, not the O form.

D is correct. Statements of the form "some F are not G" are statements of the O form. In this case, "whales" is the value for F, and "fish" is the value for G.

E is not correct. One of the statements among A-D is of the A form.

Which of the following statements is of the A form?

A. Everyone hates you. A is correct. Statements of the form "all F are G" are statements of the A form. In this case, "hates you" is the value for G, and "everyone," which means "all people" or "all persons," would make "people" or "persons" the value for F.

B is not correct. "Someone hates you" is not a statement of the A form.

C is not correct. "Nobody hates you" is not a statement of the A form.

D is not correct. "Someone doesn't hate you" is not a statement of the A form.

E is not correct. One of the statements among A-D is of the A form.

B is not correct. "Someone hates you" is not a statement of the A form.

C is not correct. "Nobody hates you" is not a statement of the A form.

D is not correct. "Someone doesn't hate you" is not a statement of the A form.

E is not correct. One of the statements among A-D is of the A form.

Which of the following statements is of the E form?

C. Nobody hates you. A is not correct. "Everyone hates you" is not a statement of the E form.

B is not correct. "Someone hates you" is not a statement of the E form.

C is correct. "Nobody hates you" is a statement of the E form. Statements of the form, "no F is G" are statements of the E form. In this case, "hates you" is the value for G, and "nobody," which means "no people" or "no persons," would make "people" or "persons" the value for F.

D is not correct. "Someone doesn't hate you" is not a statement of the E form.

E is not correct. One of the statements among A-D is of the E form.

B is not correct. "Someone hates you" is not a statement of the E form.

C is correct. "Nobody hates you" is a statement of the E form. Statements of the form, "no F is G" are statements of the E form. In this case, "hates you" is the value for G, and "nobody," which means "no people" or "no persons," would make "people" or "persons" the value for F.

D is not correct. "Someone doesn't hate you" is not a statement of the E form.

E is not correct. One of the statements among A-D is of the E form.

Which of the following statements is of the I form?

B. Someone hates you. A is not correct. "Everyone hates you" is not a statement of the I form.

B is correct. "Someone hates you" is a statement of the I form. Statements of the form "some F is G" are statements of the I form. In this case, "hates you" is the value for G, and "someone," which means "some person," would make "person" the value for F.

C is not correct. "Nobody hates you" is not a statement of the I form.

D is not correct. "Someone doesn't hate you" is not a statement of the I form.

E is not correct. One of the statements among A-D is of the I form.

B is correct. "Someone hates you" is a statement of the I form. Statements of the form "some F is G" are statements of the I form. In this case, "hates you" is the value for G, and "someone," which means "some person," would make "person" the value for F.

C is not correct. "Nobody hates you" is not a statement of the I form.

D is not correct. "Someone doesn't hate you" is not a statement of the I form.

E is not correct. One of the statements among A-D is of the I form.

Which of the following statements is of the O form?

D. Someone doesn't hate you. A is not correct. "Everyone hates you" is not a statement of the O form.

B is not correct. "Someone hates you" is a statement of the O form.

C is not correct. "Nobody hates you" is not a statement of the O form.

D is correct. "Someone doesn't hate you" is a statement of the O form. Statements of the form, "some F is not G" are statements of the O form. In this case, "hates you" is the value for G, and "someone," which means "some person," would make "person" the value for F.

E is not correct. One of the statements among A-D is of the O form.

B is not correct. "Someone hates you" is a statement of the O form.

C is not correct. "Nobody hates you" is not a statement of the O form.

D is correct. "Someone doesn't hate you" is a statement of the O form. Statements of the form, "some F is not G" are statements of the O form. In this case, "hates you" is the value for G, and "someone," which means "some person," would make "person" the value for F.

E is not correct. One of the statements among A-D is of the O form.

Which of the following pairs of statements are inconsistent with each other?

C. Some sheep are purple; no sheep are purple. A is not correct. "Some sheep are purple" is not inconsistent with the claim, "some sheep are not purple." If there were some purple sheep, along with some non-purple sheep, then both statements could be true.

B is not correct. "All sheep are purple" is not inconsistent with "no sheep are purple." If there were no sheep, then "all sheep are purple" would be true, since there would be not be any sheep that were not purple; and "no sheep are purple" would also be true, since there would not be any purple sheep.

C is correct. If some sheep are purple, then it cannot be true that no sheep are purple. "Some sheep are purple" and "no sheep are purple" are inconsistent.

D is not correct. Only one of the pairs of sentences among A-D is inconsistent.

E is not correct. One of the pairs of sentences among A-D is inconsistent.

B is not correct. "All sheep are purple" is not inconsistent with "no sheep are purple." If there were no sheep, then "all sheep are purple" would be true, since there would be not be any sheep that were not purple; and "no sheep are purple" would also be true, since there would not be any purple sheep.

C is correct. If some sheep are purple, then it cannot be true that no sheep are purple. "Some sheep are purple" and "no sheep are purple" are inconsistent.

D is not correct. Only one of the pairs of sentences among A-D is inconsistent.

E is not correct. One of the pairs of sentences among A-D is inconsistent.

Which of the following pairs of statements are inconsistent with each other?

A. All otters are quotable; some otter is not quotable. A is correct. "All otters are quotable" is inconsistent with the claim "some otters are not quotable." If all otters are quotable, then there cannot be some otter who is not quotable.

B is not correct. "No otters are quotable" is not inconsistent with the claim "some otter is not quotable." Suppose that there are otters, and that none of them are quotable. In such a case, both statements would be true.

C is not correct. "Some otters are quotable" is not inconsistent with "some otter is not quotable." Suppose there are some quotable and some non-quotable otters. In such a situation, both claims would be true.

D is not correct. "All otters are quotable" is not inconsistent with "no otters are quotable." If there were no otters, then "all otters are quotable" would be true, since there would be not be any otters that are not quotable, and "no otters are quotable" would also be true, since there would not be any quotable otters.

E is not correct. One of the pairs of sentences among A-D is inconsistent.

B is not correct. "No otters are quotable" is not inconsistent with the claim "some otter is not quotable." Suppose that there are otters, and that none of them are quotable. In such a case, both statements would be true.

C is not correct. "Some otters are quotable" is not inconsistent with "some otter is not quotable." Suppose there are some quotable and some non-quotable otters. In such a situation, both claims would be true.

D is not correct. "All otters are quotable" is not inconsistent with "no otters are quotable." If there were no otters, then "all otters are quotable" would be true, since there would be not be any otters that are not quotable, and "no otters are quotable" would also be true, since there would not be any quotable otters.

E is not correct. One of the pairs of sentences among A-D is inconsistent.

Which of the following is a valid Immediate Categorical Inference?

E. None of the above A is not correct since "Horace the sheep is brown" is not of the form A, E, I, or O.

B is not correct since "Chloe is a sheep in our field" is not of the form A, E, I, or O.

C is not correct because the argument is not valid.

D is not correct because the argument is not valid.

Therefore, E is correct

B is not correct since "Chloe is a sheep in our field" is not of the form A, E, I, or O.

C is not correct because the argument is not valid.

D is not correct because the argument is not valid.

Therefore, E is correct

Which of the following is a valid Immediate Categorical Inference?

E. None of the above.

A is not correct. From the claim, "all the quotable otters are authors," it does not follow that anything, such as Trotter, is a quotable otter.

B is not correct since "Trotter the otter is an author" is not of A, E, I, or O form.

C is not correct. From the claim, "all otters are authors," it does not follow that any otters are non-authors.

D is not correct. From the claim, "no quotable otter is an author," it does not follow that any otter is not quotable.

Therefore, E is correct.

A is not correct. From the claim, "all the quotable otters are authors," it does not follow that anything, such as Trotter, is a quotable otter.

B is not correct since "Trotter the otter is an author" is not of A, E, I, or O form.

C is not correct. From the claim, "all otters are authors," it does not follow that any otters are non-authors.

D is not correct. From the claim, "no quotable otter is an author," it does not follow that any otter is not quotable.

Therefore, E is correct.

Which of the following statements is inconsistent with "All politicians are honest"?

D. Some politicians are not honest.

A is not correct. The proposition "all politicians are honest" is consistent with itself.

B is not correct. "No politicians are honest" is not inconsistent with the claim, "all politicians are honest." To see why, suppose that there are no politicians. In such a case, the statement "all politicians are honest" would be true, since there would be no dishonest politicians. The statement, "no politicians are honest" would be true, too, since there would be no honest politicians.

C is not correct. "Some politicians are honest" is not inconsistent with "all politicians are honest."

D is correct. "All politicians are honest" is inconsistent with "some politicians are not honest."

E is not correct. One of the sentences among A-D is inconsistent with "all politicians are honest."

A is not correct. The proposition "all politicians are honest" is consistent with itself.

B is not correct. "No politicians are honest" is not inconsistent with the claim, "all politicians are honest." To see why, suppose that there are no politicians. In such a case, the statement "all politicians are honest" would be true, since there would be no dishonest politicians. The statement, "no politicians are honest" would be true, too, since there would be no honest politicians.

C is not correct. "Some politicians are honest" is not inconsistent with "all politicians are honest."

D is correct. "All politicians are honest" is inconsistent with "some politicians are not honest."

E is not correct. One of the sentences among A-D is inconsistent with "all politicians are honest."

Which of the following statements is inconsistent with "No politicians are honest"?

C. Some politicians are honest.

A is not correct. "No politicians are honest" is not inconsistent with the claim "all politicians are honest." To see why, suppose that there are no politicians. In such a case, the statement "all politicians are honest" would be true, since there would be no dishonest politicians. The statement "no politicians are honest" would be true, too, since there would be no honest politicians.

B is not correct. "No politicians are honest" is consistent with itself.

C is correct. "Some politicians are honest" is inconsistent with "no politicians are honest."

D is not correct. "some politicians are not honest" is not inconsistent with "no politicians are honest." Suppose there are politicians, and that none of them are honest. Then both statements would be true.

E is not correct. One of the sentences among A-D is inconsistent with "all politicians are honest."

A is not correct. "No politicians are honest" is not inconsistent with the claim "all politicians are honest." To see why, suppose that there are no politicians. In such a case, the statement "all politicians are honest" would be true, since there would be no dishonest politicians. The statement "no politicians are honest" would be true, too, since there would be no honest politicians.

B is not correct. "No politicians are honest" is consistent with itself.

C is correct. "Some politicians are honest" is inconsistent with "no politicians are honest."

D is not correct. "some politicians are not honest" is not inconsistent with "no politicians are honest." Suppose there are politicians, and that none of them are honest. Then both statements would be true.

E is not correct. One of the sentences among A-D is inconsistent with "all politicians are honest."

Which of the following statements is inconsistent with "Some politicians are honest"?

B. No politicians are honest.

A is not correct. "Some politicians are honest" is not inconsistent with the claim "all politicians are honest." Suppose that there are some politicians, and all of them are honest. In such a case, both statements would be true.

B is correct. "Some politicians are honest" is inconsistent with "no politicians are honest."

C is not correct. "Some politicians are honest" is consistent with itself.

D is not correct. "some politicians are not honest" is not inconsistent with "some politicians are honest." Suppose there are some honest and non-honest politicians. Then both statements would be true.

E is not correct. One of the sentences among A-D is inconsistent with "some politicians are honest."

A is not correct. "Some politicians are honest" is not inconsistent with the claim "all politicians are honest." Suppose that there are some politicians, and all of them are honest. In such a case, both statements would be true.

B is correct. "Some politicians are honest" is inconsistent with "no politicians are honest."

C is not correct. "Some politicians are honest" is consistent with itself.

D is not correct. "some politicians are not honest" is not inconsistent with "some politicians are honest." Suppose there are some honest and non-honest politicians. Then both statements would be true.

E is not correct. One of the sentences among A-D is inconsistent with "some politicians are honest."

Which of the following statements is inconsistent with "Some politicians are not honest"?

A. All politicians are honest.

A is correct. "Some politicians are not honest" is inconsistent with the claim "all politicians are honest."

B is not correct. "Some politicians are not honest" is not inconsistent with "no politicians are honest." Suppose there are some politicians, and that none of them are honest. Then both statements would be true.

C is not correct. "Some politicians are honest" is consistent with the claim, "some politicians are not honest." Suppose there are some honest and some dishonest politicians. In such a case, both statements would be true.

D is not correct. "some politicians are not honest" is consistent with itself.

E is not correct. One of the sentences among A-D is inconsistent with "some politicians are not honest."

A is correct. "Some politicians are not honest" is inconsistent with the claim "all politicians are honest."

B is not correct. "Some politicians are not honest" is not inconsistent with "no politicians are honest." Suppose there are some politicians, and that none of them are honest. Then both statements would be true.

C is not correct. "Some politicians are honest" is consistent with the claim, "some politicians are not honest." Suppose there are some honest and some dishonest politicians. In such a case, both statements would be true.

D is not correct. "some politicians are not honest" is consistent with itself.

E is not correct. One of the sentences among A-D is inconsistent with "some politicians are not honest."

Which of the following statements is inconsistent with "Some non-bankers are not honest"?

A. All non-bankers are honest.

A is correct. "Some non-bankers are not honest" is inconsistent with the claim "all non-bankers are honest." If all the non-bankers are honest, then it cannot be true that some of them are not honest.

B is not correct. "Some non-bankers are not honest" is not inconsistent with "no bankers are honest." Nothing about bankers follows from the claim about non-bankers.

C is not correct. "Some non-bankers are honest" is consistent with the claim "some non-bankers are not honest." Suppose there are some honest and some dishonest non-bankers. In such a case, both statements would be true.

D is not correct. "some non-bankers are not honest" is consistent with itself.

E is not correct. One of the sentences among A-D is inconsistent with "some non-bankers are not honest."

A is correct. "Some non-bankers are not honest" is inconsistent with the claim "all non-bankers are honest." If all the non-bankers are honest, then it cannot be true that some of them are not honest.

B is not correct. "Some non-bankers are not honest" is not inconsistent with "no bankers are honest." Nothing about bankers follows from the claim about non-bankers.

C is not correct. "Some non-bankers are honest" is consistent with the claim "some non-bankers are not honest." Suppose there are some honest and some dishonest non-bankers. In such a case, both statements would be true.

D is not correct. "some non-bankers are not honest" is consistent with itself.

E is not correct. One of the sentences among A-D is inconsistent with "some non-bankers are not honest."

Which of the following statements is inconsistent with "All non-giraffes are not mammals"?

E. None of the above

A is not correct. "All non-giraffes are not mammals" is consistent with itself.

B is not correct. "All non-giraffes are not mammals" is not inconsistent with "no non-giraffes are not mammals." Suppose that there are no non-giraffes. Then, since there would not be any mammalian non-giraffes, "all non-giraffes are not mammals" would be true. "No non-giraffes are not mammals" would be true, too, since there would not be any non-mammal non-giraffes.

C is not correct. "Some non-giraffes are not mammals" is consistent with the claim, "all non-giraffes are not mammals." Suppose there are some non-giraffes, and none of them are mammals. In such a case, both statements would be true.

D is not correct. Nothing about giraffes follows from the claim "all non-giraffes are not mammals."

E is correct. None of the sentences among A-D is inconsistent with "all non-giraffes are not mammals."

A is not correct. "All non-giraffes are not mammals" is consistent with itself.

B is not correct. "All non-giraffes are not mammals" is not inconsistent with "no non-giraffes are not mammals." Suppose that there are no non-giraffes. Then, since there would not be any mammalian non-giraffes, "all non-giraffes are not mammals" would be true. "No non-giraffes are not mammals" would be true, too, since there would not be any non-mammal non-giraffes.

C is not correct. "Some non-giraffes are not mammals" is consistent with the claim, "all non-giraffes are not mammals." Suppose there are some non-giraffes, and none of them are mammals. In such a case, both statements would be true.

D is not correct. Nothing about giraffes follows from the claim "all non-giraffes are not mammals."

E is correct. None of the sentences among A-D is inconsistent with "all non-giraffes are not mammals."

Which of the following statements is inconsistent with "No non-turtles are not female"?

A. Some non-turtles are not female.

A is correct. "No non-turtles are not female" is inconsistent with "some non-turtles are not female." If none of the non-turtles are not female, then it cannot be that some of them are not female.

B is not correct. "All turtles are female" is not inconsistent with "no non-turtles are not female." The first statement is only about turtles, while the second concerns the non-turtles.

C is not correct. "All non-turtles are not female" is consistent with the claim, "no non-turtles are not female." Suppose there are no non-turtles. In such a case, since there would not be any non-turtles, the statement "all non-turtles are not female" would be true (since there would not be any non-turtles, which were not non-female), and the statement "no non-turtles are not female" would also be true (since there would not be any non-female non-turtles).

D is not correct. "Some turtles are not female" is consistent with the claim, "no non-turtles are not female." Suppose there are some turtles, and that none of them are female, and suppose that all non-turtles are female. Then both statements would be true.

E is not correct. One of the sentences among A-D is inconsistent with "no non-turtles are not female."

A is correct. "No non-turtles are not female" is inconsistent with "some non-turtles are not female." If none of the non-turtles are not female, then it cannot be that some of them are not female.

B is not correct. "All turtles are female" is not inconsistent with "no non-turtles are not female." The first statement is only about turtles, while the second concerns the non-turtles.

C is not correct. "All non-turtles are not female" is consistent with the claim, "no non-turtles are not female." Suppose there are no non-turtles. In such a case, since there would not be any non-turtles, the statement "all non-turtles are not female" would be true (since there would not be any non-turtles, which were not non-female), and the statement "no non-turtles are not female" would also be true (since there would not be any non-female non-turtles).

D is not correct. "Some turtles are not female" is consistent with the claim, "no non-turtles are not female." Suppose there are some turtles, and that none of them are female, and suppose that all non-turtles are female. Then both statements would be true.

E is not correct. One of the sentences among A-D is inconsistent with "no non-turtles are not female."

Which of the following statements is inconsistent with "No non-turtles are honest"?

A. Some non-turtles are honest.

A is correct. "No non-turtles are honest" is inconsistent with "some non-turtles are honest."

B is not correct. "All turtles are honest" is not inconsistent with "no non-turtles are honest." Nothing about turtles follows from the claim about non-turtles.

C is not correct. "All non-turtles are honest" is consistent with the claim, "no non-turtles are honest." Suppose there are no non-turtles. In such a case, since there would not be any honest non-turtles, the statements "all non-turtles are honest" would be true along with "no non-turtles are honest."

D is incorrect. "Some turtles are not honest" is not inconsistent with the claim, "no non-turtles are honest."

E is not correct. One of the sentences among A-D is inconsistent with "no non-turtles are honest."

A is correct. "No non-turtles are honest" is inconsistent with "some non-turtles are honest."

B is not correct. "All turtles are honest" is not inconsistent with "no non-turtles are honest." Nothing about turtles follows from the claim about non-turtles.

C is not correct. "All non-turtles are honest" is consistent with the claim, "no non-turtles are honest." Suppose there are no non-turtles. In such a case, since there would not be any honest non-turtles, the statements "all non-turtles are honest" would be true along with "no non-turtles are honest."

D is incorrect. "Some turtles are not honest" is not inconsistent with the claim, "no non-turtles are honest."

E is not correct. One of the sentences among A-D is inconsistent with "no non-turtles are honest."

Which of the following statements is inconsistent with "No non-dodos are not quotable"?

C. Some non-dodos are not quotable.

A is not correct. "All non-dodos are not quotable" is consistent with "no non-dodos are not quotable." Suppose there are no non-dodos. In such a case, since there would not be any quotable non-dodos, the statement "all non-dodos are not quotable" would be true, as well as the statement "no non-dodos are not quotable."

B is not correct. "No dodos are quotable" is not inconsistent with "no non-dodos are not quotable." Nothing about dodos follows from the claim about non-dodos.

C is correct. "Some non-dodos are not quotable" is inconsistent with the claim "no non-dodos are not quotable." If some of the dodos are not quotable, then it cannot be true that none of them are not quotable.

D is not correct. "All non-dodos are quotable" is consistent with the claim "no non-dodos are not quotable."

E is not correct. One of the sentences among A-D is inconsistent with "no non-dodos are not quotable."

A is not correct. "All non-dodos are not quotable" is consistent with "no non-dodos are not quotable." Suppose there are no non-dodos. In such a case, since there would not be any quotable non-dodos, the statement "all non-dodos are not quotable" would be true, as well as the statement "no non-dodos are not quotable."

B is not correct. "No dodos are quotable" is not inconsistent with "no non-dodos are not quotable." Nothing about dodos follows from the claim about non-dodos.

C is correct. "Some non-dodos are not quotable" is inconsistent with the claim "no non-dodos are not quotable." If some of the dodos are not quotable, then it cannot be true that none of them are not quotable.

D is not correct. "All non-dodos are quotable" is consistent with the claim "no non-dodos are not quotable."

E is not correct. One of the sentences among A-D is inconsistent with "no non-dodos are not quotable."

Which of the following statements is inconsistent with "All non-otters are not authors"?

E. None of the above

A is not correct. "No otters are authors" is consistent with "all non-otters are not authors." Nothing about otters follows from the statement "all non-otters are not authors."

B is not correct. "Some non-authors are otters" is consistent with "all non-otters are not authors." Suppose that there are no authors, either among the otters or among the non-otters. In such a case, both statements could be true.

C is not correct. "No non-otters are not authors" is consistent with the claim "all non-otters are not authors." Suppose there are no non-otters. In such a case, since there would not be any non-otter authors, the statement "all non-otters are not authors" would be true, as well as the statement "no non-otters are not authors."

D is not correct. "Some non-otters are not authors" is consistent with the claim "all non-otters are not authors."

E is correct. None of the sentences among A-D is inconsistent with "all non-otters are not authors."

A is not correct. "No otters are authors" is consistent with "all non-otters are not authors." Nothing about otters follows from the statement "all non-otters are not authors."

B is not correct. "Some non-authors are otters" is consistent with "all non-otters are not authors." Suppose that there are no authors, either among the otters or among the non-otters. In such a case, both statements could be true.

C is not correct. "No non-otters are not authors" is consistent with the claim "all non-otters are not authors." Suppose there are no non-otters. In such a case, since there would not be any non-otter authors, the statement "all non-otters are not authors" would be true, as well as the statement "no non-otters are not authors."

D is not correct. "Some non-otters are not authors" is consistent with the claim "all non-otters are not authors."

E is correct. None of the sentences among A-D is inconsistent with "all non-otters are not authors."

Most men at Charles' age go bald. Therefore, Charles is bald.

A. This is an argument. This passage claims that most men at Charles' age go bald, and that this is a reason to think that Charles is bald. According to this claim, his age helps us understand why he went bald and serves as a reason to agree that Charles is bald.

Which of the following arguments is a syllogism?

D.

All dodos are birds.

Some birds are not witches.

_______________________

So all dodos are witches.

A is not correct. The connective "because" is not part of any statement with the A, E, I, or O form. Syllogisms must have two premises, each of which is a statement with the A, E, I or O form.

B is not correct. Although the premise "some funerals are lamentable" has the O form, and although the second premise, "some things are arranged by birds," has the A form, the conclusion does not connect the subject term, "funerals" with the predicate term, "arranged by birds." A syllogism must have premises of the A, E, I, or O form, and it also must have a conclusion, which links the subject term in the first premise with the predicate term in the second premise.

C is not correct. Although the conclusion, "all the movie scenes are equally terrible," follows from the premises "some movie scenes are terrible" and "no terrible thing is more terrible than any other terrible thing," there is no way to represent the validity of this argument with Venn diagrams.

D is correct. Syllogisms have two premises, each of which is a statement with the A, E, I or O form. The premise "all dodos are birds" has the A form and the premise "some birds are not witches." The conclusion, moreover, connects the subject term "dodos" with the predicate term "witches." So the argument stated in D is a syllogism. It is not a valid syllogism, of course, because the conclusion contradicts the second premise. Yet it is a syllogism nonetheless.

E is not correct. One of the arguments among A-D are syllogisms.

All dodos are birds.

Some birds are not witches.

_______________________

So all dodos are witches.

A is not correct. The connective "because" is not part of any statement with the A, E, I, or O form. Syllogisms must have two premises, each of which is a statement with the A, E, I or O form.

B is not correct. Although the premise "some funerals are lamentable" has the O form, and although the second premise, "some things are arranged by birds," has the A form, the conclusion does not connect the subject term, "funerals" with the predicate term, "arranged by birds." A syllogism must have premises of the A, E, I, or O form, and it also must have a conclusion, which links the subject term in the first premise with the predicate term in the second premise.

C is not correct. Although the conclusion, "all the movie scenes are equally terrible," follows from the premises "some movie scenes are terrible" and "no terrible thing is more terrible than any other terrible thing," there is no way to represent the validity of this argument with Venn diagrams.

D is correct. Syllogisms have two premises, each of which is a statement with the A, E, I or O form. The premise "all dodos are birds" has the A form and the premise "some birds are not witches." The conclusion, moreover, connects the subject term "dodos" with the predicate term "witches." So the argument stated in D is a syllogism. It is not a valid syllogism, of course, because the conclusion contradicts the second premise. Yet it is a syllogism nonetheless.

E is not correct. One of the arguments among A-D are syllogisms.

Which of the following arguments is a valid syllogism?

(Hint: try drawing Venn diagrams to see!)

(Hint: try drawing Venn diagrams to see!)

C.

Some dodos are authors.

All authors are witches.

____________________________

Therefore, some dodos are witches.

A is not correct. As your own diagram should show, we do not know whether there should be an "X" in the region where the circle of dodos and the circle of omnivores intersect. It could be that the none of the dodos who are quotable are also omnivores, even though some quotable non-dodos are omnivores.

B is not correct. As your own diagram should show, the Venn diagram for "some composite objects are material objects" and "some material objects are spatially locatable" does not require the part of the circle of composite objects, which is outside the circle of spatially locatable things, to be shaded out. Because that region is not shaded out, there may very well be composite objects that are not spatially locatable.

C is correct. As your own diagram should show, there is only one place to put an "X," which is both inside the circle of dodos, and also inside the circle of authors. Because that place is in the circle of witches, it follows that some dodos are witches.

D is not correct. Because D's conclusion does not connect the subject term in the first premise with the predicate term in the second premise, D is not a syllogism at all. Since D is not a syllogism at all, it cannot be a valid syllogism.

E is not correct. One of the arguments among A-D is a valid syllogism.

Some dodos are authors.

All authors are witches.

____________________________

Therefore, some dodos are witches.

A is not correct. As your own diagram should show, we do not know whether there should be an "X" in the region where the circle of dodos and the circle of omnivores intersect. It could be that the none of the dodos who are quotable are also omnivores, even though some quotable non-dodos are omnivores.

B is not correct. As your own diagram should show, the Venn diagram for "some composite objects are material objects" and "some material objects are spatially locatable" does not require the part of the circle of composite objects, which is outside the circle of spatially locatable things, to be shaded out. Because that region is not shaded out, there may very well be composite objects that are not spatially locatable.

C is correct. As your own diagram should show, there is only one place to put an "X," which is both inside the circle of dodos, and also inside the circle of authors. Because that place is in the circle of witches, it follows that some dodos are witches.

D is not correct. Because D's conclusion does not connect the subject term in the first premise with the predicate term in the second premise, D is not a syllogism at all. Since D is not a syllogism at all, it cannot be a valid syllogism.

E is not correct. One of the arguments among A-D is a valid syllogism.

Which of the following arguments is a valid syllogism?

(Hint: try drawing Venn diagrams to see!)

(Hint: try drawing Venn diagrams to see!)

D.

All otters are mammals.

All mammals are quotable.

______________________

So all otters are quotable.

A is not correct. As your diagram should show, the Venn diagram for "some authors are omnivores" and "all omnivores are material objects" does not require the circle of authors, which is outside the circle of material objects, to be shaded out. Because that region is not shaded out, there may very well be authors that are not material objects.

B is not correct. As your diagram should show, the premise "no quotable things are omnivores" requires us to shade out the intersecting region between the circle of quotable things and the circle of omnivores. The premise "some otters are quotable," however, requires us to put an "X" in the non-shaded region, where the circles of otters and quotable things intersect. Neither premise, however, requires us to shade out every region where the circle of otters intersects with the circle of omnivores. According to the diagram, there may be some omnivorous, non-quotable otters.

C is not correct. As your diagram should show, the first premise requires the region of the circle of dodos that intersects with the circle of authors, to be shaded out. The second premise requires the region of the circle of authors that intersects with the circle of witches to be shaded out. This leaves a region in the circle of dodos that intersects with the circle of witches unshaded. Since there is an unshaded portion of the circle of dodos that intersects with the circle of witches, it does not follow that no dodos are witches.

D is correct. The premise "all otters are mammals" requires the region of the circle of otters that does not intersect with the circle of mammals to be shaded out. The premise "all mammals are quotable" requires the region of mammals that does not intersect with the circle of quotable things to be shaded out as well. So the only non-shaded area in the circle of mammals is also in the circle of quotable things. So from the premises, the conclusion "all otters are quotable" follows.

E is not correct. One of the arguments among A-D is a valid syllogism.

All otters are mammals.

All mammals are quotable.

______________________

So all otters are quotable.

A is not correct. As your diagram should show, the Venn diagram for "some authors are omnivores" and "all omnivores are material objects" does not require the circle of authors, which is outside the circle of material objects, to be shaded out. Because that region is not shaded out, there may very well be authors that are not material objects.

B is not correct. As your diagram should show, the premise "no quotable things are omnivores" requires us to shade out the intersecting region between the circle of quotable things and the circle of omnivores. The premise "some otters are quotable," however, requires us to put an "X" in the non-shaded region, where the circles of otters and quotable things intersect. Neither premise, however, requires us to shade out every region where the circle of otters intersects with the circle of omnivores. According to the diagram, there may be some omnivorous, non-quotable otters.

C is not correct. As your diagram should show, the first premise requires the region of the circle of dodos that intersects with the circle of authors, to be shaded out. The second premise requires the region of the circle of authors that intersects with the circle of witches to be shaded out. This leaves a region in the circle of dodos that intersects with the circle of witches unshaded. Since there is an unshaded portion of the circle of dodos that intersects with the circle of witches, it does not follow that no dodos are witches.

D is correct. The premise "all otters are mammals" requires the region of the circle of otters that does not intersect with the circle of mammals to be shaded out. The premise "all mammals are quotable" requires the region of mammals that does not intersect with the circle of quotable things to be shaded out as well. So the only non-shaded area in the circle of mammals is also in the circle of quotable things. So from the premises, the conclusion "all otters are quotable" follows.

E is not correct. One of the arguments among A-D is a valid syllogism.

Consider the statement "The only time I read a book by Mark Twain was when I was in prison." Which of the following statements is implied by this last statement?

A. All times that I read a Mark Twain book are times that I was in prison.

A is correct. If I say, "the only time I read a book by Mark Twain was in prison," this implies that there are no times in which I have read a Mark Twain book outside of prison. In other words, of all the times in which I have read a Mark Twain book, they have all been times in prison. What the statement in the question implies, then, is "all times that I read a Mark Twain book were times that I was in prison."

B is not correct, since the statement "the only time I read a book by Mark Twain was in prison" can be true even when the speaker has spent many times in prison without a Mark Twain book.

C is not correct, since "the only time I read a book by Mark Twain was in prison" means the same thing as "all the times that I read a Mark Twain book are times that I was in prison." This statement is indeed compatible with the statement, "some of the times that I was in prison are times that I read a Mark Twain book," but they do not mean the same thing. The first statement, for instance could be true even if the speaker had never read a Mark Twain book, but the statement "some of the times that I was in prison are times that I read a Mark Twain book" could not.

The same reasoning goes for D.

E is not correct, because it directly contradicts the statement "all the times that I read a Mark Twain book are times that I was in prison." If such a statement is true, then there could not be times in which the speaker read a Mark Twain book without being in prison.

A is correct. If I say, "the only time I read a book by Mark Twain was in prison," this implies that there are no times in which I have read a Mark Twain book outside of prison. In other words, of all the times in which I have read a Mark Twain book, they have all been times in prison. What the statement in the question implies, then, is "all times that I read a Mark Twain book were times that I was in prison."

B is not correct, since the statement "the only time I read a book by Mark Twain was in prison" can be true even when the speaker has spent many times in prison without a Mark Twain book.

C is not correct, since "the only time I read a book by Mark Twain was in prison" means the same thing as "all the times that I read a Mark Twain book are times that I was in prison." This statement is indeed compatible with the statement, "some of the times that I was in prison are times that I read a Mark Twain book," but they do not mean the same thing. The first statement, for instance could be true even if the speaker had never read a Mark Twain book, but the statement "some of the times that I was in prison are times that I read a Mark Twain book" could not.

The same reasoning goes for D.

E is not correct, because it directly contradicts the statement "all the times that I read a Mark Twain book are times that I was in prison." If such a statement is true, then there could not be times in which the speaker read a Mark Twain book without being in prison.

Consider the statement "Phoebe bakes nothing but pies, but those pies taste great." Which of the following Venn Diagrams is equivalent to this last statement?

(D)

D is the correct answer. The statement, "Phoebe bakes nothing but pies, but those pies taste great" mentions three categories. One is the category of things that Phoebe bakes, one is the category of pies, and one is the category of things that taste great. Phoebe herself is not a category. Since Phoebe is not a category, neither A nor B can be correct.

The statement includes the claim, "Phoebe bakes nothing but pies," which means the same thing as "all the things that Phoebe bakes are pies." Since the statement "all the things that Pheobe bakes are pies" requires us to shade out the region in the circle of things that Pheobe bakes, which does not intersect with the circle of pies.

Moreover, the statement also includes the claim "those pies taste great." In this context, what this means is "all the pies, which are among the things that Phoebe bakes, are things that taste great." This statement requires us to shade out the region in the circle of pies, which intersects with the circle of things that Phoebe bakes, but which does not intersect with the circle of things that tastes great.

Since C fails to shade either area specified by the statement, and since D shades exactly the regions specified by the statement, D is correct.

D is the correct answer. The statement, "Phoebe bakes nothing but pies, but those pies taste great" mentions three categories. One is the category of things that Phoebe bakes, one is the category of pies, and one is the category of things that taste great. Phoebe herself is not a category. Since Phoebe is not a category, neither A nor B can be correct.

The statement includes the claim, "Phoebe bakes nothing but pies," which means the same thing as "all the things that Phoebe bakes are pies." Since the statement "all the things that Pheobe bakes are pies" requires us to shade out the region in the circle of things that Pheobe bakes, which does not intersect with the circle of pies.

Moreover, the statement also includes the claim "those pies taste great." In this context, what this means is "all the pies, which are among the things that Phoebe bakes, are things that taste great." This statement requires us to shade out the region in the circle of pies, which intersects with the circle of things that Phoebe bakes, but which does not intersect with the circle of things that tastes great.

Since C fails to shade either area specified by the statement, and since D shades exactly the regions specified by the statement, D is correct.

Consider the statement "George Washington slept in New Jersey at least once in 1789." Which of the following statements is equivalent to this last statement?

C. Some of the places that George Washington slept in 1789 are places in New Jersey.

C is correct. The statement "George Washington slept in New Jersey at least once in 1789" mentions two categories: places George Washington slept at least once in 1789, and things in New Jersey. If we represent the statement with a Venn diagram, we should have two non-shaded circles, with an X in their mutual intersection, whose labels are "places George Washington slept at least once in 1789" and "things in New Jersey." This diagram says that some of the places that George Washington slept at least once in 1789 were in New Jersey. Since this is what C says, C is correct.

B cannot be correct, since the statement in B claims something that the statement in the question did not. The statement in B claims that some of the places that George Washington slept in 1789 are not in New Jersey. This is compatible with what the statement in the question says, but it is not equivalent to it. To see why, suppose that Washington didn't sleep anywhere in 1789 besides New Jersey. Given such a supposition, the statement in the question would be true, but the statement in B would be false.

A cannot be correct either, since it claims something that contradicts the statement in the question. The statement in the question implies that some of the places that George Washington slept in 1789 are in New Jersey, but the statement in A implies that none of the places that George Washington slept in 1789 are in New Jersey.

Both D and E are compatible with the statement in question, but neither D nor E is equivalent to the statement in question. The statement in question could be true, even when D is false and George Washington slept elsewhere in 1789; and the statement in question could be true, even when E is false and there is a place in New Jersey that was not slept in by Washington during the year of 1789.

C is correct. The statement "George Washington slept in New Jersey at least once in 1789" mentions two categories: places George Washington slept at least once in 1789, and things in New Jersey. If we represent the statement with a Venn diagram, we should have two non-shaded circles, with an X in their mutual intersection, whose labels are "places George Washington slept at least once in 1789" and "things in New Jersey." This diagram says that some of the places that George Washington slept at least once in 1789 were in New Jersey. Since this is what C says, C is correct.

B cannot be correct, since the statement in B claims something that the statement in the question did not. The statement in B claims that some of the places that George Washington slept in 1789 are not in New Jersey. This is compatible with what the statement in the question says, but it is not equivalent to it. To see why, suppose that Washington didn't sleep anywhere in 1789 besides New Jersey. Given such a supposition, the statement in the question would be true, but the statement in B would be false.

A cannot be correct either, since it claims something that contradicts the statement in the question. The statement in the question implies that some of the places that George Washington slept in 1789 are in New Jersey, but the statement in A implies that none of the places that George Washington slept in 1789 are in New Jersey.

Both D and E are compatible with the statement in question, but neither D nor E is equivalent to the statement in question. The statement in question could be true, even when D is false and George Washington slept elsewhere in 1789; and the statement in question could be true, even when E is false and there is a place in New Jersey that was not slept in by Washington during the year of 1789.

Which of the following statements conveys the same information that this Venn diagram conveys?

A. There are fish, and all of them are carnivorous sharks. The only non-shaded region in the circle of fish is also in the circle of sharks and the circle of carnivores. This means that all the fish are carnivorous sharks. The fact that there is an X in the circles' intersection, moreover, means that there are also some fish. So the diagram indicates that there are some fish, and that all of them are carnivorous sharks.

Which of the following statements conveys exactly the same information that this Venn diagram conveys?

E. None of the above

The only non-shaded region in any of the three circles is their common intersection. This means that all the fish are carnivorous sharks, all sharks are carnivorous fish, all carnivores are both sharks and fish, and that there are some fish.

Statement A is not equivalent to this claim, since it leaves out the claim that all sharks are carnivorous fish, and that all carnivores are both fish and sharks.

Statement B is not equivalent to the claim expressed by the diagram, since it leaves out the claim that all fish are carnivorous sharks, the claim that all sharks are fish, the claim that all fish are carnivorous sharks, and the claim that all carnivores are both sharks and fish.

Statement C is not equivalent to the claim expressed by the diagram, since it leaves out all three claims of the "A" form.

Statement D is incorrect, since it leaves out the claim that all fish are carnivorous sharks, and the claim that all sharks are carnivorous fish. Since none of A-D are correct, E is the correct answer.

The only non-shaded region in any of the three circles is their common intersection. This means that all the fish are carnivorous sharks, all sharks are carnivorous fish, all carnivores are both sharks and fish, and that there are some fish.

Statement A is not equivalent to this claim, since it leaves out the claim that all sharks are carnivorous fish, and that all carnivores are both fish and sharks.

Statement B is not equivalent to the claim expressed by the diagram, since it leaves out the claim that all fish are carnivorous sharks, the claim that all sharks are fish, the claim that all fish are carnivorous sharks, and the claim that all carnivores are both sharks and fish.

Statement C is not equivalent to the claim expressed by the diagram, since it leaves out all three claims of the "A" form.

Statement D is incorrect, since it leaves out the claim that all fish are carnivorous sharks, and the claim that all sharks are carnivorous fish. Since none of A-D are correct, E is the correct answer.

Consider this argument:

All parties are fun ____________________________________

Therefore, all non-fun events are not parties.

Which of these Venn diagrams conveys the same information that the above argument conveys?

All parties are fun ____________________________________

Therefore, all non-fun events are not parties.

Which of these Venn diagrams conveys the same information that the above argument conveys?

(B) To represent the statement "all parties are fun" with a Venn diagram, we just need two circles: one for parties, and one for events that are fun. We then shade out the portion of the circle of parties, which are outside the circle of events that are fun. This shows that all parties are fun, and that, consequently, no non-fun things are parties. Since this is what B shows, B is the correct answer.

Consider this argument:

All fun events are parties _______________________________

Therefore, all non-parties are not fun.

Which of these Venn diagrams conveys the same information that the above argument conveys?

All fun events are parties _______________________________

Therefore, all non-parties are not fun.

Which of these Venn diagrams conveys the same information that the above argument conveys?

(C) To represent the statement "all fun events are parties" with a Venn diagram, we just need two circles: one for parties, and one for events that are fun. We then shade out the portion of the circle of fun events that is outside the circle of parties. This shows that all fun events are parties. Since there is no space anywhere on the diagram for non-parties, it follows that there are no non-parties that are fun. So the conclusion, namely that no non-parties are fun events, follows. Since this is what C shows, so C is the correct answer.

Which of the following bits of the English language sometimes expresses a truth-functional connective?

None of the above. A truth-functional connective is one that forms a proposition, whose truth can be determined merely by determining the truth of its parts. If, for some propositions p and q, where p is true and q is true, can it be determined whether "p therefore q" is true? Can we tell whether "p therefore q" is true, simply on the basis of telling whether p is true, and telling whether q is true? No, we cannot. So A is false.

Nor can we tell whether "necessarily, whenever p, q" is true, simply on the basis of telling whether p is true, and telling whether q is true. So B is false.

Nor can we determine the truth of "it is possible that p," simply on the basis of knowing whether p is true. So C is false.

Nor can we tell whether "I believe that p" is true, simply on the basis of telling whether p is true. The question of whether p is true does not settle the question of whether it is believed by anyone. So D is false.

None of the English expressions in the available answers to this question form propositions, whose truth can be determined merely by knowing the truth of their parts. So E is correct.

Nor can we tell whether "necessarily, whenever p, q" is true, simply on the basis of telling whether p is true, and telling whether q is true. So B is false.

Nor can we determine the truth of "it is possible that p," simply on the basis of knowing whether p is true. So C is false.

Nor can we tell whether "I believe that p" is true, simply on the basis of telling whether p is true. The question of whether p is true does not settle the question of whether it is believed by anyone. So D is false.

None of the English expressions in the available answers to this question form propositions, whose truth can be determined merely by knowing the truth of their parts. So E is correct.

Which of the following bits of the English language sometimes expresses a truth-functional connective?

"furthermore." "In contrast" is not a truth-functional connective. For any claim, p, one cannot determine the truth of "in contrast, p" merely by knowing whether p is true. The truth of p does not settle the question of whether p is in contrast to anything.

"Unfortunately" is not a truth-functional connective. For any claim, p, one cannot determine the truth of "unfortunately p" merely by knowing whether p is true. The truth of p does not settle the question of whether it is unfortunate that p.

The word "furthermore" is sometimes used as an ordinary English synonym for "and." "And" is a truth-functional connective, since, for any two claims p and q, the truth of "p and q" can be determined merely by looking at the truth of whether p, and the truth of whether q. If p is true, and q is true, then so must be "p and q." If p is true and q is false, however, or if p is false and q is true, then "p and q" must be false. If p and q are both false, then so is "p and q."

"As I said before" is not a truth-functional connective. For any claim, p, one cannot determine the truth of "as I said before, p" merely on the basis of knowing whether p is true. The truth of p does not settle the question of whether p was said before.

"Unfortunately" is not a truth-functional connective. For any claim, p, one cannot determine the truth of "unfortunately p" merely by knowing whether p is true. The truth of p does not settle the question of whether it is unfortunate that p.

The word "furthermore" is sometimes used as an ordinary English synonym for "and." "And" is a truth-functional connective, since, for any two claims p and q, the truth of "p and q" can be determined merely by looking at the truth of whether p, and the truth of whether q. If p is true, and q is true, then so must be "p and q." If p is true and q is false, however, or if p is false and q is true, then "p and q" must be false. If p and q are both false, then so is "p and q."

"As I said before" is not a truth-functional connective. For any claim, p, one cannot determine the truth of "as I said before, p" merely on the basis of knowing whether p is true. The truth of p does not settle the question of whether p was said before.

Which of the following asserts a conjunction of two or more propositions?

I have been seeing Peter but not Paul. A expresses a disjunction of two propositions connected by the word "or." Answer A does not express a conjunction.

B expresses a pair of questions. Questions, however, are not propositions, since they cannot be true or false. So B does not express a conjunction of two or more propositions.

C expresses a pair of commands. Commands, however, are not propositions, since they cannot be true or false. So C does not express a conjunction of two or more propositions.

D is correct. The proposition expressed by D asserts the conjunction of two claims: "I have been seeing Peter AND I have not been seeing Paul."

E is incorrect. One of the English expressions in A-D expresses a conjunction of two or more propositions.

B expresses a pair of questions. Questions, however, are not propositions, since they cannot be true or false. So B does not express a conjunction of two or more propositions.

C expresses a pair of commands. Commands, however, are not propositions, since they cannot be true or false. So C does not express a conjunction of two or more propositions.

D is correct. The proposition expressed by D asserts the conjunction of two claims: "I have been seeing Peter AND I have not been seeing Paul."

E is incorrect. One of the English expressions in A-D expresses a conjunction of two or more propositions.

Which of the following asserts a conjunction of two or more propositions?

I drove to New York City and back. A is correct. The proposition represented in (a) could be phrased as the conjunction of two claims: "I drove to New York City and I drove back." None of B-D, however, need to be interpreted as asserting two or more propositions.

B does not express a conjunction of two or more propositions. It expresses a single proposition about what one said.

C does not express a conjunction of two or more propositions. It expresses a single proposition about what one thought.

D does not express a conjunction of two or more propositions. It expresses a single proposition, which concerns the explanation of what happened.

E is incorrect. One of the English sentences in A-D asserts a conjunction of two or more propositions

B does not express a conjunction of two or more propositions. It expresses a single proposition about what one said.

C does not express a conjunction of two or more propositions. It expresses a single proposition about what one thought.

D does not express a conjunction of two or more propositions. It expresses a single proposition, which concerns the explanation of what happened.

E is incorrect. One of the English sentences in A-D asserts a conjunction of two or more propositions

Which of the following arguments is valid?

Alice and Jane are both talking.

____________________________

Therefore, Alice is talking

A is the only valid argument. From the fact that Alice and Jane are both talking, it logically follows that Alice is talking.

B is not valid. From the fact that Alice and Jane are both talking, it does not logically follow that nobody else is talking.

C is not valid. From the fact that Alice is talking, it does not logically follow that Jane is talking.

D is not valid. From the fact that Alice is talking but Jane is not, it does not logically follow that Alice and Jane are both talking.

E is incorrect. One of the arguments listed in A-D is logically valid.

____________________________

Therefore, Alice is talking

A is the only valid argument. From the fact that Alice and Jane are both talking, it logically follows that Alice is talking.

B is not valid. From the fact that Alice and Jane are both talking, it does not logically follow that nobody else is talking.

C is not valid. From the fact that Alice is talking, it does not logically follow that Jane is talking.

D is not valid. From the fact that Alice is talking but Jane is not, it does not logically follow that Alice and Jane are both talking.

E is incorrect. One of the arguments listed in A-D is logically valid.

A conjunction introduction argument is a valid argument in which the conclusion conjoins two or more of the premises. Which of the following arguments is a conjunction introduction argument?

None of the above. The argument in A does not have two or more premises. Since a conjunction introduction argument requires two or more premises, the argument in A is not a conjunction introduction argument.

The argument in B does not have two or more premises. Since a conjunction introduction argument requires two or more premises, the argument in B is not a conjunction introduction argument.

The argument in C does not have two or more premises. Since a conjunction introduction argument requires two or more premises, the argument in D is not a conjunction introduction argument.

The argument in D does not have two or more premises. Since a conjunction introduction argument requires two or more premises, the argument in D is not a conjunction introduction argument.

E is correct. None of the above arguments have two or more premises. Since a conjunction introduction argument requires two or more premises, none of the arguments are conjunction introduction arguments.

The argument in B does not have two or more premises. Since a conjunction introduction argument requires two or more premises, the argument in B is not a conjunction introduction argument.

The argument in C does not have two or more premises. Since a conjunction introduction argument requires two or more premises, the argument in D is not a conjunction introduction argument.

The argument in D does not have two or more premises. Since a conjunction introduction argument requires two or more premises, the argument in D is not a conjunction introduction argument.

E is correct. None of the above arguments have two or more premises. Since a conjunction introduction argument requires two or more premises, none of the arguments are conjunction introduction arguments.

A conjunction elimination argument is a valid argument in which the conclusion is a conjunct of some conjunction that appears in the premises. Which of the following arguments is a conjunction elimination argument?

Alice and Jane are both talking.

_____________________________

Therefore, Alice is talking.

A is a conjunction elimination argument. The proposition "Alice and Jane are both talking" asserts the following conjunction: "Alice is talking and Jane is talking." One of these conjuncts, which A places at the conclusion, is "Alice is talking."

B is not a conjunction elimination argument. The proposition "no one else is talking" is not one of the conjuncts of the premise "Alice and Jane are both talking."

C is not a conjunction elimination argument. A conjunction elimination argument must have a conjunction in its premises. C does not have a conjunction in its premises.

D is not a conjunction elimination argument. The proposition "Alice is not talking" is not one of the conjuncts of the premise "Alice is talking but Jane is not."

E is incorrect. One of the arguments in A-D is a conjunction elimination argument.

_____________________________

Therefore, Alice is talking.

A is a conjunction elimination argument. The proposition "Alice and Jane are both talking" asserts the following conjunction: "Alice is talking and Jane is talking." One of these conjuncts, which A places at the conclusion, is "Alice is talking."

B is not a conjunction elimination argument. The proposition "no one else is talking" is not one of the conjuncts of the premise "Alice and Jane are both talking."

C is not a conjunction elimination argument. A conjunction elimination argument must have a conjunction in its premises. C does not have a conjunction in its premises.

D is not a conjunction elimination argument. The proposition "Alice is not talking" is not one of the conjuncts of the premise "Alice is talking but Jane is not."

E is incorrect. One of the arguments in A-D is a conjunction elimination argument.

Suppose you had to fill in the rightmost column of the truth table for conjunction:

p.....q..........p&q

T.....T..........

T.....F..........

F.....T..........

F.....F..........

Going from top to bottom, how would you fill in that column?

p.....q..........p&q

T.....T..........

T.....F..........

F.....T..........

F.....F..........

Going from top to bottom, how would you fill in that column?

TFFF

A is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

B is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

C is correct. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

D is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

E is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

A is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

B is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

C is correct. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

D is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

E is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

Now suppose you had to fill in the rightmost column of the following truth table for conjunction:

p.....q..........p&q

F.....F..........

T.....T..........

F.....T..........

T.....F..........

Now, going from top to bottom, how would you fill in that column?

p.....q..........p&q

F.....F..........

T.....T..........

F.....T..........

T.....F..........

Now, going from top to bottom, how would you fill in that column?

FTFF

A is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

B is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

C is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

D is correct. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

E is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

A is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

B is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

C is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

D is correct. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

E is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

And now suppose you had to fill in the rightmost column of the following truth table for conjunction:

p.....q..........p&q

F.....F..........

F.....T..........

T.....T..........

T.....F..........

Now, going from top to bottom, how would you fill in that column?

p.....q..........p&q

F.....F..........

F.....T..........

T.....T..........

T.....F..........

Now, going from top to bottom, how would you fill in that column?

FFTF

A is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

B is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

C is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

D is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

E is correct. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

A is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

B is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

C is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

D is incorrect. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

E is correct. A conjunction can only be true if both of its conjuncts are true. Otherwise, it is false.

Which of the following arguments is valid?

Two or more of the above.

A is incorrect. A valid argument is one whose conclusion logically follows from the premises. From the claim that Yamamoto or Jiabao will be the US president in 2016, it does not follow that Yamamoto will be the US president in 2016. A is invalid.

B is incorrect. The argument expressed in B is valid. Yet the argument stated in B is not the only valid one among A-D. From the claim that Jiabao will be the US president in 2016, it follows that either Jiabao or Yamamoto will be the US president in 2016. There is another valid argument among A-D, however.

C is incorrect. From the claim that Yamamoto will be the US president in 2016, it does not follow that Jiabao will never be the US president again. C is invalid.

D is incorrect. The argument expressed in D is valid. Yet the argument stated in D is not the only valid one among A-D. From the claim that either Jiabao or Yamamoto will be the US president in 2016, and from the claim that Jiabao will not be the US president in 2016, it follows that Yamamoto will be the US president in 2016. There is another valid argument among A-D, however.

E is correct. Both B and D express valid arguments.

A is incorrect. A valid argument is one whose conclusion logically follows from the premises. From the claim that Yamamoto or Jiabao will be the US president in 2016, it does not follow that Yamamoto will be the US president in 2016. A is invalid.

B is incorrect. The argument expressed in B is valid. Yet the argument stated in B is not the only valid one among A-D. From the claim that Jiabao will be the US president in 2016, it follows that either Jiabao or Yamamoto will be the US president in 2016. There is another valid argument among A-D, however.

C is incorrect. From the claim that Yamamoto will be the US president in 2016, it does not follow that Jiabao will never be the US president again. C is invalid.

D is incorrect. The argument expressed in D is valid. Yet the argument stated in D is not the only valid one among A-D. From the claim that either Jiabao or Yamamoto will be the US president in 2016, and from the claim that Jiabao will not be the US president in 2016, it follows that Yamamoto will be the US president in 2016. There is another valid argument among A-D, however.

E is correct. Both B and D express valid arguments.

A disjunction introduction argument is a valid argument, in which the conclusion disjoins one of the premises with some other proposition. Which of the following arguments is a disjunction introduction argument?

Jiabao will be the US president in 2016.

______________________________________________

Therefore, Yamamoto or Jiabao will be the US president in 2016.

A is incorrect. To disjoin one proposition with another is to connect those two propositions with the connective "or." The conclusion of A is not the result of disjoining a proposition with its premise.

B is correct. The conclusion of B is the result of disjoining a proposition with its premise.

C is incorrect. The conclusion of C is not the result of disjoining a proposition with its premise.

D is incorrect. The conclusion of D is not the result of disjoining a proposition with its premise. The conclusion of D validly follows from its premises, but the conclusion is not the result of disjoining a proposition with any of the premises.

E is incorrect. Only one of the arguments stated in A-D is a disjunction introduction argument.

______________________________________________

Therefore, Yamamoto or Jiabao will be the US president in 2016.

A is incorrect. To disjoin one proposition with another is to connect those two propositions with the connective "or." The conclusion of A is not the result of disjoining a proposition with its premise.

B is correct. The conclusion of B is the result of disjoining a proposition with its premise.

C is incorrect. The conclusion of C is not the result of disjoining a proposition with its premise.

D is incorrect. The conclusion of D is not the result of disjoining a proposition with its premise. The conclusion of D validly follows from its premises, but the conclusion is not the result of disjoining a proposition with any of the premises.

E is incorrect. Only one of the arguments stated in A-D is a disjunction introduction argument.

A disjunction elimination argument is a valid argument in which the conclusion is a disjunct of some disjunction that appears in the premises. Which of the following arguments is a disjunction elimination argument?

Romney or Obama will be the US president in 2013.

Obama will not be the US president in 2013.

______________________________________________

Therefore, Romney will be the US president in 2013.

A is incorrect. Although the conclusion of A is a disjunct of a disjunction that appears in its premises, A is not a valid argument. From the fact that Yamamoto or Jiabao will be the US president in 2016, it does not follow that Yamamoto will be the US president in 2016.

B is incorrect. A disjunction elimination argument is a valid argument, whose conclusion is a disjunct of some disjunction that appears in its premises. B only has one premise, and it is not a disjunction. So B cannot be a disjunction elimination argument.

C is incorrect. A disjunction elimination argument is a valid argument, whose conclusion is a disjunct of some disjunction that appears in its premises. C only has one premise, and it is not a disjunction. So C cannot be a disjunction elimination argument.

D is correct. Only D is valid, while at the same time having a conclusion that is a disjunct of a disjunction in one of its premises. If Yamamoto or Jiabao will be the US president in 2016, and if Jiabao will not be the US president in 2016, then it follows that Yamamoto will be the US president in 2013.

E is incorrect. Only one of the arguments among A-D is a disjunction elimination argument.

Obama will not be the US president in 2013.

______________________________________________

Therefore, Romney will be the US president in 2013.

A is incorrect. Although the conclusion of A is a disjunct of a disjunction that appears in its premises, A is not a valid argument. From the fact that Yamamoto or Jiabao will be the US president in 2016, it does not follow that Yamamoto will be the US president in 2016.

B is incorrect. A disjunction elimination argument is a valid argument, whose conclusion is a disjunct of some disjunction that appears in its premises. B only has one premise, and it is not a disjunction. So B cannot be a disjunction elimination argument.

C is incorrect. A disjunction elimination argument is a valid argument, whose conclusion is a disjunct of some disjunction that appears in its premises. C only has one premise, and it is not a disjunction. So C cannot be a disjunction elimination argument.

D is correct. Only D is valid, while at the same time having a conclusion that is a disjunct of a disjunction in one of its premises. If Yamamoto or Jiabao will be the US president in 2016, and if Jiabao will not be the US president in 2016, then it follows that Yamamoto will be the US president in 2013.

E is incorrect. Only one of the arguments among A-D is a disjunction elimination argument.

Suppose you had to fill in the rightmost column of the truth table for disjunction:

p.....q..........pVq

T.....T..........

T.....F..........

F.....T..........

F.....F..........

Going from top to bottom, how would you fill in that column?

p.....q..........pVq

T.....T..........

T.....F..........

F.....T..........

F.....F..........

Going from top to bottom, how would you fill in that column?

TTTF

A is correct. A disjunction is true exactly when at least one of its disjuncts is true, and it is only false when all of its disjuncts are false.

A is correct. A disjunction is true exactly when at least one of its disjuncts is true, and it is only false when all of its disjuncts are false.

Now suppose you had to fill in the rightmost column of the truth table for disjunction:

p.....q..........pVq

F.....F..........

T.....T..........

F.....T..........

T.....F..........

Now, going from top to bottom, how would you fill in that column?

p.....q..........pVq

F.....F..........

T.....T..........

F.....T..........

T.....F..........

Now, going from top to bottom, how would you fill in that column?

FTTT

D is correct. A disjunction is true exactly when at least one of its disjuncts is true, and is only false when both of its disjuncts are false.

D is correct. A disjunction is true exactly when at least one of its disjuncts is true, and is only false when both of its disjuncts are false.

And now suppose you had to fill in the rightmost column of the truth table for disjunction:

p.....q..........pVq

F.....F..........

F.....T..........

T.....T..........

T.....F..........

Now, going from top to bottom, how would you fill in that column?

p.....q..........pVq

F.....F..........

F.....T..........

T.....T..........

T.....F..........

Now, going from top to bottom, how would you fill in that column?

FTTT

D is correct. A disjunction is true exactly when at least one of its disjuncts is true, and is only false when both of its disjuncts are false.

D is correct. A disjunction is true exactly when at least one of its disjuncts is true, and is only false when both of its disjuncts are false.

Which of the following asserts the negation of a proposition?

I do not like green eggs and ham.

The proposition asserted by C is the negation of the claim, "I like green eggs and ham." It reads, "it is NOT the case that I like green eggs and ham."

The proposition asserted by C is the negation of the claim, "I like green eggs and ham." It reads, "it is NOT the case that I like green eggs and ham."

Which of the following asserts the negation of a proposition?

Neither Dali nor Picasso was a musician.

The proposition asserted by E is the following: "It is NOT the case that EITHER Dali OR Picasso was a musician."

The proposition asserted by E is the following: "It is NOT the case that EITHER Dali OR Picasso was a musician."

Which of the following arguments is valid?

It's not true that Joe is a plumber.

-------------------

Therefore, Joe is a non-plumber.

From the falsity of the claim that Joe is a plumber, it follows that Joe is a non-plumber.

-------------------

Therefore, Joe is a non-plumber.

From the falsity of the claim that Joe is a plumber, it follows that Joe is a non-plumber.

Which of the following arguments is valid?

It is not true that Joe is a plumber.

_____________________

Therefore, Joe is a non-plumber.

A is incorrect. From the proposition, "Joe is not a plumber," it does not follow that it is not true that Joe is a non-plumber.

B is incorrect. From the two propositions, "either Joe is a plumber or Bob is a builder" and "Joe is not a plumber," it does not follow that Bob is not a builder.

C is correct. If the proposition, "Joe is a plumber," is not true, it follows that Joe is a non-plumber.

D is incorrect. The disjunction, "either Joe is a plumber and Bob is a builder or both Joe and Bob are plumbers," does not imply that Bob is not a builder.

E is incorrect. One of the arguments among A-D is valid.

_____________________

Therefore, Joe is a non-plumber.

A is incorrect. From the proposition, "Joe is not a plumber," it does not follow that it is not true that Joe is a non-plumber.

B is incorrect. From the two propositions, "either Joe is a plumber or Bob is a builder" and "Joe is not a plumber," it does not follow that Bob is not a builder.

C is correct. If the proposition, "Joe is a plumber," is not true, it follows that Joe is a non-plumber.

D is incorrect. The disjunction, "either Joe is a plumber and Bob is a builder or both Joe and Bob are plumbers," does not imply that Bob is not a builder.

E is incorrect. One of the arguments among A-D is valid.

Which of the following arguments is valid?

Walter is a teacher.

Joe is a plumber.

Plumbers are not teachers.

_______________________

Therefore, Joe is not Walter.

A is correct. From the three propositions, "plumbers are not teachers," "Joe is a plumber," and "Walter is a teacher," it follows that Joe is not Walter. To see why, suppose that the three premises are true, and that "Joe is not Walter" is false. If Joe is Walter, then from the truth of "Joe is a plumber" and "Walter is a teacher," it would follow that there is a plumber, Joe, who was also a teacher. This would contradict the other premise, however, which is the proposition "plumbers are not teachers." So if all three premises are true, it must follow that Joe is not Walter.

B is incorrect. From the three propositions, "plumbers are sometimes not teachers," "Joe is a plumber," and "Walter is a teacher," it does not follow that Joe is not a teacher. Joe, who is a plumber, may also be a teacher, even though plumbers are sometimes not teachers.

C is incorrect. From the three propositions, "someone who is not a plumber might also not be a teacher," "Joe is not a plumber," and "Walter is not a teacher," it does not follow that Joe is Walter. Joe and Walter could be two different people, even though one is not a teacher, while the other is not a plumber.

D is incorrect. From the two propositions, "either Walter is not a teacher or Joe is not a plumber" and "it is not true that Joe is not a plumber," it does not follow that Walter is a plumber. Walter might be neither a teacher not a plumber.

E is incorrect. One of the arguments stated in A-D is valid.

Joe is a plumber.

Plumbers are not teachers.

_______________________

Therefore, Joe is not Walter.

A is correct. From the three propositions, "plumbers are not teachers," "Joe is a plumber," and "Walter is a teacher," it follows that Joe is not Walter. To see why, suppose that the three premises are true, and that "Joe is not Walter" is false. If Joe is Walter, then from the truth of "Joe is a plumber" and "Walter is a teacher," it would follow that there is a plumber, Joe, who was also a teacher. This would contradict the other premise, however, which is the proposition "plumbers are not teachers." So if all three premises are true, it must follow that Joe is not Walter.

B is incorrect. From the three propositions, "plumbers are sometimes not teachers," "Joe is a plumber," and "Walter is a teacher," it does not follow that Joe is not a teacher. Joe, who is a plumber, may also be a teacher, even though plumbers are sometimes not teachers.

C is incorrect. From the three propositions, "someone who is not a plumber might also not be a teacher," "Joe is not a plumber," and "Walter is not a teacher," it does not follow that Joe is Walter. Joe and Walter could be two different people, even though one is not a teacher, while the other is not a plumber.

D is incorrect. From the two propositions, "either Walter is not a teacher or Joe is not a plumber" and "it is not true that Joe is not a plumber," it does not follow that Walter is a plumber. Walter might be neither a teacher not a plumber.

E is incorrect. One of the arguments stated in A-D is valid.

Which of the following arguments is valid?

None of the above.

A is incorrect. From the two propositions, "Joe is not John" and "John is not a plumber," it does not follow that Joe is not a plumber. Joe and John could be two different plumbers.

B is incorrect. From the two propositions, "James and Susan are not related" and "Susan and Alice are not related," it does not follow that James and Alice are not related. James and Alice could be related to each other, even though neither one is related to Susan.

C is incorrect. From the two propositions, "James is not a nonconformist" and "either Susan is not a nonconformist or James is a conformist," it does not follow that Susan is not a nonconformist. Susan could be a nonconformist while James, who is a conformist, is also not a nonconformist.

D is incorrect. From the four propositions, "either Susan is a conformist or James is not a nonconformist," "either Alice is a nonconformist or Susan is not a nonconformist," "either James is a conformist or Susan is not a nonconformist," and "Alice is not a nonconformist or she is not," it does not follow that Susan is a conformist. Susan could be conformist while Alice is a nonconformist, and while James, who is not a nonconformist, is a conformist.

E is correct. None of the arguments in A-D have conclusions that logically follow from their premises.

A is incorrect. From the two propositions, "Joe is not John" and "John is not a plumber," it does not follow that Joe is not a plumber. Joe and John could be two different plumbers.

B is incorrect. From the two propositions, "James and Susan are not related" and "Susan and Alice are not related," it does not follow that James and Alice are not related. James and Alice could be related to each other, even though neither one is related to Susan.

C is incorrect. From the two propositions, "James is not a nonconformist" and "either Susan is not a nonconformist or James is a conformist," it does not follow that Susan is not a nonconformist. Susan could be a nonconformist while James, who is a conformist, is also not a nonconformist.

D is incorrect. From the four propositions, "either Susan is a conformist or James is not a nonconformist," "either Alice is a nonconformist or Susan is not a nonconformist," "either James is a conformist or Susan is not a nonconformist," and "Alice is not a nonconformist or she is not," it does not follow that Susan is a conformist. Susan could be conformist while Alice is a nonconformist, and while James, who is not a nonconformist, is a conformist.

E is correct. None of the arguments in A-D have conclusions that logically follow from their premises.

Suppose you had to fill in the rightmost column of the following truth-table:

p.....q..........p&q..........-(p&q)

T.....T

T.....F

F.....T

F.....F

Going from top to bottom, how would you fill in that column?

p.....q..........p&q..........-(p&q)

T.....T

T.....F

F.....T

F.....F

Going from top to bottom, how would you fill in that column?

FTTT

E is correct. A conjunction is only true if both of its conjuncts are true, and it is false otherwise. The negation of a conjunction, therefore, will only be false when both of the conjuncts are true, and will be true otherwise.

E is correct. A conjunction is only true if both of its conjuncts are true, and it is false otherwise. The negation of a conjunction, therefore, will only be false when both of the conjuncts are true, and will be true otherwise.

Suppose you had to fill in the rightmost column of the following truth-table:

p.....q..........pVq..........-(pVq)

T.....T

T.....F

F.....T

F.....F

Going from top to bottom, how would you fill in that column?

p.....q..........pVq..........-(pVq)

T.....T

T.....F

F.....T

F.....F

Going from top to bottom, how would you fill in that column?

FFFT

E is correct. A disjunction is only false if both of its disjuncts are false, and it is true otherwise. The negation of a disjunction, therefore, will only be true when both of the disjuncts are false, and will be false otherwise.

E is correct. A disjunction is only false if both of its disjuncts are false, and it is true otherwise. The negation of a disjunction, therefore, will only be true when both of the disjuncts are false, and will be false otherwise.

Suppose you had to fill in the rightmost column of the following truth-table:

p.....q.....-p.....-q..........-pV-q

T.....T

T.....F

F.....T

F.....F

Going from top to bottom, how would you fill in that column?

p.....q.....-p.....-q..........-pV-q

T.....T

T.....F

F.....T

F.....F

Going from top to bottom, how would you fill in that column?

FTTT

E is correct. A disjunction is only false if both of its disjuncts are false, and it is true otherwise. A disjunction of two negations, therefore, will only be false if both of the disjuncts, without their negations, are true. So the disjunction, "-pV-q" will only be false if both p and q are true.

E is correct. A disjunction is only false if both of its disjuncts are false, and it is true otherwise. A disjunction of two negations, therefore, will only be false if both of the disjuncts, without their negations, are true. So the disjunction, "-pV-q" will only be false if both p and q are true.

Suppose you had to fill in the rightmost column of the following truth-table:

p.....q.....r.....pVq.....-(pVq).....-r.....-(pVq)&(-r)

T.....T.....T

T.....T.....F

T.....F.....T

T.....F.....F

F.....T.....T

F.....T.....F

F.....F.....T

F.....F.....F

Going from top to bottom, how would you fill in that column?

p.....q.....r.....pVq.....-(pVq).....-r.....-(pVq)&(-r)

T.....T.....T

T.....T.....F

T.....F.....T

T.....F.....F

F.....T.....T

F.....T.....F

F.....F.....T

F.....F.....F

Going from top to bottom, how would you fill in that column?

FFFFFFFT

A is correct. The rightmost proposition, -(pVq)&(-r), is a conjunction. It can only be true if both the conjuncts are true. For the first conjunct to be true, the disjunction "-(pVq)" has to be true, which means that "pVq" has to be false. For "pVq" to be false, however, both p and q have to be false, since a disjunction is only false when both of its disjuncts are. So if the rightmost proposition is true, then both p and q have to be false. In addition, however, for the rightmost proposition to be true, the second conjunct, "-r," also has to be true. For "-r" to be true, however, r has to be false. So, in order for the rightmost proposition to be true, p, q, and r all have to be false. The whole rightmost proposition is true exactly when p, q and r are all false; and the whole proposition is false otherwise.

A is correct. The rightmost proposition, -(pVq)&(-r), is a conjunction. It can only be true if both the conjuncts are true. For the first conjunct to be true, the disjunction "-(pVq)" has to be true, which means that "pVq" has to be false. For "pVq" to be false, however, both p and q have to be false, since a disjunction is only false when both of its disjuncts are. So if the rightmost proposition is true, then both p and q have to be false. In addition, however, for the rightmost proposition to be true, the second conjunct, "-r," also has to be true. For "-r" to be true, however, r has to be false. So, in order for the rightmost proposition to be true, p, q, and r all have to be false. The whole rightmost proposition is true exactly when p, q and r are all false; and the whole proposition is false otherwise.

Suppose you had to fill in the rightmost column of the following truth-table:

p q p⊃q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

p q p⊃q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

D. TFTT A conditional is only false when its antecedent is true and its consequent is false. It is true otherwise.

Suppose you had to fill in the rightmost column of the following truth-table:

p q pVq (pVq)⊃q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

p q pVq (pVq)⊃q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

D. TFTT A conditional is only false when its antecedent is true and its consequent is false. It is true otherwise. If the antecedent and the consequent are both false, then the whole conditional is still true.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p&q -(p&q) r⊃-(p&q)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill in that column?

p q r p&q -(p&q) r⊃-(p&q)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill in that column?

D. FTTTTTTT For the rightmost proposition to be false, its antecedent, r, has to be true while its consequent, -(p&q), has to be false. Otherwise, the rightmost proposition will come true. For the consequent to be false, however, the conjunction p&q must be true, which in turn requires both p to be true and q to be true. So it turns out that there is only one way for the rightmost proposition to be false. The rightmost proposition is only false when p, q, and r are all true. Otherwise, the rightmost proposition is true.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r q⊃p (q⊃p)⊃r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill in that column?

p q r q⊃p (q⊃p)⊃r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill in that column?

E. none of the above In order for the rightmost proposition to be false, the antecedent must be true and the consequent must be false. The antecedent is q⊃p, and the consequent is r. The antecedent, q⊃p, is only false, however, if q is true and p is false. Otherwise, the antecedent is true. So in order for the rightmost proposition to be false, r has to be false, and, in addition, either q has to be false or p has to be true. If r is true, or if q is true while p is false, then the rightmost proposition is true. Otherwise, the rightmost proposition is false. The correct answer should be "TFTFTTTF," which is not listed in (a)-(d).

Which of the following asserts a material conditional?

If John is not in his office, then he's at home.

A material conditional contains two complete propositions, which are separated by the English words "if... then." The proposition expressed by (a) is "If John is not in his office, then John is at home."

A material conditional contains two complete propositions, which are separated by the English words "if... then." The proposition expressed by (a) is "If John is not in his office, then John is at home."

Which of the following asserts a material conditional?

The teams will play this season only if the strike ends.

A material conditional contains two complete propositions, which are separated by the English words "if... then." The proposition expressed by (d) is "If the teams will play this season, then the strike ends."

A material conditional contains two complete propositions, which are separated by the English words "if... then." The proposition expressed by (d) is "If the teams will play this season, then the strike ends."

Which of the following arguments is valid?

If John goes, then Jim will stay.

John will go.

Jim will stay.

If an argument contains both a conditional premise, as well as a premise that is the conditional's antecedent, then it is valid to infer that conditional's consequent. It is NOT valid to infer the antecedent of a conditional from the conditional and its consequent, however.

John will go.

Jim will stay.

If an argument contains both a conditional premise, as well as a premise that is the conditional's antecedent, then it is valid to infer that conditional's consequent. It is NOT valid to infer the antecedent of a conditional from the conditional and its consequent, however.

Which of the following arguments is valid?

If you are not rich, then you are sad.

You are not sad.

Therefore, you are rich.

If an argument contains a conditional as one of its premises, along with the negation of that conditional's consequent, then it is valid to infer the negation of the antecedent.

You are not sad.

Therefore, you are rich.

If an argument contains a conditional as one of its premises, along with the negation of that conditional's consequent, then it is valid to infer the negation of the antecedent.

Which of the following asserts a biconditional?

He wins if, and only if, every other player loses.

A biconditional, as the name suggests, links two propositions with a pair of material conditionals, so that each one is the material antecedent, and the material consequent, of the other. A biconditional, in other words, uses the phrase "if and only if." Other, equivalent phrases include "exactly when" and "just in case."

A biconditional, as the name suggests, links two propositions with a pair of material conditionals, so that each one is the material antecedent, and the material consequent, of the other. A biconditional, in other words, uses the phrase "if and only if." Other, equivalent phrases include "exactly when" and "just in case."

Suppose you had to fill in the rightmost column of the following truth-table:

p q p≡q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

p q p≡q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

TFFT

A biconditional is only true when both of its propositions have the same truth value. If a biconditional's antecedent and consequent are both true, for instance, or if they are both false, then the whole biconditional is true. Otherwise, the whole biconditional is false.

A biconditional is only true when both of its propositions have the same truth value. If a biconditional's antecedent and consequent are both true, for instance, or if they are both false, then the whole biconditional is true. Otherwise, the whole biconditional is false.

Suppose you had to fill in the rightmost column of the following truth-table:

p q p&q -p -p≡(p&q)

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

p q p&q -p -p≡(p&q)

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

FTFF

For a biconditional to be true, both of its component propositions must have the same truth-value. So the only way for the rightmost proposition to be true, then, is if -p and p&q have the same truth-value. If -p and p&q are both true, then, or both false, then the biconditional -p≣(p&q) will also be true. Otherwise, the biconditional will be false. There is no way for both sides of the biconditional to be true, however, since the truth of -p requires the falsity of p&q. So the only way for the rightmost proposition to come out true is for both sides of the biconditional to come out false. There is only one truth-value assignment, however, that renders both sides of the biconditional false: it is when p is true and q is false. The answer, then, should be "FTFF."

For a biconditional to be true, both of its component propositions must have the same truth-value. So the only way for the rightmost proposition to be true, then, is if -p and p&q have the same truth-value. If -p and p&q are both true, then, or both false, then the biconditional -p≣(p&q) will also be true. Otherwise, the biconditional will be false. There is no way for both sides of the biconditional to be true, however, since the truth of -p requires the falsity of p&q. So the only way for the rightmost proposition to come out true is for both sides of the biconditional to come out false. There is only one truth-value assignment, however, that renders both sides of the biconditional false: it is when p is true and q is false. The answer, then, should be "FTFF."

Some valid arguments have true premises and a true conclusion.

True. Consider this argument: "Our solar system contains fewer than seventy-eight planets. Therefore, our solar system contains fewer than seventy-nine planets." The premise and conclusion are both true. The argument is valid because it is not possible for the conclusion to be false when the premise is true.

Some valid arguments have true premises and a false conclusion.

False. If the premises are actually true and the conclusion is actually false, then it is at least possible that the premises are true and the conclusion is false. So the argument cannot be valid, according to our definition of validity.

Some valid arguments have false premises and a true conclusion.

True. Consider this argument: "Our solar system contains fewer than five planets. Therefore, our solar system contains fewer than fifty planets." The premise is false, and conclusion is true. Moreover, the argument is valid because it is not possible for the conclusion to be false when the premise is true.

Some valid arguments have false premises and a false conclusion.

True. Consider this argument: "Our solar system contains fewer than four planets. Therefore, our solar system contains fewer than five planets." The premise and conclusion are both false. Nonetheless, this argument is still valid because it is not possible for the conclusion to be false when the premise is true.

Most professors agree that they are paid too little, so they are.

Invalid argument. It is possible for the premise to be true and the conclusion false if most professors believe that they deserve to be paid more than they really do deserve to be paid. This possibility shows that the argument is invalid.

Susan is smart and strong, so she is smart.

Valid argument. It is not possible for anyone to have both of two qualities without having the first of the two qualities. Notice that it does not matter who Susan is.

Sara is either smart or strong, so she is smart.

Invalid argument. If Sara is strong but smart, then the premise is true when the conclusion is false. This possibility shows that the argument is invalid.

Washington is in the United States. I live in Washington. So I live in the United States.

Valid argument. There is no possible way for the conclusion to be false when both premises are true. Notice that I can know this relation between the premises and conclusion, even if I do not know whether the premises are true, and even if I do not know whether the argument is about Washington State or Washington, DC.

Washington is in the United States. I live in the United States. So I live in Washington.

Invalid argument. If I live in Texas, then both premises are true, but the conclusion is false. This possibility shows that the argument is invalid.

Washington is in the United Kingdom. I live in Washington. So I live in the United Kingdom.

Valid argument. There is no possible way for the conclusion to be false when both premises are true. It does not matter that the first premise is actually false, because validity depends on what is possible.

There is no largest six-digit number, because six-digit numbers are numbers and there is no largest number.

Invalid argument. The premises are both true and the conclusion is false. That shows that it is possible for the premises to be true and the conclusion false. Hence, the argument must be invalid, even if it is not clear why.

Every argument with a true conclusion is sound.

False

Every argument with a false conclusion is unsound.

True

Every argument that is sound has a true conclusion.

True

Every argument that is unsound has a false conclusion.

False

All my children are teenagers. All teenagers are students. Therefore, All my children are students.

valid and sound

All my children are students. All teenagers are students. Therefore, All my children are teenagers.

invalid and unsound

All teenagers are my children. All my children are students. Therefore, All teenagers are students.

valid but unsound

Philadelphia is RICH IN HISTORY, but it is not now the capital of the United States, so the United States Congress must meet somewhere else.

Remove. The fact that Philadelphia has a rich past is irrelevant to the conclusion about where the United States Congress meets at present.

Not everybody whom you invited is going to come to your party. SOME OF THEM WONT COME. So this room should be big enough.

Remove. To say that some won't come is the same as to say that not all of them will come. Thus, the second sentence repeats the first, so it can be removed without weakening the argument.

I know that my wife is at home, since I just called her there and spoke to her. WE TALKED ABOUT OUR DINNER PLANS.

Remove. The fact that the husband and wife talked about dinner plans is an irrelevant tangent. What matters to the argument is only that she spoke from home, not what they spoke about.

SOME students could not concentrate on the lecture, because they did not eat lunch before class.

Do not remove. The guarding term "some" cannot be dropped in this case because it is an essential part of the argument and the speaker is not asserting that "all" or "most" of the students could not concentrate.

THE MOST SURPRISING NEWS OF ALL is that Johnson dropped out of the race because he thought his opponent was better qualified than he was for the office.

Remove. Whether or not such news is surprising does not change the truth of the premises or the force of the argument, and thus it can be dropped.

The challenger in this election is likely to win, since EXPERTS AGREE that more women support him.

Remove. The assuring phrase "experts agree that" can be dropped because it is not their agreement that makes the Democrat likely to win. It is the support of the women.

Married people are happier, so marriage must be a good thing, OR AT LEAST I THINK SO.

Remove. The guarding phrase "or at least I think so" can be dropped, because what makes marriage good is that it makes people happy, not whether the speaker thinks so.

James is on the chess and football teams. Football is a team sport. So James plays a team sport.

Can the first premise be broken up into (a) and (b)?

(a) James is on the chess team.

(b) James is on the football team.

(c) Football is a team sport.

_____________________________

∴ (d) James plays a team sport.

Can the first premise be broken up into (a) and (b)?

(a) James is on the chess team.

(b) James is on the football team.

(c) Football is a team sport.

_____________________________

∴ (d) James plays a team sport.

Yes. To say that James is on both teams is to say both that he is on the chess team and also that he is on the football team. The argument is not distorted by breaking these parts into two separate premises. Indeed, breaking them up shows that only one of them (the one about football) is really relevant to the conclusion.

Mary is either a junior or a senior, so she is almost ready to graduate.

Can this premise be broken up into (a) and (b)?

(a) Mary is a junior.

(b) Mary is a senior.

________________________________

∴ (c) Mary is almost ready to graduate.

Can this premise be broken up into (a) and (b)?

(a) Mary is a junior.

(b) Mary is a senior.

________________________________

∴ (c) Mary is almost ready to graduate.

No. The person giving this argument does not claim that Mary is a junior and also does not claim that Mary is a senior. All the arguer claims is that Mary is either one or the other. Thus, it distorts the argument to include these claims as separate premises.

Mercury is the only common metal that is liquid at room temperature, so a pound of mercury would be liquid in this room.

Can this premise be broken up into (a) and (b)?

(a) Mercury is the only common metal.

(b) Mercury is liquid at room temperature.

____________________________________________

∴ (c) a pound of mercury would be liquid in this room.

Can this premise be broken up into (a) and (b)?

(a) Mercury is the only common metal.

(b) Mercury is liquid at room temperature.

____________________________________________

∴ (c) a pound of mercury would be liquid in this room.

No. Breaking up the premise in in this way turns a true premise ("Mercury is the only common metal that is liquid at room temperature") into a set of premises with one member that is false ("Mercury is the only common metal"). Thus, this division distorts and weakens the argument.

Since he won the lottery, he's rich and lucky, so he'll probably do well in the stock market, too, unless his luck runs out.

Can this conclusion be broken up into (c) and (d)?

(a) He won the lottery.

______________________

∴ (b) He's rich and lucky.

(c) His luck will not run out.

_______________________________________

∴ (d) He'll probably do well in the stock market.

Can this conclusion be broken up into (c) and (d)?

(a) He won the lottery.

______________________

∴ (b) He's rich and lucky.

(c) His luck will not run out.

_______________________________________

∴ (d) He'll probably do well in the stock market.

No. The arguer does not actually say that his luck will not run out, so it distorts the argument to include premise (c).

Since many newly emerging nations do not have the capital resources that are necessary for sustained growth, they will continue to need help from industrial nations to avoid mass starvation.

Can this premise be broken up into (a) and (b)?

(a) Adequate capital resources are necessary for sustained growth.

(b) Many newly emerging nations do not have adequate capital resources.

___________________________________________________________________________________________

∴ (c) Many newly emerging nations will continue to need help from industrial nations to avoid mass starvation.

Can this premise be broken up into (a) and (b)?

(a) Adequate capital resources are necessary for sustained growth.

(b) Many newly emerging nations do not have adequate capital resources.

___________________________________________________________________________________________

∴ (c) Many newly emerging nations will continue to need help from industrial nations to avoid mass starvation.

Yes. Here (a) and (b) make independent claims, since either one can be true without the other being true, so it does not distort the argument to separate them.

ARGUMENT: I know Pat can't be a father, because she is not a male. So she can't be a grandfather either.

(1) Pat is not male.

_____________________

∴ (2) Pat can't be a father. (from 1)

_____________________

∴ (3) Pat can't be a grandfather. (from 2)

Arrangement (A) fails to break up the premise with the argument marker "because." Arrangements (A) and (B) both fail to capture the role of "Pat is not male" as a premise to support "Pat can't be a father" in the first sentence. Arrangement (D) suggests that "Pat can't be a grandfather" is a reason to believe "Pat can't be a father," but that argument is not valid.

_____________________

∴ (2) Pat can't be a father. (from 1)

_____________________

∴ (3) Pat can't be a grandfather. (from 2)

Arrangement (A) fails to break up the premise with the argument marker "because." Arrangements (A) and (B) both fail to capture the role of "Pat is not male" as a premise to support "Pat can't be a father" in the first sentence. Arrangement (D) suggests that "Pat can't be a grandfather" is a reason to believe "Pat can't be a father," but that argument is not valid.

Which kind of structure does the argument above have?

linear structure. In Answer (C) of the previous question, (1) is a reason for (2) which is in turn a reason for (3).

ARGUMENT: Our team can't win this Saturday, both because they are no good and also because they are not going to play this Saturday.

(1) Our team is no good.

_____________________________

∴ (2) Our team can't win this Saturday. (from 1)

(3) Our team is not going to play this Saturday.

_____________________________

∴ (2) Our team can't win this Saturday. (from 3)

Arrangement (A) suggests that both premises are needed to work together to support the conclusion, whereas actually each premise by itself is enough to support the conclusion. Arrangement (B) suggests that "Our team is no good" is a reason to believe "Our team is not going to play this Saturday," but that can't be right, because even bad teams play sometimes on Saturday (unless it is the playoffs, which is not mentioned). Similarly, arrangement (C) suggests that "Our team is not going to play this Saturday" is a reason to believe "Our team is no good," but that can't be right, because even good teams do not play every Saturday.

_____________________________

∴ (2) Our team can't win this Saturday. (from 1)

(3) Our team is not going to play this Saturday.

_____________________________

∴ (2) Our team can't win this Saturday. (from 3)

Arrangement (A) suggests that both premises are needed to work together to support the conclusion, whereas actually each premise by itself is enough to support the conclusion. Arrangement (B) suggests that "Our team is no good" is a reason to believe "Our team is not going to play this Saturday," but that can't be right, because even bad teams play sometimes on Saturday (unless it is the playoffs, which is not mentioned). Similarly, arrangement (C) suggests that "Our team is not going to play this Saturday" is a reason to believe "Our team is no good," but that can't be right, because even good teams do not play every Saturday.

What kind of structure does the argument in the question above have?

branching structure. In Answer (D) in the previous question, (1) is a reason for (2), and (3) is another reason for the same conclusion.

ARGUMENT: Either Jack is a fool or Mary is a crook, because she ended up with all of his money.

(1) Mary ended up with all of Jack's money.

_____________________________

∴ (2) Either Jack is a fool or Mary is a crook. (from 1)

The word "because" is a premise marker instead of a conclusion marker, so arrangement (B) cannot be accurate. Arrangements (C) and (D) break up "Either Jack is a fool or Mary is a crook" into the two parts "Jack is a fool" and "Mary is a crook," but either-or sentences should not be broken into parts like this.

_____________________________

∴ (2) Either Jack is a fool or Mary is a crook. (from 1)

The word "because" is a premise marker instead of a conclusion marker, so arrangement (B) cannot be accurate. Arrangements (C) and (D) break up "Either Jack is a fool or Mary is a crook" into the two parts "Jack is a fool" and "Mary is a crook," but either-or sentences should not be broken into parts like this.

What kind of structure does the argument in the question above have?

None of the above. If we do not break the either-or sentence into parts, then there is only one premise and one conclusion, so the structure cannot be linear, branching, or joint.

ARGUMENT: Gold is a metal, so it must conduct electricity, since all metals do.

(1) Gold is a metal.

(2) All metals conduct electricity.

____________________

∴ (3) Gold must conduct electricity. (from 1-2)

Arrangement (A) cannot be accurate because the conclusion contains an argument marker "since" that joins two parts that need to be separated. Arrangement (B) cannot be accurate because the fact that gold is a metal is no reason to believe that ALL metals conduct electricity in the step from (1) to (2) in (B). Arrangement (C) cannot be accurate because the premise "Gold is a metal" would not be enough by itself to show that "Gold must conduct electricity" if it were not also true that "All metals conduct electricity." Similarly, the premise "All metals conduct electricity" would not be enough by itself to show that "Gold must conduct electricity" if it were not also true that "Gold is a metal." The two premises need to work together.

(2) All metals conduct electricity.

____________________

∴ (3) Gold must conduct electricity. (from 1-2)

Arrangement (A) cannot be accurate because the conclusion contains an argument marker "since" that joins two parts that need to be separated. Arrangement (B) cannot be accurate because the fact that gold is a metal is no reason to believe that ALL metals conduct electricity in the step from (1) to (2) in (B). Arrangement (C) cannot be accurate because the premise "Gold is a metal" would not be enough by itself to show that "Gold must conduct electricity" if it were not also true that "All metals conduct electricity." Similarly, the premise "All metals conduct electricity" would not be enough by itself to show that "Gold must conduct electricity" if it were not also true that "Gold is a metal." The two premises need to work together.

What kind of structure does the argument in the question above have?

joint structure. The premise "Gold is a metal" would not be enough by itself to show that "Gold must conduct electricity" if it were not also true that "All metals conduct electricity." Similarly, the premise "All metals conduct electricity" would not be enough by itself to show that "Gold must conduct electricity" if it were not also true that "Gold is a metal." Thus, the argument needs both premises to work together in order to support the conclusion. That mutual dependence makes this a joint structure.

CONTEXT: You and Susan are working on a project together, and she is not pulling her weight.

ARGUMENT: Susan refuses to work on Sundays, which shows that she is lazy and inflexible.

ARGUMENT: Susan refuses to work on Sundays, which shows that she is lazy and inflexible.

Anyone who refuses to work on Sundays is both lazy and inflexible. The answer is (B) Anyone who refuses to work on Sundays is both lazy and inflexible.

(D) is not enough to make the argument valid, because it is too vague. (A) is also not enough to make the argument valid, because the conclusion says that Susan is BOTH lazy AND inflexible. However, suppressed premises are supposed to make the argument valid if possible. (C) is stronger than is necessary to make the argument valid, because it is about ALL days, not just Sundays. However, suppressed premises should not claim more than is needed to make the argument valid.

(D) is not enough to make the argument valid, because it is too vague. (A) is also not enough to make the argument valid, because the conclusion says that Susan is BOTH lazy AND inflexible. However, suppressed premises are supposed to make the argument valid if possible. (C) is stronger than is necessary to make the argument valid, because it is about ALL days, not just Sundays. However, suppressed premises should not claim more than is needed to make the argument valid.

CONTEXT: You are trying to decide whether to visit a sick friend in the hospital.

ARGUMENT: You promised to visit her, so you should visit her.

ARGUMENT: You promised to visit her, so you should visit her.

You should keep your promises.(A) is enough to make the argument valid and show how and why the premise supports the conclusion. (B) and (C) might be true, but they do not mention promises, and this explicit argument is about promises. (D) is clearly false, but we should try to avoid adding clearly false suppressed premises.

CONTEXT: We want to figure out whether Mildred is older than the other candidate for a job.

ARGUMENT: Mildred must be over forty-three, since she has a daughter who is thirty-six years old.

ARGUMENT: Mildred must be over forty-three, since she has a daughter who is thirty-six years old.

Parents must be more than seven years older than their daughters. The answer is (D) Parents must be more than seven years older than their daughters.

(A) is clearly false, but we should avoid adding clearly false suppressed premises if possible. (B) is too vague to make the argument valid, but suppressed premises should make the argument valid if possible. (C) claims more than is needed to make the argument valid, but suppressed premises should not claim more than is needed to make the argument valid.

(A) is clearly false, but we should avoid adding clearly false suppressed premises if possible. (B) is too vague to make the argument valid, but suppressed premises should make the argument valid if possible. (C) claims more than is needed to make the argument valid, but suppressed premises should not claim more than is needed to make the argument valid.

CONTEXT: A boat sank, and authorities have been searching for survivors for over a week.

ARGUMENT: There must not be any survivors, since, if there were any, they would have been found by now.

ARGUMENT: There must not be any survivors, since, if there were any, they would have been found by now.

No survivors have been found by now. The answer is (C) No survivors have been found by now.

(A) approximately repeats the conclusion, so it cannot show the path from the explicit premise to the conclusion, as a suppressed premise should. (B) Plus, the explicit premise would not support the conclusion. Indeed, (B) would refute that conclusion. (D) is clearly false, but we should avoid adding clearly false suppressed premises, if possible.

(A) approximately repeats the conclusion, so it cannot show the path from the explicit premise to the conclusion, as a suppressed premise should. (B) Plus, the explicit premise would not support the conclusion. Indeed, (B) would refute that conclusion. (D) is clearly false, but we should avoid adding clearly false suppressed premises, if possible.

CONTEXT: You are listening to a lecture in a traditional college classroom, but one person in the audience looks older than the rest. Her name is Mary.

ARGUMENT: Mary can't be a student, because she is a professor, and professors must already have degrees.

ARGUMENT: Mary can't be a student, because she is a professor, and professors must already have degrees.

Students don't already have degrees. The answer is (D) Students don't already have degrees.

(A) and (B) do not make the argument valid, because the argument is not about age. Suppressed premises are supposed to show the path from the explicit premises to the conclusion instead of creating a different path to that conclusion. (C) is enough to get us straight from the premise "Mary is a professor" to the conclusion "Mary can't be a student," but then it cannot explain why the arguer went on to add "Professors must already have degrees," so it misses part of the argument. (D) enables us to understand all parts of the argument: Mary can't be a student, because she is a professor, and professors must already have degrees. Then: Students don't already have degrees, so Mary can't be a student. Notice that (D) is false, but it still might be the most plausible suppressed premise to make sense of what the arguer probably had in mind.

(A) and (B) do not make the argument valid, because the argument is not about age. Suppressed premises are supposed to show the path from the explicit premises to the conclusion instead of creating a different path to that conclusion. (C) is enough to get us straight from the premise "Mary is a professor" to the conclusion "Mary can't be a student," but then it cannot explain why the arguer went on to add "Professors must already have degrees," so it misses part of the argument. (D) enables us to understand all parts of the argument: Mary can't be a student, because she is a professor, and professors must already have degrees. Then: Students don't already have degrees, so Mary can't be a student. Notice that (D) is false, but it still might be the most plausible suppressed premise to make sense of what the arguer probably had in mind.

If you DO find a reconstruction that is sound, then the conclusion of the argument that you reconstructed is:

True. A sound argument has true premises and is valid in the sense that it is not possible for its conclusion to be false when its premises are true, so its conclusion must be true as well.

If you do NOT find a reconstruction that is sound, then he argument that you reconstructed:

could be either sound or unsound. The fact that you fail to find a sound reconstruction could show only that you lack imagination and skill, so the argument that you were trying to reconstruct still might be sound, even though you could not figure out how.

If you find a reconstruction that is not sound, then the argument that you reconstructed:

could be either sound or unsound. The fact that ONE reconstruction is unsound does not show that ALL reconstructions are unsound. There still might be ANOTHER reconstruction that is sound, even if you find one that is not sound. Admittedly, the goal is to find a sound reconstruction, if there is one, but one still might find a reconstruction that is not good in that way, because it is not sound, even though another reconstruction is better because it is sound.

His natural talents were not enough; he lost the match because he had not practice sufficiently. You need either great natural talent or hard work to become a winner.

(1) His natural talents were not enough to win without practicing.

(2) He had not practiced sufficiently.

(3) People lose matches if they do not either practice sufficiently or have enough natural talent to win without practicing.

___________________________

∴ (4) He lost the match. (from 1-3)

(A) is not a good reconstruction because "His natural talents were not enough" is not a reason why "He had not practiced sufficiently." (B) is not a good reconstruction because the passage suggests that inadequate natural talent is not enough to explain the loss if one practices sufficiently, and inadequate practice is not enough to explain the loss if one has enough natural talent.

(2) He had not practiced sufficiently.

(3) People lose matches if they do not either practice sufficiently or have enough natural talent to win without practicing.

___________________________

∴ (4) He lost the match. (from 1-3)

(A) is not a good reconstruction because "His natural talents were not enough" is not a reason why "He had not practiced sufficiently." (B) is not a good reconstruction because the passage suggests that inadequate natural talent is not enough to explain the loss if one practices sufficiently, and inadequate practice is not enough to explain the loss if one has enough natural talent.

I took lots of mathematics, so I know that 81 is not a prime number, because 81 is divisible by 3. Indeed, 81 = 3 to the fourth power. Any idiot knows that.

(1) 81 is evenly divisible by 3.

(2) 81 is evenly divisible by 1.

(3) 81 is evenly divisible by 81.

(4) 1, 3, and 81 are three different numbers.

________________________

∴ (5) 81 is evenly divisible by three different numbers. (from 1-4)

(6) Every prime number is evenly divisible by no more than two different numbers.

(7) Three different numbers is more than two different numbers.

________________________

∴ (8) 81 is not a prime number. (from 5-7)

Reconstruction (A) is valid, but it is still not adequate, because it lacks suppressed premises that show how to get from the premise to the conclusion. Reconstruction (C) is also inadequate, because it claims that "I know that 81 is not a prime number" is the same as "81 is not a prime number," although these are not equivalent, since the latter can be true when when the former is not.

(2) 81 is evenly divisible by 1.

(3) 81 is evenly divisible by 81.

(4) 1, 3, and 81 are three different numbers.

________________________

∴ (5) 81 is evenly divisible by three different numbers. (from 1-4)

(6) Every prime number is evenly divisible by no more than two different numbers.

(7) Three different numbers is more than two different numbers.

________________________

∴ (8) 81 is not a prime number. (from 5-7)

Reconstruction (A) is valid, but it is still not adequate, because it lacks suppressed premises that show how to get from the premise to the conclusion. Reconstruction (C) is also inadequate, because it claims that "I know that 81 is not a prime number" is the same as "81 is not a prime number," although these are not equivalent, since the latter can be true when when the former is not.

We really got ripped off by that security company. This burglar alarm won't work unless we are lucky or the burglar uses the front door, so we can't count on it. I think we need a new alarm.

1) This burglar alarm won't work unless we are lucky or the burglar uses the front door.

(2) We can't count on being lucky.

(3) We can't count on the burglar using the front door.

(4) We can't count on a burglar alarm that won't work.

______________________

∴ (5) We can't count on this burglar alarm. (from 1-3)

(6) This burglar alarm is the only one that we have.

(7) We need a burglar alarm that we can count on.

______________________

∴ (8) We need a new burglar alarm. (from 1)

Reconstruction (B) is inadequate because the explicit argument does not say that we are not lucky or that the burglar won't use the front door. These are risks rather than definite occurrences. Reconstruction (C) is inadequate because neither subargument (from (1) to (2) and from (3) to (4)) is valid, and because "so" is a conclusion marker, so "We can't count on this burglar alarm" is a conclusion rather than a premise.

(2) We can't count on being lucky.

(3) We can't count on the burglar using the front door.

(4) We can't count on a burglar alarm that won't work.

______________________

∴ (5) We can't count on this burglar alarm. (from 1-3)

(6) This burglar alarm is the only one that we have.

(7) We need a burglar alarm that we can count on.

______________________

∴ (8) We need a new burglar alarm. (from 1)

Reconstruction (B) is inadequate because the explicit argument does not say that we are not lucky or that the burglar won't use the front door. These are risks rather than definite occurrences. Reconstruction (C) is inadequate because neither subargument (from (1) to (2) and from (3) to (4)) is valid, and because "so" is a conclusion marker, so "We can't count on this burglar alarm" is a conclusion rather than a premise.

Joe is not a freshman, since he lives in a fraternity, and freshmen are not allowed to live in fraternities. He also can't be a senior, since he has not declared a major. And he can't be a junior, because I never met him before today, and I would have met him before now if he were a junior. So Joe must be a sophomore.

(1) Joe lives in a fraternity.

(2) Joe is allowed to live where he lives. (suppressed premise)

(3) Freshmen (first-year students) are not allowed to live in fraternities.

______________________

∴ (4) Joe is not a freshman.

(5) Joe has not declared a major.

(6) All seniors (fourth-year students) have declared a major. (suppressed premise)

______________________

∴ (7) Joe is not a senior.

(8) I never met Joe before today.

(9) If Joe were a junior (third-year student), then I would have met him before today.

______________________

∴ (10) Joe is not a junior.

(11) Joe is either a freshman, a sophomore, a junior, or a senior. (suppressed premise)

______________________

∴ (12) Joe is a sophomore (second-year student). (from 4, 7, 10, and 11)

Reconstruction (A) merely lists the premises and does not show how they fit together into a structure. Reconstruction (C) falsely suggests that the argument gives three separate reason why Joe is a sophomore, because fact that Joe is not a freshman alone shows that he is a sophomore (in the step from (4) to (5)), and similarly for junior (in the step from (12) to (13)) and senior (in the step from (8) to (9)). However, the argument can conclude that he is a sophomore only after ALL other possibilities have been excluded. That is why this form of argument is often called "process of elimination."

(2) Joe is allowed to live where he lives. (suppressed premise)

(3) Freshmen (first-year students) are not allowed to live in fraternities.

______________________

∴ (4) Joe is not a freshman.

(5) Joe has not declared a major.

(6) All seniors (fourth-year students) have declared a major. (suppressed premise)

______________________

∴ (7) Joe is not a senior.

(8) I never met Joe before today.

(9) If Joe were a junior (third-year student), then I would have met him before today.

______________________

∴ (10) Joe is not a junior.

(11) Joe is either a freshman, a sophomore, a junior, or a senior. (suppressed premise)

______________________

∴ (12) Joe is a sophomore (second-year student). (from 4, 7, 10, and 11)

Reconstruction (A) merely lists the premises and does not show how they fit together into a structure. Reconstruction (C) falsely suggests that the argument gives three separate reason why Joe is a sophomore, because fact that Joe is not a freshman alone shows that he is a sophomore (in the step from (4) to (5)), and similarly for junior (in the step from (12) to (13)) and senior (in the step from (8) to (9)). However, the argument can conclude that he is a sophomore only after ALL other possibilities have been excluded. That is why this form of argument is often called "process of elimination."

Charles went bald because most men his age go bald.

This is an argument. This sentence claims that the fact that most men his age go bald is a reason that explains why Charles went bald. According to this claim, his age helps us understand why he went bald and makes it less surprising that he went bald.

Charles went bald, and most men his age go bald.

This is not an argument. This sentence says only that both facts are true and does not explicitly say that one is reason for the other.

My roommate likes to ski, so I do, too.

This is an argument. This sentence claims that the fact that my roommate likes to ski is a reason that explains why I like to ski. According to this claim, my roommates' likes helps us understand why I like to ski and makes it less surprising that I like to ski.

My roommate likes to ski, and so do I.

This is not an argument. This sentence says only that both facts are true and does not explicitly say that one is a reason for the other. The word "so" here is simply short for "also" and does not function as an argument marker.

I have been busy since Tuesday.

This is not an argument. The sentence says only that I have been busy after the time or day when it was Tuesday. It does not say that I have been busy because that day was Tuesday. Indeed, the sentence that I have been busy since Tuesday simply states one single fact, so there is not a premise separate from a conclusion. An argument requires at least one premise and also a conclusion, so this single sentence cannot be an argument.

I am busy, since my teacher assigned lots of homework.

This is an argument. This sentence claims that the assignment is a reason why I am busy.

conclusion markers

indicate that the sentence following is a conclusion (so, therefore, thus, hence, accordingly)

premise markers

indicate that the sentence following is a premise (because, for, as, for the reason that)

For questions 7-11, indicate whether the underlined phrase is a premise marker, conclusion marker, or neither.

He apologized, SO you should forgive him.

He apologized, SO you should forgive him.

Conclusion marker. The word "so" indicates that the sentence following "so" is a conclusion. Thus, this sentence means that "you should forgive him" is a conclusion supported by the premise "he apologized." This function of the word "so" is shown by the fact that the meaning does not change if we substitute "therefore" for "so": He apologized. Therefore, you should forgive him.

IN VIEW OF THE FACT THAT he apologizes, you should forgive him.

Premise marker. The phrase "In view of the fact that" indicates that the sentence immediately following that phrase is a premise. Thus, this sentence means that the fact that he apologized is a reason for the conclusion that you should forgive him. This function of the phrase is shown by the fact that the meaning does not change if we substitute "because" for "In view of the fact that": Because he apologized, you should forgive him.

He apologized. ACCORDINGLY, you should forgive him.

Conclusion marker. The word "accordingly" indicates that the sentence following that phrase is a conclusion. Thus, this sentence means that "you should forgive him" is a conclusion supported by the premise "he apologized." This function of the word is shown by the fact that the meaning does not change if we substitute "therefore" for "accordingly": He apologized. Therefore, you should forgive him.

AFTER he apologizes, you should forgive him.

Neither. The word "after" indicates only a temporal relation, not a relation of reason or premise to conclusion. Thus, the meaning does change if we substitute "because" (or "therefore") for "after": Because he apologized, you should forgive him. Notice that this sentence might suggest (or conversationally imply) that his apology is a reason to forgive him, but the sentence does not make that claim explicitly, so "after" is neither a reason marker nor a conclusion marker.

SEEING AS he apologized, you should forgive him.

The phrase "Seeing as" indicates that the sentence immediately following that phrase is a premise. Thus, this sentence means that the fact that he apologized is a reason for the conclusion that you should forgive him. This function of the phrase is shown by the fact that the meaning does not change if we substitute "because" for "Seeing as": Because he apologized, you should forgive him.

Things are lot quieter because Jesse James left town.

Jesse James left town.

_____________________

∴ Things are a lot quieter.

The word "because" is a premise marker, so the following sentence, "Jesse James left town," is a premise.

_____________________

∴ Things are a lot quieter.

The word "because" is a premise marker, so the following sentence, "Jesse James left town," is a premise.

Because Jesse James left town, things are a lot quieter.

Jesse James left town.

_____________________

∴ Things are a lot quieter.

The word "because" is a premise marker, so the following sentence, "Jesse James left town," is a premise.

_____________________

∴ Things are a lot quieter.

The word "because" is a premise marker, so the following sentence, "Jesse James left town," is a premise.

The hour is up, so you must hand in your exams.

The hour is up.

_________________________

∴ You must hand in your exams.

The word "so" here is a conclusion marker, so the following sentence, "You must hand in your exams," is a conclusion.

_________________________

∴ You must hand in your exams.

The word "so" here is a conclusion marker, so the following sentence, "You must hand in your exams," is a conclusion.

Other airlines will carry more passengers, for United Airlines is on strike.

United Airlines is on strike.

____________________________________

∴ Other airlines will carry more passengers.

The word "for" here is a premise marker, so the following sentence, "United Airlines is on strike," is a premise.

____________________________________

∴ Other airlines will carry more passengers.

The word "for" here is a premise marker, so the following sentence, "United Airlines is on strike," is a premise.

Since Chicago is north of Washington, and Washington is north of Charleston, Chicago is north of Charleston.

Chicago is north of Washington.

Washington is north of Charleston.

_____________________________

∴ Chicago is north of Charleston.

The word "since" here is a premise marker, so the following sentences, "Chicago is north of Washington" and "Washington is north of Charleston," are premises.

Washington is north of Charleston.

_____________________________

∴ Chicago is north of Charleston.

The word "since" here is a premise marker, so the following sentences, "Chicago is north of Washington" and "Washington is north of Charleston," are premises.

Texas has a greater area than Topeka, and Topeka has a greater area than the Bronx Zoo. Thus, Texas has a greater area than the Bronx Zoo.

Texas has a greater area than Topeka.

Topeka has a greater area than the Bronx Zoo.

_______________________________________

∴ Texas has a greater area than the Bronx Zoo.

The word "Thus" is a conclusion marker, so the following sentence, "Texas has a greater area than the Bronx Zoo," is a conclusion.

Topeka has a greater area than the Bronx Zoo.

_______________________________________

∴ Texas has a greater area than the Bronx Zoo.

The word "Thus" is a conclusion marker, so the following sentence, "Texas has a greater area than the Bronx Zoo," is a conclusion.

standard form

(1) premise

(2) premise

_____________

∴ (3) conclusion (from 1-2)

(2) premise

_____________

∴ (3) conclusion (from 1-2)

Ram asks Walter which team will win the basketball tournament, and Walter says, "Duke, and I can prove it: Duke is going to win, so Duke will win. Now, prove that I am wrong!"

The argument is circular. The premise ("Duke is going to win") is equivalent to the conclusion ("Duke will win"). That makes the argument circular in the sense that it ends up where it started. It does not get anywhere. Circular arguments like this are illegitimate, because if they worked, then they could be used to prove anything, including the opposite conclusion (as in "Duke is going to lose, so Duke will lose"). That shows why such circular arguments are objectionable.

Ram asks Walter which team will win the basketball tournament, and Walter says, "Duke, and I can prove it: Duke will score more points than all of its opponents, so Duke will win." Then Ram asks, "But how do you know that Duke will score more points than its opponents?" Walter answers, "I don't know. I just hope so."

The premise is unjustified. The premise ("Duke will score more points than all of its opponents") is unjustified if Walter is just hoping (or guessing), as he admits. He has no reason to believe that it is true. If we were allowed to use premises that are unjustified like this, then we could use them to prove anything, including absurdities. That is why such unjustified premises are not allowed.

argumentative moves

assuring

guarding: making the premises of your argument weaker so that it is harder to object to them

discounting: citing a possible objection in order to reject it or counter it

guarding: making the premises of your argument weaker so that it is harder to object to them

discounting: citing a possible objection in order to reject it or counter it

ways to assure

1. authoritative

2. reflexive (talking about yourself)

3. abusive (they abuse you to get you to agree)

2. reflexive (talking about yourself)

3. abusive (they abuse you to get you to agree)

I ASSURE YOU THAT you can trust my sister.

Assuring term. The phrase "I assure you" is clearly intended to assure you that you can trust my sister, so it functions as an assuring term.

You CERTAINLY can trust my sister.

Assuring term. The term "certainly" here means that it is certain (or you can be certain) that you can trust my sister. This assures you. This function is shown by the fact that the meaning does not change significantly if you replace "certainly" with "assuredly" to get "You assuredly can trust my sister." Notice also that it would be odd to add "... but I am not sure" at the end to get this oxymoron: You certainly can trust my sister, but I am not sure that you can trust her.

Imagine that you want to go to a soccer game tonight, so you ask me whether the local team is playing a home game tonight. I respond, "I BELIEVE THAT they are playing a home game tonight."

Not assuring term. If I knew, then I would say simply that they are playing a home game tonight. Thus, the fact that I add the qualification "I believe that ..." suggests uncertainty, so it does not assure you that I am right. This role is shown by the naturalness of adding "... but I am not sure" at the end to get this: "I believe that they are playing a home game tonight, but I am not sure."

I KNOW THAT you can trust my sister.

Assuring term. The phrase "I know that ..." here implies that the following claim is true and justified or reliable. The force of this utterance does not change significantly if you replace "I know that ..." with "I am sure that ..." to get "I am sure that you can trust my sister." Notice also that it would be odd to add "... but I am not sure" at the end to get this oxymoron: I know that can trust my sister, but I am not sure that you can trust her.

ONLY AN IDIOT WOULD DENY THAT you can trust my sister.

Assuring term. This example is abusive assuring. It tries to make you feel assured by abusing the very idea of questioning what I claim, which is that you can trust my sister.

SOME people have raised questions about her program, so we should oppose it.

Guarding term. By saying that some people have raised questions instead of saying that everyone finds the program questionable, the speaker weakens this claim so that it is easier to defend. That is what guarding does. Notice that, even if some people (say, 1 percent) have raised questions, there still might be a lot more people (99 percent!) who do not find her program questionable at all.

MANY people agree that her program won't work, so we should oppose it.

Guarding term. By saying that many people agree instead of saying that all people agree, the speaker weakens this claim so that it is easier to defend. That is what guarding does. Notice that, even if many people agree that her program won't work, there still might be many—even more—other people who disagree and think that her program will work.

Her program will lead to many problems, so we should oppose it.

Not guarding term. No reasonable person would think that her program will lead to all problems—that is, to every problem in the world—so it makes no sense to understand this premise that her program will lead to many problems as a weakened version of the claim that it will lead to all problems. Thus, this use of the term "many" does not function as a guarding term. This example shows that we cannot identify guarding in a purely mechanical way simple by looking for certain words. Instead, we need to think about how the term functions in the argument.

Her program MIGHT lead to problems, so we should oppose it.

Guarding term. By saying that her program "might" lead to problems instead of saying that it "will" lead to problems, the speaker weakens this claim so that it is easier to defend. That is what guarding does. Notice that, even if her program might lead to problems, it still might be much more likely that her program will not lead to any problems at all.

I SUSPECT THAT her program will lead to problems, so we should oppose it.

Guarding term. By saying "I suspect" instead of "I know," the speaker weakens this claim so that it is easier to defend. That is what guarding does. Notice that the speaker can suspect problems even if the speaker has no reason at all to suspect those problems, so what the speaker literally says is true even if the speaker has no reason for any suspicions.

ALTHOUGH it is raining, I am going for a walk anyway.

Discounting term. If the speaker merely said, "I am going for a walk," then the audience might object, "You should not go for a walk now because it is raining." The initial clause, "although it is raining," shows that the speaker is aware of the rain and recognizes it as a possible objection to going for a walk. Thus, the speaker heads off that potential objection. That is what discounting does: it avoids or responds to objections.

Notice that, if the speaker had instead said, "It is raining and I am going for a walk," this new utterance would not indicate any awareness that rain is a reason not to go for a walk, so this new utterance would not discount that objection. This comparison shows that the word "although" functions as a discounting term in a way that the word "and" does not.

Notice that, if the speaker had instead said, "It is raining and I am going for a walk," this new utterance would not indicate any awareness that rain is a reason not to go for a walk, so this new utterance would not discount that objection. This comparison shows that the word "although" functions as a discounting term in a way that the word "and" does not.

I am going for a walk EVEN if it is raining.

Discounting term. As before, the clause "even if it is raining" shows that the speaker is aware of the rain and recognizes it as a possible objection to going for a walk. The clause discounts that potential objection.

The comparison between this example and the previous one shows that discounting can occur either at the end or the beginning of a sentence.

The comparison between this example and the previous one shows that discounting can occur either at the end or the beginning of a sentence.

It is raining. STILL, I am going for a walk.

Discounting term. The word "still" shows that the speaker recognizes the rain as a possible objection to going for a walk. The clause with "still" discounts that potential objection. This function is shown by the fact that the meaning does not change if we replace "still" with "however" (as in "However, I am going for a walk").

The comparison between this example and the previous two shows that the discounting term can be located either before the objection that is discounted (as in "although it is raining" and "even if it is raining") or before the clause that the objection objects to (as in "Still, I am going for a walk anyway") or in the middle of that clause (as in "I am still going for a walk").

The comparison between this example and the previous two shows that the discounting term can be located either before the objection that is discounted (as in "although it is raining" and "even if it is raining") or before the clause that the objection objects to (as in "Still, I am going for a walk anyway") or in the middle of that clause (as in "I am still going for a walk").

It is STILL raining, and I am going for a walk.

Not discounting term. Here the word "still" merely indicates that the rain continues. It does not show awareness of (or respond to) a potential objection, so here its function is not discounting. This is supported by the fact that it makes no sense to replace "still" with "however" as in "It is however raining, and I am going for a walk." The comparison between this example and the previous one shows that the same word "still" can be used for discounting in some cases and not in others.

Sure, I will get wet from walking in the rain. I will have fun, NEVERTHELESS, and I want to get some exercise.

Discounting term. The word "nevertheless" shows that the speaker recognizes getting wet as a possible objection to going for a walk. The clause with "nevertheless" discounts that potential objection by suggesting that it is more important to have fun and get some exercise. This function is shown by the fact that the meaning does not change significantly if we replace "nevertheless" with "however."

This example shows that the discounting term can be located in the middle of the clause that the objection objects to.

This example shows that the discounting term can be located in the middle of the clause that the objection objects to.

For the questions in this exercise, indicate whether the underlined term is a positive evaluative term (E+), a negative evaluative term (E-), or neither (N).

Janet is an EXCELLENT golfer.

Janet is an EXCELLENT golfer.

E+. To say Janet is excellent at golf implies that she is very good at golf. This implication shows that "excellent" is evaluative in the same way as "good," which is a paradigm case of a positive evaluative term.

The band was playing very LOUDLY.

Neither. Although some people think that loud music is bad, other people think that loud music is good. It even makes sense to say, "Their playing was very loud, which made it even better." This shows that "loud" does not imply "bad" (or "good"). Hence, "loudly" is not an evaluative term.

The band was playing TOO loudly.

E-. People who think that the loudness level is good will deny that it is too loud. Hence, "too loud" implies "bad" or "not good" in this respect, so "too loudly" is a negatively evaluative phrase. The comparison with the previous example shows that the word "too" is what makes this phrase negatively evaluative

They turned THE RIGHT WAY at the intersection.

E+. Here "the right way" means "the correct way" in contrast with "the wrong way," so this use of the term "right" is positively evaluative.

They mistakenly turned RIGHT at the intersection.

Neither. Here the word "right" contrasts with "left" rather than with "wrong." There is nothing good or bad in itself about turning right, so here the term "right" is not evaluative. The comparison between this example and the previous one shows that the same word can be evaluative in some cases but not in others, so we cannot decide mechanically whether a word is evaluative just by the shape of its letters.

They turned JUST before the other car.

Neither. Here the phrase "just before" contrasts with "long before" rather than with "unjust" or "unfair," so here the word "just" is not evaluative.

Their punishment was UNJUST.

E-. Here the word "unjust" means "unfair." Unfairness is wrong or bad, since we would not call something unfair if we thought it was good. Thus, "unjust" is negatively evaluative.

For the questions in this exercise, use the following labels to indicate the function of each of the bold words or phrases in this passage from "A Piece of God's Handiwork," by Robert Redford (paragraph 3).

The BLM says its hands are tied. Why? Because these lands were set aside subject to "valid existing rights," and Conoco has a lease that gives it the right to drill. Sure Conoco has a lease--more than one, in fact--but those leases were originally issued without sufficient environmental study or public input, As a result, none of them conveyed a valid right to drill. What's more, in deciding to issue a permit to drill now, the BLM did not conduct a full analysis of the environmental impacts of drilling in these incomparable lands, but instead determined there would be no significant environmental harm on the basis of an abbreviated review that didn't even look at drilling on the other federal leases.

THE BLM SAYS its hands are tied.

The BLM says its hands are tied. Why? Because these lands were set aside subject to "valid existing rights," and Conoco has a lease that gives it the right to drill. Sure Conoco has a lease--more than one, in fact--but those leases were originally issued without sufficient environmental study or public input, As a result, none of them conveyed a valid right to drill. What's more, in deciding to issue a permit to drill now, the BLM did not conduct a full analysis of the environmental impacts of drilling in these incomparable lands, but instead determined there would be no significant environmental harm on the basis of an abbreviated review that didn't even look at drilling on the other federal leases.

THE BLM SAYS its hands are tied.

None of the above. This opening might seem to be assuring, because it cites the BLM, and some would see the BLM as an authority on whether its hands are tied. However, Redford is not agreeing with the BLM. Instead, he is reporting what they say in an attempt to excuse their inaction. Hence, Redford is not assuring his audience that the BLM's hands really are tied. More generally, when the author does not believe that a claim is true, then the author (if honest) usually does not assure the audience that the claim is true.

BECAUSE these lands were set aside subject to "valid existing rights," and Conoco has a lease that gives it the right to drill.

Premise marker. The word "because" marks the following sentence "these lands were set aside ..." as a premise that gives a reason for the conclusion stated in the preceding sentence "its hands are tied." The fact that the lands were set aside is supposed to explain why the BLM cannot stop the drilling.

SURE Conoco has a lease--more than one, in fact--but those leases were originally issued without sufficient environmental study or public input.

Assuring term. By using the word "sure," Redford is admitting a point made by the BLM: Conoco has a lease. The fact that both sides of the debate agree on this point is supposed to assure the audience that it is true that Conoco has a lease. This move enables the discussion to move on to more controversial issues.

Sure Conoco has a lease--more than one, IN FACT--but those leases were originally issued without sufficient environmental study or public input.

Assuring term. To call something a fact instead of an opinion or theory is to assure the audience that the claim is true.

Sure Conoco has a lease--more than one, in fact--BUT those leases were originally issued without sufficient environmental study or public input.

Discounting term. The word "but" is a paradigm case of a discounting term. The objection that is discounted here is the BLM's claim that Conoco has a lease, so its hands are tied. The suggested response to this objection is that those leases could be rejected as invalid—or at least questioned—because they "were originally issued without sufficient environmental study or public input." If these leases could be rejected as invalid, then the hands of the BLM are not really tied, as they claim.

AS A RESULT, none of them conveyed a valid right to drill. What's more, in deciding to issue a permit to drill now, the BLM did not conduct a full analysis of the environmental impacts of drill in these incomparable lands, but instead determined there would be no significant environmental harm on the basis of an abbreviated review that didn't even look at drilling on the other federal leases.

Conclusion marker. The "result" here is a result of an argument—a conclusion—so this phrase marks the following sentence as a conclusion. The premise is that "those leases were originally issued without sufficient environmental study or public input," and the conclusion is "none of them conveyed a valid right to drill." The phrase "as a result" indicates that this premise is supposed to provide a reason for this conclusion.

WHAT'S MORE, in deciding to issue a permit to drill now, the BLM did not conduct a full analysis of the environmental impacts of drilling in these incomparable lands, but instead determined there would be no significant environmental harm on the basis of an abbreviated review that didn't even look at drilling on the other federal leases.

Premise marker. Since the preceding sentences gave one reason why the BLM's hands were not tied, the phrase "what's more" indicates that more reasons—and more arguments—are coming now. The preceding reasons concerned the circumstances of Conoco's lease, whereas the following reasons concern the circumstances of the permit. The suggested argument is that (Premise) "the BLM did not conduct a full analysis," so (Conclusion) the permit is not valid (or at least could be questioned by the BLM, so its hands are not tied). Since the sentence immediately after "What's more" is a premise, that phrase is a premise marker. (It is also possible to interpret the phrase "What's more, ..." as saying only "And here's another point" instead of "And here's another reason." On that alternative interpretation, the answer here would be "None of the above." It is not completely clear whether Redford meant to indicate another reason, but it seems that he did.)

What's more, in deciding to issue a permit to drill now, the BLM did not conduct a full analysis of the environmental impacts of drilling in these INCOMPARABLE lands, but instead determined there would be no significant environmental harm on the basis of an abbreviated review that didn't even look at drilling on the other federal leases.

None of the above. Redford is clearly suggesting that these lands are incomparable because they are so beautiful. Nonetheless, the word "incomparable" by itself does not mean "incomparably beautiful" or "incomparably good." All that the word "incomparable" by itself literally means is "unable to be compared." In that literal sense, something could be incomparably bad or incomparable in some evaluatively neutral sense (such as incomparably hot or incomparably small). Thus, the term "incomparable" by itself is not evaluative.

What's more, in deciding to issue a permit to drill now, the BLM did not conduct a full analysis of the environmental impacts of drilling in these incomparable lands, BUT instead determined there would be no significant environmental harm on the basis of an abbreviated review that didn't even look at drilling on the other federal leases.

Discounting term. Again, the word "but" is a paradigm case of a discounting term. The objection that is discounted here is that the BLM did do some analysis or review of environmental impacts of drilling. That review might be taken by objectors to establish that the permit to drill is valid. Redford's response to this objection is that the review was "abbreviated" rather than "full" and "didn't even look at drilling on the other federal leases." He is arguing that the permit is not valid and does not establish a valid right to drill.

What's more, in deciding to issue a permit to drill now, the BLM did not conduct a full analysis of the environmental impacts of drilling in these incomparable lands, but instead determined there would be no significant environmental HARM on the basis of an abbreviated review that didn't even look at drilling on the other federal leases.

Negative evaluative term. Harm is bad. If some effect is not bad, then it is not a harm. Notice that the complete phrase "no significant environmental harm" is not negative, but that is because of the negative word "no." The word "harm" still signals negative evaluation even when that evaluation is denied by another word like "no." The whole phrase is not negative even though a part of the phrase is.

For the questions in this exercise, use the following labels to indicate the function of each of the bold words or phrases in this passage from "A Piece of God's Handiwork," by Robert Redford (paragraphs 6-8).

What we're talking about is, in the words of President Clinton, a small piece of "God's handiwork." Almost 4 1/2 million acres of irreplaceable red rock wilderness remain outside the monument. Let us at least protect what is within it. The many roadless areas within the monument should remain so—protected as wilderness. The monument's designation means little if a pattern of exploitation is allowed to continue.

Environmentalists—including the Southern Utah Wilderness Alliance, the Natural Resources Defense Council, and the Wilderness Society—appealed BLM's decision to the Interior Department's Board of Land Appeals. This appeal, however, was rejected earlier this month. This is a terrible mistake. We shouldn't be drilling in our national monuments. Period. As President Clinton said when dedicating the new monument, "Sometimes progress is measured in mastering frontiers, but sometimes we must measure progress in protecting frontiers for our children and children to come."

Allowing drilling to go forward in the Grand Staircase-Escalante Monument would permanently stain what might otherwise have been a defining legacy of the Clinton presidency.

ALMOST 4 1/2 million acres of irreplaceable red rock wilderness remain outside the monument.

What we're talking about is, in the words of President Clinton, a small piece of "God's handiwork." Almost 4 1/2 million acres of irreplaceable red rock wilderness remain outside the monument. Let us at least protect what is within it. The many roadless areas within the monument should remain so—protected as wilderness. The monument's designation means little if a pattern of exploitation is allowed to continue.

Environmentalists—including the Southern Utah Wilderness Alliance, the Natural Resources Defense Council, and the Wilderness Society—appealed BLM's decision to the Interior Department's Board of Land Appeals. This appeal, however, was rejected earlier this month. This is a terrible mistake. We shouldn't be drilling in our national monuments. Period. As President Clinton said when dedicating the new monument, "Sometimes progress is measured in mastering frontiers, but sometimes we must measure progress in protecting frontiers for our children and children to come."

Allowing drilling to go forward in the Grand Staircase-Escalante Monument would permanently stain what might otherwise have been a defining legacy of the Clinton presidency.

ALMOST 4 1/2 million acres of irreplaceable red rock wilderness remain outside the monument.

Guarding term. The term "almost" contrasts with "exactly," so it weakens this claim and thereby makes this premise easier to defend. It prevents opponents from denying the premise by arguing that the monument really contains only 4.4 million acres.

The many ROADLESS areas within the monument should remain so--protected as wilderness.

None of the above. Although Redford seems to think that it is good for lands to be roadless, the term "roadless" by itself does not literally imply "good" (or "bad"). That is shown by the fact that opponents can sensibly respond, "Yes, those areas are roadless, and that is too bad. They would be better if they had roads." Hence, "roadless" is not literally an evaluative term in itself, even though Redford uses it in this context to suggest his evaluation.

The many roadless areas within the monument SHOULD remain so-- protected as wilderness.

Positive evaluative term. The term "should" is a paradigm case of a positive evaluative term. It is good to do actions that we should do, and it is good when events that should happen do happen. If it were bad or neutral for the roadless areas within the monument to remain roadless, then it would not be true that they should remain so. Thus, to say that they SHOULD remain so, is to imply that their remaining so is not bad or neutral but, instead, is good. That implication of value makes the term "should" positively evaluative.

The monument's designation means little IF a pattern of exploitation is allowed to continue.

None of the above. The term "if" indicates a conditional (or "if..., then..." relation). Here the conditional is: If a pattern of exploitation is allowed to continue, then the monument's designation means little. However, a conditional is not yet enough for an argument unless the "if" clause is asserted. There is no argument in "If I run 100 meters in 9 seconds, then I will win the Olympics" because I cannot run 100 meters in 9 seconds, so I cannot reasonably assert the antecedent. In Redford's conditional, the "if" clause or antecedent is "a pattern of exploitation is allowed to continue." He is doing his best to keep that "if" clause or antecedent from becoming true, so he is not asserting that it IS true. For these reasons, the word "if" by itself is not an argument marker here (or anywhere).

This appeal, HOWEVER, was rejected earlier this month.

Discounting term. The word "however" is a paradigm case of a discounting term. Here Redford discounts the objection that the appeal by environmentalists makes the issue moot and removes the danger to the monument from drilling. Redford's response to this objection is that the appeal was rejected. That response is supposed to show that the danger continues.

This is a TERRIBLE MISTAKE.

Negative evaluative term. The phrase "terrible mistake" is clearly a negative evaluation. If something is good or neutral, then it is not a terrible mistake.

We SHOULDN'T be drilling inner national monuments.

Negative evaluative term. Since the term "should" is a paradigm case of a positive evaluative term (see Question 3 above), the phrase "should not" is a negative evaluative phrase. It is bad to do what we should not do, and it is bad when what should not happen does happen. If drilling in our national monuments were good or neutral, then it would not be true that we should not be drilling in them. Thus, to say that we SHOULD NOT be drilling there implies that it is bad to drill there. This implication of (dis)value is what makes this term (negatively) evaluative.

We shouldn't be drilling in our national monuments. PERIOD.

Assuring term. The term "period" refers to the punctuation at the end of a sentence when no more words are added or need to be added to complete the sentence. This colloquial phrase indicates that the preceding point is so clear that no more words—specifically, no more support or qualification—needs to be added. By saying that the preceding point is that clear, Redford is assuring readers that the preceding claim is true. The meaning would be substantially the same if he replaced "Period" with "That is clearly true." (Still, as in many other cases of close analysis, it is not completely clear what Redford means by this word, so it might be marked as "None of the above.")

As President Clinton said when dedicating the new monument, "Sometimes progress is measured in mastering frontiers, but SOMETIMES we must measure progress in protecting frontiers for our children and children to come."

Guarding term. The term "some" is a paradigm case of a guarding term because it contrasts with "all." Here "sometimes" contrasts with "always." By saying "sometimes we must measure progress in protecting frontiers ..." instead of "we must always measure progress in protecting frontiers ...," Redford is weakening his premise in order to make it more defensible. In particular, he prevents opponents from refuting his premise by responding that sometimes we should not protect frontiers. That response would refute "we must always measure progress in protecting frontiers ..." but would not refute his guarded premise: "sometimes we must measure progress in protecting frontiers ...."

Allowing drilling to go forward in the Grand Staircase-Escalante Monument would permanently stain what MIGHT otherwise have been a defining legacy of the Clinton presidency.

Guarding term. The term "might" weakens Redford's claim in order to make it easier to defend. If he claimed that the monument definitely would have been a defining legacy of the Clinton presidency, then opponents could refute his claim by pointing to other acts by Clinton that might define his legacy. In contrast, this response would not refute Redford's guarded claim that the monument "might otherwise have been a defining legacy of the Clinton presidency." Thus, this word "might" guards that claim by protecting it against refutation.

All Letters to the Editor give arguments.

Answer: False.

Some letters to the editor just give thanks, as in the example in the lecture.

Some letters to the editor just give thanks, as in the example in the lecture.

Arguments are explicit ways to formulate reasons.

Answer: True.

Arguments give different kinds of reasons, but they always give some kind of reason. This will be explained in detail in the next three lectures.

Arguments give different kinds of reasons, but they always give some kind of reason. This will be explained in detail in the next three lectures.

This course will be fun.

Answer: True.

You will see.

You will see.

Arguments are verbal fights.

Answer: False.

People in fights are trying to hurt each other, but people who give arguments are often trying to help each other.

People in fights are trying to hurt each other, but people who give arguments are often trying to help each other.

Every argument includes a conclusion.

Answer: True.

Arguments are defined so that they must always have a conclusion.

Arguments are defined so that they must always have a conclusion.

All arguments are made up of (or expressed in) language

Answer: True.

Premises are sentences, statements, or propositions. Sentences and statements are made up of language, and propositions are expressed by language. Notice that no particular language is required, so it would be false to say that all arguments are in English.

Premises are sentences, statements, or propositions. Sentences and statements are made up of language, and propositions are expressed by language. Notice that no particular language is required, so it would be false to say that all arguments are in English.

Every argument is intended to establish its conclusion.

Answer: False.

Sometimes the conclusion is already established as true, and the point of the argument is only to explain why it is true. Hence, Monty Python is not always right.

Sometimes the conclusion is already established as true, and the point of the argument is only to explain why it is true. Hence, Monty Python is not always right.

Every argument succeeds in giving good reasons for its conclusion.

Answer: False.

Although people who give arguments always intend to give some kind of reason, they often fail to fulfill that intention. Bad arguments fail to give good reasons.

Although people who give arguments always intend to give some kind of reason, they often fail to fulfill that intention. Bad arguments fail to give good reasons.

For questions 6-10, indicate whether the following sentences are arguments.

Megafauna: n. very large animals.

Megafauna: n. very large animals.

Answer: False.

The defined term is not a full sentence, statement, or proposition, so it cannot be a conclusion. The definition is not a reason for the term that is defined.

The defined term is not a full sentence, statement, or proposition, so it cannot be a conclusion. The definition is not a reason for the term that is defined.

Reptiles include turtles, alligators, crocodiles, snakes, lizards, and the tuatara.

Answer: Not an argument.

One word in a list does not give a reason for the other words. One could argue, "This is a turtle, so it is a reptile," but the list by itself does not explicitly state that argument or any argument.

One word in a list does not give a reason for the other words. One could argue, "This is a turtle, so it is a reptile," but the list by itself does not explicitly state that argument or any argument.

World War II occurred after World War I occurred.

Answer: Not an argument.

This sentence is about historical or chronological order rather than rational order. It does not explicitly claim that World War I gives a reason for World War II.

This sentence is about historical or chronological order rather than rational order. It does not explicitly claim that World War I gives a reason for World War II.

World War II occurred because World War I occurred.

Answer: Yes.

The word "because" makes this sentence claim that World War I gives a reason why World War II occurred.

The word "because" makes this sentence claim that World War I gives a reason why World War II occurred.

The sides of this right triangle are 1 meter long, so its hypotenuse is 2 meters long.

Answer: Yes.

The word "so" indicates that the first sentence is supposed to be a reason for the second sentence. This argument is very bad, since the hypotenuse must be square root of 2 meters long instead of 2 meters long. But bad arguments are still arguments.

The word "so" indicates that the first sentence is supposed to be a reason for the second sentence. This argument is very bad, since the hypotenuse must be square root of 2 meters long instead of 2 meters long. But bad arguments are still arguments.

argument

a) a series of sentences, statements, or propositions

b) where some are the premises

c) and one is the conclusion

d) where the premises are intended to give a reason for the conclusion

b) where some are the premises

c) and one is the conclusion

d) where the premises are intended to give a reason for the conclusion

We need to understand the purpose of an artifact in order to understand that artifact.

Answer: True.

Artifacts are created for special purposes, and different artifacts (such as a screwdriver and a spatula) are distinguished by their purposes, among other things.

Artifacts are created for special purposes, and different artifacts (such as a screwdriver and a spatula) are distinguished by their purposes, among other things.

Whether you succeed in persuading someone depends on what effect your argument has on that person.

Answer: True.

To persuade or convince someone is to make that person believe, so the argument persuades the audience only if it makes the audience believe the conclusion.

To persuade or convince someone is to make that person believe, so the argument persuades the audience only if it makes the audience believe the conclusion.

persuading

making people believe or do something that they would not otherwise believe or do (goal: to change your belief or actions)

Whether you succeed in justifying a conclusion depends on what effect your argument has on the audience.

Answer: False.

To justify a conclusion is to give a reason for that conclusion, but the audience might not understand or accept that reason, even when it is a good reason. Then the argument might have no effect (or an unintended effect) on the audience.

To justify a conclusion is to give a reason for that conclusion, but the audience might not understand or accept that reason, even when it is a good reason. Then the argument might have no effect (or an unintended effect) on the audience.

justifying

showing someone a reason to believe the conclusion (goal: to give good reasons to believe something)

Whenever you are trying to justify a conclusion, you are trying to persuade someone.

Answer: False.

You can try to give a reason for the conclusion even when you know that the audience will not understand or accept that reason. The point might be to show them that you have a reason, even if they don't accept your reason.

You can try to give a reason for the conclusion even when you know that the audience will not understand or accept that reason. The point might be to show them that you have a reason, even if they don't accept your reason.

If your argument does not persuade your audience, it is no good.

False.

An argument can still be good for the purposes of justification and explanation even if nobody is persuaded by it. It can also help one's audience understand one's position, even if they are not persuaded, as Judith said in the lecture.

An argument can still be good for the purposes of justification and explanation even if nobody is persuaded by it. It can also help one's audience understand one's position, even if they are not persuaded, as Judith said in the lecture.

An argument that does not give any good reason to believe its conclusion can still persuade someone to believe its conclusion.

True.

People can get fooled by bad reasons.

People can get fooled by bad reasons.

Sometimes people cannot be persuaded because they refuse to give up beliefs that they should not give up.

True.

Refusing to give up a strongly held belief can prevent someone from being persuaded, as Jessica says in the lecture. Sometimes those strongly held beliefs are false or unjustified, but sometimes they are true and justified. In this latter case, these people should not give up those true and justified beliefs.

Refusing to give up a strongly held belief can prevent someone from being persuaded, as Jessica says in the lecture. Sometimes those strongly held beliefs are false or unjustified, but sometimes they are true and justified. In this latter case, these people should not give up those true and justified beliefs.

When people use arguments, they always intend to have some effect on other people.

False.

Sometimes we formulate arguments in private in order to figure out what to believe ourselves without telling anyone else.

Sometimes we formulate arguments in private in order to figure out what to believe ourselves without telling anyone else.

Can any argument persuade every person in the world?

No.

Every argument has to be formulated in some language, but there will always be someone in the world who does not speak that language or understand that argument, so they cannot be persuaded by it. Babies are also people who cannot be persuaded by arguments, at least if they do not yet speak any language.

Every argument has to be formulated in some language, but there will always be someone in the world who does not speak that language or understand that argument, so they cannot be persuaded by it. Babies are also people who cannot be persuaded by arguments, at least if they do not yet speak any language.

The goal of explanation is understanding.

Answer: True

A reason why something happened helps us understand why it happened.

A reason why something happened helps us understand why it happened.

causal explanation

Why did the bridge collapse? The earthquake shook it.

teleological explanation

Why did Joe go to the grocery store? To buy milk.

formal explanation

Why doesn't this peg fit in that round hole? The peg is square.

material explanation

Why is this golf club light? It is made of graphite.

All explanations are given in the form of arguments.

Answer: False

Some explanations are given in the form of arguments, but other explanations are given in the form of narratives or stories (such as a story about why I moved to Duke).

Some explanations are given in the form of arguments, but other explanations are given in the form of narratives or stories (such as a story about why I moved to Duke).

Whenever you predict that something will happen, you explain why it happens.

Answer: False.

Bode's law (discussed in the lecture) was used to predict the existence of Neptune, but it did not explain why Neptune existed or why Neptune was at a certain distance from the Sun.

Bode's law (discussed in the lecture) was used to predict the existence of Neptune, but it did not explain why Neptune existed or why Neptune was at a certain distance from the Sun.

The population of India explains why it won more medals than the United States in the 2012 Olympics.

Answer: False.

India did not win more Olympic medals than the US in 2012, so there is no phenomenon to be explained in this example. Explanations generally assume that the event to be explained did happen.

India did not win more Olympic medals than the US in 2012, so there is no phenomenon to be explained in this example. Explanations generally assume that the event to be explained did happen.

Why did he add more sugar? To make the cake sweeter.

Answer: teleological.

A teleological explanation gives the purpose or function of the phenomenon that it explains, and this explanation says that his purpose was to make the cake sweeter.

A teleological explanation gives the purpose or function of the phenomenon that it explains, and this explanation says that his purpose was to make the cake sweeter.

Why is the pillow so soft? Because it is filled with duck feathers.

Answer: material

A material explanation cites the material that makes up the thing to be explained, and this explanation cites the material in the pillow.

A material explanation cites the material that makes up the thing to be explained, and this explanation cites the material in the pillow.

Why doesn't the screwdriver work with the slotted screw? Because it is a Phillips screwdriver (with a cross-shaped end).

Answer: formal

A formal explanation cites the shape or form of the thing to be explained, and this explanation cites the shape of the screwdriver.

A formal explanation cites the shape or form of the thing to be explained, and this explanation cites the shape of the screwdriver.

Why did the tire on her car go flat? Because it was punctured by a nail.

Answer: causal

A causal explanation cites the event that brought about (or sustains) the thing to be explained, and this explanation cites the event of being punctured.

A causal explanation cites the event that brought about (or sustains) the thing to be explained, and this explanation cites the event of being punctured.

Purpose of arguments

1. persuasion

2. justification

3. explanation

2. justification

3. explanation

The English language could use the word "death" to refer to life.

True. Linguistic meaning can vary with social convention among all speakers of the language.

I can make the word "baboon" in the public English language refer to my sister's friends by stipulating that I will use the word "baboon" to refer to my sister's friends.

False. The meaning of a word in a public natural language like English depends on shared social conventions rather than on idiosyncratic stipulations by individuals. Stipulating that the word "baboon" means or refers to my sister's friends does not magically turn her friends into baboons, and it also does not magically change the English language or the meaning of the word "baboon" in the public English language.

People are always aware of the rules that they follow when they speak.

False. English speakers pronounce "finger", "singer", and "plunger" properly without knowing the rule that governs these pronunciations.

People are never aware of the rules that they follow when they speak.

False. People who are just learning a language often follow rules consciously because they become fluent.

Language is...

1. important

2. conventional

3. representational

4. social

2. conventional

3. representational

4. social

Arguments are fallacious when they are spoken in heavy accents.

Answer: False

Accent is an aspect of language that does not affect truth or rationality, so it does not affect whether arguments are good as arguments.

Accent is an aspect of language that does not affect truth or rationality, so it does not affect whether arguments are good as arguments.

The meaning of the phrase "my car" is the object that is my car.

Answer: False

The meaning is not the object because, if it were, then it would be meaningless to say "I do not own any car", but this sentence is not meaningless. The phrase "my car" refers to my car, but reference is not the same as meaning.

The meaning is not the object because, if it were, then it would be meaningless to say "I do not own any car", but this sentence is not meaningless. The phrase "my car" refers to my car, but reference is not the same as meaning.

We can describe the meaning of a word or sentence by specifying how it is used.

Answer: True

This is what Wittgenstein meant by his slogan "Meaning is use."

This is what Wittgenstein meant by his slogan "Meaning is use."

To say "I am sorry" is to apologize to someone even if that person does not forgive you, so apologizing is

Answer: a speech act

A speech act occurs even if the intended effect does not occur.

A speech act occurs even if the intended effect does not occur.

To utter a series of words that are meaningful together is to perform

Answer: a linguistic act

A linguistic act is just the utterance of a meaningful sentence.

A linguistic act is just the utterance of a meaningful sentence.

To alert someone to a danger is to make that person aware of that danger, so alerting is

Answer: a conversational act

Explanation: A conversational act is the bringing about of an effect.

Explanation: A conversational act is the bringing about of an effect.

X The old man the ship.

Answer: Meaningful.

If you read "the old man" as a noun phrase, then you will look for a verb and not find one. That makes this garden path sentence seem meaningless. However, "the old" can be a noun by itself referring to old people, and "man" can be a verb referring to managing the ship, and then the sentence means "The old people manage the ship."

If you read "the old man" as a noun phrase, then you will look for a verb and not find one. That makes this garden path sentence seem meaningless. However, "the old" can be a noun by itself referring to old people, and "man" can be a verb referring to managing the ship, and then the sentence means "The old people manage the ship."

X I feel as much like I did yesterday as I do today.

Answer: Not meaningful.

This sentence might seem grammatical, but what does it mean? This example comes from Daniel Wegner.

This sentence might seem grammatical, but what does it mean? This example comes from Daniel Wegner.

X The cotton clothing is made of grows in Mississippi.

Answer: Meaningful

If you read "the cotton clothing" as a noun phrase and "is made of" as a verb phrase, then you will expect these words to be followed by a noun describing what the cotton clothing is made of. That makes this garden path sentence seem meaningless. In contrast, the right way to interpret the sentence is so that "the cotton" is the subject, "grows in Mississippi" is the verb, and "clothing is made of" is a dependent clause specifying which cotton is being talked about, so the whole sentence means "The cotton [that] clothing is made of grows in Mississippi.

If you read "the cotton clothing" as a noun phrase and "is made of" as a verb phrase, then you will expect these words to be followed by a noun describing what the cotton clothing is made of. That makes this garden path sentence seem meaningless. In contrast, the right way to interpret the sentence is so that "the cotton" is the subject, "grows in Mississippi" is the verb, and "clothing is made of" is a dependent clause specifying which cotton is being talked about, so the whole sentence means "The cotton [that] clothing is made of grows in Mississippi.

X The square root of pine is tree.

Answer: Not meaningful.

This sentence might seem grammatical, but what does it mean? It does not mean "The square root of nine is three", even though they sound similar.

This sentence might seem grammatical, but what does it mean? It does not mean "The square root of nine is three", even though they sound similar.

X In order to pass the thereby test, it must always be true that "If I say 'I _____,' then I thereby _____."

Answer: False

This needs to be true only in appropriate circumstances. That was why strangers do not get married just whenever anyone pronounces them husband and wife.

This needs to be true only in appropriate circumstances. That was why strangers do not get married just whenever anyone pronounces them husband and wife.

X Whenever a verb passes the thereby test (that is, whenever the verb fits into the blank in "If I say 'I _____' in appropriate circumstances, then I thereby _____"), the verb names a speech act.

Answer: True

This is the point of the thereby test.

This is the point of the thereby test.

X Concluding is a speech act.

Answer: True

If I say 'I conclude that Pluto is not a planet' in appropriate circumstance (including the presence of premises and perhaps authority), then I thereby conclude that Pluto is not a planet. Thus, "conclude" passes the thereby test, so this verb names a speech act.

If I say 'I conclude that Pluto is not a planet' in appropriate circumstance (including the presence of premises and perhaps authority), then I thereby conclude that Pluto is not a planet. Thus, "conclude" passes the thereby test, so this verb names a speech act.

X When I say, "I order you to leave" in appropriate circumstances, then I thereby order you to leave.

Answer: True

Ordering is a speech act.

Ordering is a speech act.

X When I say, "I ordered you to leave" in appropriate circumstances, then I thereby ordered you to leave.

Answer: False

When I say, "I ordered you to leave", I am referring to the past rather than the present. Imagine that on Monday I order you to leave, and then on Tuesday I say, "I ordered you to leave." What I say on Tuesday is true, but I do not thereby order you, since the word "thereby" in this context claims that I order you at the time in the very act of saying that I order you.

When I say, "I ordered you to leave", I am referring to the past rather than the present. Imagine that on Monday I order you to leave, and then on Tuesday I say, "I ordered you to leave." What I say on Tuesday is true, but I do not thereby order you, since the word "thereby" in this context claims that I order you at the time in the very act of saying that I order you.

X When I say, "My sister orders you to leave" in appropriate circumstances, then I thereby order you to leave.

Answer: False

This sentence is referring to what my sister does, so the order given by her is not accomplished thereby—that is, in the very act of uttering these words.

This sentence is referring to what my sister does, so the order given by her is not accomplished thereby—that is, in the very act of uttering these words.

X When I say, "I apologize for hurting you" in appropriate circumstances, then I thereby apologize for hurting you.

Answer: True

Apologizing is a speech act.

Apologizing is a speech act.

X When I say, "I am sorry for hurting you" in appropriate circumstances, then I thereby am sorry for hurting you.

Answer: False

This sentence describes how I feel, but that feeling does not occur in the very act of uttering these words, since I could feel sorry without uttering the words, and I could utter the words without really feeling sorry. The sentence does perform the speech act of apologizing, but feeling sorry is not a speech act.

This sentence describes how I feel, but that feeling does not occur in the very act of uttering these words, since I could feel sorry without uttering the words, and I could utter the words without really feeling sorry. The sentence does perform the speech act of apologizing, but feeling sorry is not a speech act.

X When I say, "I advise you to keep trying" in appropriate circumstances, then I thereby advise you to keep trying.

Answer: True

Advising is a speech act.

Advising is a speech act.

X When I say, "I convince you to keep trying" in appropriate circumstances, then I thereby convince you to keep trying.

Answer: False

Convincing requires an effect in the audience (here that the audience does keep trying). That effect occurs at a later time after the speech act, so this effect does not occur in the very act of saying "I convince you to keep trying."

Convincing requires an effect in the audience (here that the audience does keep trying). That effect occurs at a later time after the speech act, so this effect does not occur in the very act of saying "I convince you to keep trying."

X When I say, "I warn you of the danger" in appropriate circumstances, then I thereby warn you of the danger.

Answer: True

Warning is a speech act.

Warning is a speech act.

X When I say, "I make you aware of the danger" in appropriate circumstances, then I thereby make you aware of the danger.

Answer: False

Making you aware requires an effect in the audience (that the audience does become aware). That effect occurs at a later time after the speech act, so this effect does not occur in the very act of saying "I make you aware of the danger."

Making you aware requires an effect in the audience (that the audience does become aware). That effect occurs at a later time after the speech act, so this effect does not occur in the very act of saying "I make you aware of the danger."

X Verbs that name conversational acts pass the thereby test.

Answer: False

A conversational act requires an effect that occurs at a later time after the utterance, so this effect does not occur during or in the very act of speaking. Hence, the speech does not thereby (in itself) perform the conversational act. That is why conversational acts fail the thereby test.

A conversational act requires an effect that occurs at a later time after the utterance, so this effect does not occur during or in the very act of speaking. Hence, the speech does not thereby (in itself) perform the conversational act. That is why conversational acts fail the thereby test.

X Informing is a conversational act.

Answer: True

Informing refers to making the audience informed or aware of certain information, which is an effect of the utterance.

Informing refers to making the audience informed or aware of certain information, which is an effect of the utterance.

X A sentence can be true even when it conversationally implies something false.

Answer: True

This is what distinguishes conversational implication from logical implication or entailment, as discussed in the lecture.

This is what distinguishes conversational implication from logical implication or entailment, as discussed in the lecture.

X For the following pairs of verbs, which one names a conversational act?

argue, convince

argue, convince

Answer: convince

I convince someone only if they believe or do what I say.

I convince someone only if they believe or do what I say.

X For the following pairs of verbs, which one names a conversational act?

alert, warn

alert, warn

Answer: alert

I alert someone to a danger only if I make them aware of the danger.

I alert someone to a danger only if I make them aware of the danger.

X Be relevant.

Answer: R = Grice's maxim of relevance

The words say it all

The words say it all

X Don't say too little or too much (to serve the shared purpose of conversation).

Answer: S = Grice's maxim of strength or quantity

Too much and too little are quantities, so Grice gave that name to this rule. Our name "Strength" is just a variant to avoid having two rules whose names begin with the letter "Q".

Too much and too little are quantities, so Grice gave that name to this rule. Our name "Strength" is just a variant to avoid having two rules whose names begin with the letter "Q".

X Don't say what you don't believe or what you have no reason to believe.

Answer: Q = Grice's maxim of quality

Explanation: Truth and rational support make a belief high in quality, which is why Grice gave that name to this rule.

Explanation: Truth and rational support make a belief high in quality, which is why Grice gave that name to this rule.

X Be brief and orderly and not obscure or ambiguous.

Answer: M = Grice's maxim of manner

Explanation: It is not clear why Grice gave this name to this rule, but it might be because to order things wrongly or to speak obscurely, vaguely, or too briefly is to speak in the wrong manner.

Explanation: It is not clear why Grice gave this name to this rule, but it might be because to order things wrongly or to speak obscurely, vaguely, or too briefly is to speak in the wrong manner.

Suppose you had to fill in the rightmost column of the truth table below:

p q -p&q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

p q -p&q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

FFTF.

In order for a conjunction to be true, both conjuncts must be true. If one of the conjuncts is a negation, like "-p," then the sentence that is negated (in this case, p) has to be false in order for the conjunct "-p" to be true. So the conjunction "-p&q" is only true when p is false and q is true.

In order for a conjunction to be true, both conjuncts must be true. If one of the conjuncts is a negation, like "-p," then the sentence that is negated (in this case, p) has to be false in order for the conjunct "-p" to be true. So the conjunction "-p&q" is only true when p is false and q is true.

Suppose you had to fill in the rightmost column of the truth table below:

p q -p⊃q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

p q -p⊃q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

TTTF

In order for a conditional to be false, the antecedent must be true and the consequent must be false. If the antecedent is a negation, like -p, then the sentence that is negated (in this case, p) has to be false for the antecedent to be true. So the conditional "-p⊃q" is only false when p is false and q is false, and it is true otherwise.

In order for a conditional to be false, the antecedent must be true and the consequent must be false. If the antecedent is a negation, like -p, then the sentence that is negated (in this case, p) has to be false for the antecedent to be true. So the conditional "-p⊃q" is only false when p is false and q is false, and it is true otherwise.

Suppose you had to fill in the rightmost column of the truth table below:

p q r -p&(q&r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill in that column?

p q r -p&(q&r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill in that column?

FFFFTFFF

In order for a conjunction to be true, both conjuncts must be true. In order for a negation to be true, the sentence, which is being negated, must be false. So for the conjunction "-p&(q&r)" to be true, p must be false, q must be true, and r must be true. This is the only assignment on which the whole conjunction can be true. Otherwise, it is false.

In order for a conjunction to be true, both conjuncts must be true. In order for a negation to be true, the sentence, which is being negated, must be false. So for the conjunction "-p&(q&r)" to be true, p must be false, q must be true, and r must be true. This is the only assignment on which the whole conjunction can be true. Otherwise, it is false.

Which of these claims follows from -p&q?

-p

Given a conjunction, one can infer either of its conjuncts. So, from the conjunction -p&q, one can infer -p and one can infer q. Both -p and q follow from -p&q. None of the other options among (a)-(d) follow, however.

Given a conjunction, one can infer either of its conjuncts. So, from the conjunction -p&q, one can infer -p and one can infer q. Both -p and q follow from -p&q. None of the other options among (a)-(d) follow, however.

Which of these claims follows from -p⊃q?

none of the above.

From a conditional, and nothing else, one cannot infer its antecedent, nor can one infer its consequent. One can only infer the consequent if the premises include both the conditional and its antecedent, and one can only infer the negation of the antecedent if the premises include both the conditional and the negation of the consequent. Since all we have is the conditional, none of (a)-(d) follow.

From a conditional, and nothing else, one cannot infer its antecedent, nor can one infer its consequent. One can only infer the consequent if the premises include both the conditional and its antecedent, and one can only infer the negation of the antecedent if the premises include both the conditional and the negation of the consequent. Since all we have is the conditional, none of (a)-(d) follow.

Which of these claims follows from -p&(q&r)?

-p

Suppose you had to fill in the rightmost column of the following truth-table:

p q r pVq -(pVq) -r -(pVq)&-r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r pVq -(pVq) -r -(pVq)&-r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

FFFFFFFT

c) is correct. In order for the rightmost conjunction to be true, both of its conjuncts have to be true. This means that -r has to be true, which requires r to be false. It also requires -(pVq) to be true, which requires pVq to be false. The only way for a disjunction to be false, however, is for both its disjuncts to be false. So, in order for the rightmost conjunction to be true, p, q, and r ALL have to be false. Otherwise, the rightmost conjunction is false.

c) is correct. In order for the rightmost conjunction to be true, both of its conjuncts have to be true. This means that -r has to be true, which requires r to be false. It also requires -(pVq) to be true, which requires pVq to be false. The only way for a disjunction to be false, however, is for both its disjuncts to be false. So, in order for the rightmost conjunction to be true, p, q, and r ALL have to be false. Otherwise, the rightmost conjunction is false.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p&q -(p&q)Vr -((p&q)Vr)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p&q -(p&q)Vr -((p&q)Vr)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

FFFTFTFT

(c) is correct. In order for the rightmost proposition to be true, the disjunction of (p&q)Vr has to be false. A disjunction is false, however, only when both of its disjuncts is false. For both of the disjuncts of (p&q)Vr to be false, however, r has to be false, and at least one p or q has to be false, too. (This is because the first disjunct, p&q, is a conjunction, which is false when either one of its conjuncts is false.) So, in order for the rightmost conjunction to be true, both r and either p or q must be false. So if r is true, or if both p and q are true, then the rightmost proposition is false. Otherwise, the rightmost proposition is true.

(c) is correct. In order for the rightmost proposition to be true, the disjunction of (p&q)Vr has to be false. A disjunction is false, however, only when both of its disjuncts is false. For both of the disjuncts of (p&q)Vr to be false, however, r has to be false, and at least one p or q has to be false, too. (This is because the first disjunct, p&q, is a conjunction, which is false when either one of its conjuncts is false.) So, in order for the rightmost conjunction to be true, both r and either p or q must be false. So if r is true, or if both p and q are true, then the rightmost proposition is false. Otherwise, the rightmost proposition is true.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p⊃q r⊃(p⊃q)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p⊃q r⊃(p⊃q)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

none of the above.

(e) is correct. In order for the rightmost proposition to be false, the antecedent must be true and the consequent must be false. The antecedent is r, and the consequent is p⊃q. For p⊃q to be false, however, p has to be true and q has to be false. So the only way for the rightmost proposition to be false is if both r and p are true, but q is false. So the correct answer would be TTFTTTTT, which is not one of the options listed in (a)-(d).

(e) is correct. In order for the rightmost proposition to be false, the antecedent must be true and the consequent must be false. The antecedent is r, and the consequent is p⊃q. For p⊃q to be false, however, p has to be true and q has to be false. So the only way for the rightmost proposition to be false is if both r and p are true, but q is false. So the correct answer would be TTFTTTTT, which is not one of the options listed in (a)-(d).

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p≡q p≡r (p≡q)≡(p≡r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p≡q p≡r (p≡q)≡(p≡r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

none of the above.

(e) is correct. In order for the rightmost proposition to be false, the propositions on either side of the main biconditional have to possess different truth-values. For the rightmost proposition to be false, in other words, either p≡q must be true while p≡r is false, or else p≡q must be false while p≡r is true. Both conditions can only happen, however, in two ways: either p and q are both true while r is false, or else p and r are both true while q is false. If p and q are both true while r is false, then the rightmost proposition is false. If p and r are both true while q is false, then the rightmost proposition is false. Otherwise, the rightmost proposition is true. The correct answer, then, should be "TFFTTFFT," which is not available among (a)-(d).

(e) is correct. In order for the rightmost proposition to be false, the propositions on either side of the main biconditional have to possess different truth-values. For the rightmost proposition to be false, in other words, either p≡q must be true while p≡r is false, or else p≡q must be false while p≡r is true. Both conditions can only happen, however, in two ways: either p and q are both true while r is false, or else p and r are both true while q is false. If p and q are both true while r is false, then the rightmost proposition is false. If p and r are both true while q is false, then the rightmost proposition is false. Otherwise, the rightmost proposition is true. The correct answer, then, should be "TFFTTFFT," which is not available among (a)-(d).

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p≡q p≡r -(p≡q)≡(p≡r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p≡q p≡r -(p≡q)≡(p≡r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

none of the above.

e) is correct. For the rightmost proposition to be true, both sides of the biconditional must have the same truth value. This means that the rightmost proposition is only true if both the biconditional, (p≡r), and the negation, -(p≡q), are true, or else when both the biconditional and the negation are both false. On one hand, for both the biconditional (p≡r) and the negation -(p≡q) to be true, p and r have to agree in their truth-value when p and q disagree in their truth value. So if p and r are both true while q is false, or p and r are both false while q is true, then the rightmost proposition is true. On the other hand, for both the biconditional (p≡r) and the negation -(p≡q) to be false, p and r have to disagree in truth value while p and q agree. So if p and q are both true while r is false, or p and q are both false while r is true, then the rightmost proposition is true. So the correct answer should be "FTTFFTTF," which is not available among (a)-(d).

e) is correct. For the rightmost proposition to be true, both sides of the biconditional must have the same truth value. This means that the rightmost proposition is only true if both the biconditional, (p≡r), and the negation, -(p≡q), are true, or else when both the biconditional and the negation are both false. On one hand, for both the biconditional (p≡r) and the negation -(p≡q) to be true, p and r have to agree in their truth-value when p and q disagree in their truth value. So if p and r are both true while q is false, or p and r are both false while q is true, then the rightmost proposition is true. On the other hand, for both the biconditional (p≡r) and the negation -(p≡q) to be false, p and r have to disagree in truth value while p and q agree. So if p and q are both true while r is false, or p and q are both false while r is true, then the rightmost proposition is true. So the correct answer should be "FTTFFTTF," which is not available among (a)-(d).

The truth-functional connective SHMORG has the following truth-table.

p q SHMORG(p,q)

T T F

T F F

F T F

F F T

Which of the following has the same truth-table as SHMORG?

p q SHMORG(p,q)

T T F

T F F

F T F

F F T

Which of the following has the same truth-table as SHMORG?

none of the above.

(e) is correct. SHMORG has the opposite properties of a disjunction. A disjunction is only false when both of the disjuncts are false, and true otherwise. A SHMORG proposition, however, is only true when both of the disjuncts are false, and false otherwise. An ordinary English phrase that would best capture SHMORG is "neither...nor".

(e) is correct. SHMORG has the opposite properties of a disjunction. A disjunction is only false when both of the disjuncts are false, and true otherwise. A SHMORG proposition, however, is only true when both of the disjuncts are false, and false otherwise. An ordinary English phrase that would best capture SHMORG is "neither...nor".

Which of the following would be true whenever SHMORG(p,q) is true?

Neither p nor q is true. (d) is correct. Again, a SHMORG proposition is true exactly when NEITHER of its parts are true. From this, it follows that either p or q is false.

Which of the following arguments is valid?

none of the above.

(e) is correct. Given that SHMORG(p,q) is true exactly when neither p nor q are true, it does not follow from p that SHMORG(p,q). Nor does SHMORG(p,q) follow from -p, or p⊃q.

(e) is correct. Given that SHMORG(p,q) is true exactly when neither p nor q are true, it does not follow from p that SHMORG(p,q). Nor does SHMORG(p,q) follow from -p, or p⊃q.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r SHMORG(p,q) SHMORG(r, SHMORG (p,q))

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r SHMORG(p,q) SHMORG(r, SHMORG (p,q))

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

FTFTFTFF.

(b) is correct. (b). Since SHMORG(p,q) means "neither p nor q" (or, equivalently, "both not-p and not-q"), SHMORG(r, SHMORG(p,q)) is the same thing as, "neither r nor (neither p nor q)". The negation of the conjunction of -p and -q, however, is the same as the disjunction, pVq. So the rightmost conjunction means the same thing as, -r&(pVq). Such a conjunction is only true, however, when r is false while either p or q are true.

(b) is correct. (b). Since SHMORG(p,q) means "neither p nor q" (or, equivalently, "both not-p and not-q"), SHMORG(r, SHMORG(p,q)) is the same thing as, "neither r nor (neither p nor q)". The negation of the conjunction of -p and -q, however, is the same as the disjunction, pVq. So the rightmost conjunction means the same thing as, -r&(pVq). Such a conjunction is only true, however, when r is false while either p or q are true.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r SHMORG(p,q) SHMORG(pVr,SHMORG(p,q))

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r SHMORG(p,q) SHMORG(pVr,SHMORG(p,q))

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

FFFFFTFF.

(d) is correct. Since SHMORG means the same thing as "neither...nor," SHMORG(pVr,SHMORG(p,q)) means the same thing as "it is not the case that pVr and it is also not the case that it is not the case that pVq." The rightmost proposition, in other words, is the following: (-(pVr))&(pVq). This proposition is only false, however, under two conditions: when either p or r are true, or when both p and q are false. Such conditions are met, however, under every truth-value assignment except one. The one truth-value assignment, according to which the rightmost proposition comes out true, is when p and r are both false but q is true. So the correct answer is (d).

(d) is correct. Since SHMORG means the same thing as "neither...nor," SHMORG(pVr,SHMORG(p,q)) means the same thing as "it is not the case that pVr and it is also not the case that it is not the case that pVq." The rightmost proposition, in other words, is the following: (-(pVr))&(pVq). This proposition is only false, however, under two conditions: when either p or r are true, or when both p and q are false. Such conditions are met, however, under every truth-value assignment except one. The one truth-value assignment, according to which the rightmost proposition comes out true, is when p and r are both false but q is true. So the correct answer is (d).

Suppose your premise is (pVq)&[(p⊃q)⊃r]. Which of the following conclusions can be deduced from that premise?

None of the above.

(a) is incorrect. To see why, suppose r is false. Since the truth of the conjunction requires the truth of (p⊃q)⊃r, which is one of its conjuncts, the falsity of r would require the falsity of p⊃q. The falsity of p⊃q requires p to be true and q to be false. If p is true, q is false, and r is false, however, the whole premise is still true. So r does not follow from the premise; the premise can be true while r is false.

(b) is incorrect. If p is false, the first conjunct, pVq, can only be true if q is true. If p is false and q is true, however, then the truth of (p⊃q)⊃r, the premise's other conjunct, would require r to be true as well. So if p is false, q is true, and r is true, the premise could still be true. The truth of the premise does not require the truth of p. That's why (b) is incorrect.

The reason why (c) is incorrect is that, if p is true and q is false, then the whole premise would still be true. Yet if p is true and q is false, p⊃q is also false. So one cannot deduce p⊃q from the premise.

(d) is incorrect. The premise does not require q to be true. The premise could be true, even while q is false.

(e) is correct. From the conjunction (pVq)&[(p⊃q)⊃r], it does not follow that r is true. For suppose r is false. Since the truth of the conjunction requires the truth of (p⊃q)⊃r, which is one of its conjuncts, the falsity of r would require the falsity of p⊃q. The falsity of p⊃q requires p to be true and q to be false. If p is true, q is false, and r is false, however, the whole premise is still true. So r does not follow from the premise; the premise can be true while r is false. (a) is wrong. Likewise for (b). If p is false, the first conjunct, pVq, can only be true if q is true. If p is false and q is true, however, then the truth of (p⊃q)⊃r, the premise's other conjunct, would require r to be true as well. So if p is false, q is true, and r is true, the premise could still be true. The truth of the premise does not require the truth of p. That's why (b) is wrong. The reason why (c) is wrong is that, if p is true and q is false, then the whole premise would still be true. Yet if p is true and q is false, p⊃q is also false. So one cannot deduce p⊃q from the premise. Finally, for the same reason that (c) is wrong, so is (d). The premise does not require q to be true. The premise could be true, even while q is false.

(a) is incorrect. To see why, suppose r is false. Since the truth of the conjunction requires the truth of (p⊃q)⊃r, which is one of its conjuncts, the falsity of r would require the falsity of p⊃q. The falsity of p⊃q requires p to be true and q to be false. If p is true, q is false, and r is false, however, the whole premise is still true. So r does not follow from the premise; the premise can be true while r is false.

(b) is incorrect. If p is false, the first conjunct, pVq, can only be true if q is true. If p is false and q is true, however, then the truth of (p⊃q)⊃r, the premise's other conjunct, would require r to be true as well. So if p is false, q is true, and r is true, the premise could still be true. The truth of the premise does not require the truth of p. That's why (b) is incorrect.

The reason why (c) is incorrect is that, if p is true and q is false, then the whole premise would still be true. Yet if p is true and q is false, p⊃q is also false. So one cannot deduce p⊃q from the premise.

(d) is incorrect. The premise does not require q to be true. The premise could be true, even while q is false.

(e) is correct. From the conjunction (pVq)&[(p⊃q)⊃r], it does not follow that r is true. For suppose r is false. Since the truth of the conjunction requires the truth of (p⊃q)⊃r, which is one of its conjuncts, the falsity of r would require the falsity of p⊃q. The falsity of p⊃q requires p to be true and q to be false. If p is true, q is false, and r is false, however, the whole premise is still true. So r does not follow from the premise; the premise can be true while r is false. (a) is wrong. Likewise for (b). If p is false, the first conjunct, pVq, can only be true if q is true. If p is false and q is true, however, then the truth of (p⊃q)⊃r, the premise's other conjunct, would require r to be true as well. So if p is false, q is true, and r is true, the premise could still be true. The truth of the premise does not require the truth of p. That's why (b) is wrong. The reason why (c) is wrong is that, if p is true and q is false, then the whole premise would still be true. Yet if p is true and q is false, p⊃q is also false. So one cannot deduce p⊃q from the premise. Finally, for the same reason that (c) is wrong, so is (d). The premise does not require q to be true. The premise could be true, even while q is false.

Which of the following pairs of statements is consistent?

All wizards are tall; no wizards are tall.

(a) is correct. If there are no wizards, then the statements "all wizards are tall" and "no wizards are tall" can both be true. "All wizards are tall" would be true because there would not be any non-tall wizards, and "no wizards are tall" would be true because there would not be any tall wizards, either.

(b) is incorrect. If all wizards are tall, then it cannot be true that some wizards are not tall.

(c) is incorrect. If some wizards are tall, then it cannot be true that no wizards are tall.

(d) is incorrect. Not all of the pairs of sentences

among (a)-(c) are consistent.

(e) is incorrect. At least one of the pairs of sentences among (a)-(c) is consistent.

(a) is correct. If there are no wizards, then the statements "all wizards are tall" and "no wizards are tall" can both be true. "All wizards are tall" would be true because there would not be any non-tall wizards, and "no wizards are tall" would be true because there would not be any tall wizards, either.

(b) is incorrect. If all wizards are tall, then it cannot be true that some wizards are not tall.

(c) is incorrect. If some wizards are tall, then it cannot be true that no wizards are tall.

(d) is incorrect. Not all of the pairs of sentences

among (a)-(c) are consistent.

(e) is incorrect. At least one of the pairs of sentences among (a)-(c) is consistent.

Which of the following Venn Diagrams represents the statement that all wizards are tall?

C

(a) is not correct. If we place an X in the region of the circle of wizards that does not intersect with the circle of tall things, and if we shade out the region of the two circles' intersection, this represents the claim, "no wizards are tall and some wizards are not tall."

(b) is not correct. If we put Xs in each of the non-intersecting regions of the circles of wizards and tall things, this represents the claim, "some wizards are not tall and some tall things are not wizards."

(c) is correct. If we shade out the region of the circle of wizards, which is outside of the circle of tall things, and if we leave the rest of the circle of wizards unshaded, then the resultant diagram represents that all wizards are tall.

(d) is not correct. If we shade out the region of the circle of tall things, which does not intersect with the circle of wizards, and if we leave the rest of the circle of tall things unshaded, this represents the claim that all tall things are wizards.

(e) is not correct. One of the diagrams among (a)-(d) represents the claim that all wizards are tall.

(a) is not correct. If we place an X in the region of the circle of wizards that does not intersect with the circle of tall things, and if we shade out the region of the two circles' intersection, this represents the claim, "no wizards are tall and some wizards are not tall."

(b) is not correct. If we put Xs in each of the non-intersecting regions of the circles of wizards and tall things, this represents the claim, "some wizards are not tall and some tall things are not wizards."

(c) is correct. If we shade out the region of the circle of wizards, which is outside of the circle of tall things, and if we leave the rest of the circle of wizards unshaded, then the resultant diagram represents that all wizards are tall.

(d) is not correct. If we shade out the region of the circle of tall things, which does not intersect with the circle of wizards, and if we leave the rest of the circle of tall things unshaded, this represents the claim that all tall things are wizards.

(e) is not correct. One of the diagrams among (a)-(d) represents the claim that all wizards are tall.

Which of the following Venn Diagrams represents the statement that some dogs are not silly?

B

(a) is not correct. If we shade out the region of the circle of silly things, which does not intersect with the circle of dogs, and if we do not shade anything else, then this represents the claim that all silly things are dogs.

(b) is correct. To represent the claim that some dogs are not silly, all we need to do is place an X in the region of the circle of dogs, which is outside the circle of silly things.

(c) is not correct. If we shade out the intersection between the circle of dogs and the circle of silly things, this represents the claim that no dogs are silly.

(d) is not correct. If we place an X in the region of the circle of silly things, which is outside the circle of dogs, this represents the claim that some silly things are not dogs.

(e) is not correct. One of the diagrams among (a)-(d) represents the claim that some dogs are not silly.

(a) is not correct. If we shade out the region of the circle of silly things, which does not intersect with the circle of dogs, and if we do not shade anything else, then this represents the claim that all silly things are dogs.

(b) is correct. To represent the claim that some dogs are not silly, all we need to do is place an X in the region of the circle of dogs, which is outside the circle of silly things.

(c) is not correct. If we shade out the intersection between the circle of dogs and the circle of silly things, this represents the claim that no dogs are silly.

(d) is not correct. If we place an X in the region of the circle of silly things, which is outside the circle of dogs, this represents the claim that some silly things are not dogs.

(e) is not correct. One of the diagrams among (a)-(d) represents the claim that some dogs are not silly.

Which of the following statements represents the information contained in the following Venn Diagram?

All wizards are tall, and all tall things are wizards.

(a) is not correct. If the diagram showed that some wizards are tall, there would be an X in the circle of wizards, where it intersects with the circle of tall things. There is no such X.

(b) is not correct. If the diagram showed that some wizards are not tall, there would be an X in the region of the circle of wizards, which is outside the circle of tall things. There is no such X.

(c) is correct. Because the diagram shades out the region of the circle of wizards that does not intersect with the circle of tall things, the diagram shows that all wizards are tall. Moreover, since the diagram shades out the region of the circle of tall things that does not intersect with the circle of wizards, the diagram also shows that all tall things are wizards.

(d) is not only incorrect; it is also inconsistent. There is no way for a single Venn diagram to simultaneously represent the claims "no wizards are tall" and "some tall things are wizards."

(e) is not correct. One of the diagrams among (a)-(d) represents the information contained in the Venn diagram.

(a) is not correct. If the diagram showed that some wizards are tall, there would be an X in the circle of wizards, where it intersects with the circle of tall things. There is no such X.

(b) is not correct. If the diagram showed that some wizards are not tall, there would be an X in the region of the circle of wizards, which is outside the circle of tall things. There is no such X.

(c) is correct. Because the diagram shades out the region of the circle of wizards that does not intersect with the circle of tall things, the diagram shows that all wizards are tall. Moreover, since the diagram shades out the region of the circle of tall things that does not intersect with the circle of wizards, the diagram also shows that all tall things are wizards.

(d) is not only incorrect; it is also inconsistent. There is no way for a single Venn diagram to simultaneously represent the claims "no wizards are tall" and "some tall things are wizards."

(e) is not correct. One of the diagrams among (a)-(d) represents the information contained in the Venn diagram.

Which of the following statements represents the information contained in the following Venn Diagram?

No wizards are tall.

(a) is not correct. To represent the claim that some wizards are tall, a Venn diagram would have to place an X in the intersection between the circle of wizards and the circle of tall things. Yet there is no such X in our diagram.

(b) is not correct. To represent that all wizards are tall, and that all tall things are wizards, a Venn diagram would have to shade out everything except the intersection between the circle of wizards and the circle of tall things. Yet that is not how this diagram is shaded.

(c) is correct. To represent that no wizards are tall, a Venn diagram needs to shade out the intersection between the circle of wizards and the circle of tall things.

(d) is not correct. To represent that some wizards are not tall, a Venn diagram would need an X that is inside the circle of wizards but outside the circle of tall things. Yet there is no such X in the diagram.

(e) is not correct. One of the options among (a)-(d) states what the Venn diagram shows.

(a) is not correct. To represent the claim that some wizards are tall, a Venn diagram would have to place an X in the intersection between the circle of wizards and the circle of tall things. Yet there is no such X in our diagram.

(b) is not correct. To represent that all wizards are tall, and that all tall things are wizards, a Venn diagram would have to shade out everything except the intersection between the circle of wizards and the circle of tall things. Yet that is not how this diagram is shaded.

(c) is correct. To represent that no wizards are tall, a Venn diagram needs to shade out the intersection between the circle of wizards and the circle of tall things.

(d) is not correct. To represent that some wizards are not tall, a Venn diagram would need an X that is inside the circle of wizards but outside the circle of tall things. Yet there is no such X in the diagram.

(e) is not correct. One of the options among (a)-(d) states what the Venn diagram shows.

Which of the following is valid?

Not all dodos are quotable.

Therefore, some dodos are not quotable

(a) is correct. From the claim that not all dodos are quotable, it immediately follows that some dodos are not quotable.

(b) is not correct. If some dodos are quotable, this does not tell us whether there are any non-quotable dodos. Suppose there are some dodos, and that all of them are quotable. Then "some dodos are quotable" would be true, even though "some dodos are not quotable" would be false.

(c) is not correct. If no dodos are quotable, this does not imply that there are no non-quotable dodos. Suppose there are some dodos, and that none of them are quotable. In such a case, the claim "no dodos are quotable" would be true, while the claim "no dodos are not quotable" would be false.

(d) is not correct. If all dodos are quotable, then it cannot be true that some dodos are not quotable.

(e) is not correct. One of the immediate categorical inferences among (a)-(d) is valid.

Therefore, some dodos are not quotable

(a) is correct. From the claim that not all dodos are quotable, it immediately follows that some dodos are not quotable.

(b) is not correct. If some dodos are quotable, this does not tell us whether there are any non-quotable dodos. Suppose there are some dodos, and that all of them are quotable. Then "some dodos are quotable" would be true, even though "some dodos are not quotable" would be false.

(c) is not correct. If no dodos are quotable, this does not imply that there are no non-quotable dodos. Suppose there are some dodos, and that none of them are quotable. In such a case, the claim "no dodos are quotable" would be true, while the claim "no dodos are not quotable" would be false.

(d) is not correct. If all dodos are quotable, then it cannot be true that some dodos are not quotable.

(e) is not correct. One of the immediate categorical inferences among (a)-(d) is valid.

Which of the following is valid?

All dodos are quotable.

All quotable things are witches.

Therefore, all dodos are witches.

(a) is not valid. As the diagram shows, "all dodos are quotable" requires us to shade out the circle of dodos that does not intersect with the circle of quotable things. The statement "some quotable things are witches," moreover, requires us to place an X where the circle of quotable things intersects with the circle of witches. It does not follow, however, that there are any dodos who are witches. If some witches were also quotable non-dodos, and if all the witches were quotable non-dodos, then the premises of (a) could be true while the conclusion is false.

(b) is not valid. As the diagram shows, the fact that some dodos are quotable, and the fact that some quotable things are witches, does not entail that some dodos are witches. If some witches were also quotable non-dodos, and if all the witches were quotable non-dodos, then the premises of (b) could be true while the conclusion is false

(c) is not valid. As the diagram shows, "no dodos are quotable" requires us to shade out the circle of dodos where it intersects with the circle of quotable things. The statement "no quotable things are witches," moreover, requires us to shade out the circle of quotable things where it intersects with the circle of witches. Together, these two shading patterns do not show that no dodos are witches, since the intersection between the circle of dodos and the circle of witches is not shaded.

(d) is correct. As the diagram shows, "all dodos are quotable" requires us to shade out the circle of dodos, except where it intersects with the circle of quotable things. The statement "all quotable things are witches," moreover, requires us to shade out the circle of quotable things, except where it intersects with the circle of witches. Together, these two shading patterns show that all dodos are witches.

All quotable things are witches.

Therefore, all dodos are witches.

(a) is not valid. As the diagram shows, "all dodos are quotable" requires us to shade out the circle of dodos that does not intersect with the circle of quotable things. The statement "some quotable things are witches," moreover, requires us to place an X where the circle of quotable things intersects with the circle of witches. It does not follow, however, that there are any dodos who are witches. If some witches were also quotable non-dodos, and if all the witches were quotable non-dodos, then the premises of (a) could be true while the conclusion is false.

(b) is not valid. As the diagram shows, the fact that some dodos are quotable, and the fact that some quotable things are witches, does not entail that some dodos are witches. If some witches were also quotable non-dodos, and if all the witches were quotable non-dodos, then the premises of (b) could be true while the conclusion is false

(c) is not valid. As the diagram shows, "no dodos are quotable" requires us to shade out the circle of dodos where it intersects with the circle of quotable things. The statement "no quotable things are witches," moreover, requires us to shade out the circle of quotable things where it intersects with the circle of witches. Together, these two shading patterns do not show that no dodos are witches, since the intersection between the circle of dodos and the circle of witches is not shaded.

(d) is correct. As the diagram shows, "all dodos are quotable" requires us to shade out the circle of dodos, except where it intersects with the circle of quotable things. The statement "all quotable things are witches," moreover, requires us to shade out the circle of quotable things, except where it intersects with the circle of witches. Together, these two shading patterns show that all dodos are witches.

Statements of the form "GINK A's are B's" are represented by means of Venn Diagrams that look like the one below. Which of the following statements is inconsistent with "GINK sheep are in the field"?

Some sheep are not in the field.

(a) is incorrect. "All the sheep are in the field" is consistent with "GINK sheep are in the field."

(b) is incorrect. If no sheep are not in the field, then it must be that all the sheep are in the field. However, as the diagram shows, such a claim is consistent with "GINK sheep are in the field."

(c) is incorrect. As the diagram shows, "GINK sheep are in the field" does not entail that some sheep are in the field, since there is no X where the circle of sheep intersects with the circle of things in the field. However, because the intersecting region is not shaded, "some sheep are in the field" is still consistent with "GINK sheep are in the field."

(d) is correct. As the diagram shows, if GINK sheep are in the field, this means that all the sheep are in the field with something that is not a sheep. Yet if all the sheep are in the field with a non-sheep, the claim "some sheep are not in the field" cannot be true. So "some sheep are not in the field" is inconsistent with "GINK sheep are in the field."

(e) is incorrect. Only one of the statements among (a)-(d) is inconsistent with "GINK sheep are in the field."

(a) is incorrect. "All the sheep are in the field" is consistent with "GINK sheep are in the field."

(b) is incorrect. If no sheep are not in the field, then it must be that all the sheep are in the field. However, as the diagram shows, such a claim is consistent with "GINK sheep are in the field."

(c) is incorrect. As the diagram shows, "GINK sheep are in the field" does not entail that some sheep are in the field, since there is no X where the circle of sheep intersects with the circle of things in the field. However, because the intersecting region is not shaded, "some sheep are in the field" is still consistent with "GINK sheep are in the field."

(d) is correct. As the diagram shows, if GINK sheep are in the field, this means that all the sheep are in the field with something that is not a sheep. Yet if all the sheep are in the field with a non-sheep, the claim "some sheep are not in the field" cannot be true. So "some sheep are not in the field" is inconsistent with "GINK sheep are in the field."

(e) is incorrect. Only one of the statements among (a)-(d) is inconsistent with "GINK sheep are in the field."

Assuming the definition of "GINK" from the previous question, which of the following statements is consistent with "GINK sheep are in the field"?

two or more of the above

(a) is incorrect. Although "all the sheep are in the field" is consistent with "GINK sheep are in the field," it is not the only such statement among (a)-(d).

(b) is incorrect. If no sheep are not in the field, then it must be that all the sheep are in the field. Although such a claim is consistent with "GINK sheep are in the field," it is not the only such claim among (a)-(d).

(c) is incorrect. Although "some sheep are in the field" is consistent with "GINK sheep are in the field," it it is not the only such statement among (a)-(d). As the diagram shows, "GINK sheep are in the field" does not entail that some sheep are in the field, since there is no X where the circle of sheep intersects with the circle of things in the field. However, because the intersecting region is not shaded, "some sheep are in the field" is still consistent with "GINK sheep are in the field."

(d) is incorrect. As the diagram shows, if GINK sheep are in the field, this means that all the sheep are in the field with something that is not a sheep. Yet if all the sheep are in the field with a non-sheep, the claim "some sheep are not in the field" cannot be true. So "some sheep are not in the field" is inconsistent with "GINK sheep are in the field."

(e) is correct. More than one of the statements among (a)-(d) is inconsistent with "GINK sheep are in the field."

(a) is incorrect. Although "all the sheep are in the field" is consistent with "GINK sheep are in the field," it is not the only such statement among (a)-(d).

(b) is incorrect. If no sheep are not in the field, then it must be that all the sheep are in the field. Although such a claim is consistent with "GINK sheep are in the field," it is not the only such claim among (a)-(d).

(c) is incorrect. Although "some sheep are in the field" is consistent with "GINK sheep are in the field," it it is not the only such statement among (a)-(d). As the diagram shows, "GINK sheep are in the field" does not entail that some sheep are in the field, since there is no X where the circle of sheep intersects with the circle of things in the field. However, because the intersecting region is not shaded, "some sheep are in the field" is still consistent with "GINK sheep are in the field."

(d) is incorrect. As the diagram shows, if GINK sheep are in the field, this means that all the sheep are in the field with something that is not a sheep. Yet if all the sheep are in the field with a non-sheep, the claim "some sheep are not in the field" cannot be true. So "some sheep are not in the field" is inconsistent with "GINK sheep are in the field."

(e) is correct. More than one of the statements among (a)-(d) is inconsistent with "GINK sheep are in the field."

Which of the following is valid?

It is not the case that no otters are authors.

Therefore, some otters are authors.

(a) is not correct. If all otters are authors, then it cannot be true that some otters are not authors.

(b) is not correct. If no otters are authors, it does not follow that no otters are not authors. Suppose there are some otters, all of whom are non-authors. In such a case, the premise would be true and the conclusion would be false.

(c) is not correct. If some otters are authors, it does not follow that there are any non-author otters. It could be that there are some otters, all of whom are authors. In such a case, "some otters are authors" would be true, even though "some otters are not authors" would be false.

(d) is correct. If it is not the case that no otters are authors, it immediately follows that some otters are authors.

(e) is not correct. One of the immediate categorical inferences among (a)-(d) is valid.

Therefore, some otters are authors.

(a) is not correct. If all otters are authors, then it cannot be true that some otters are not authors.

(b) is not correct. If no otters are authors, it does not follow that no otters are not authors. Suppose there are some otters, all of whom are non-authors. In such a case, the premise would be true and the conclusion would be false.

(c) is not correct. If some otters are authors, it does not follow that there are any non-author otters. It could be that there are some otters, all of whom are authors. In such a case, "some otters are authors" would be true, even though "some otters are not authors" would be false.

(d) is correct. If it is not the case that no otters are authors, it immediately follows that some otters are authors.

(e) is not correct. One of the immediate categorical inferences among (a)-(d) is valid.

Which of the following is valid?

All otters are authors.

All authors are fodder.

Therefore, all otters are fodder.

(a) is correct. As the diagram shows, "all otters are authors" requires us to shade out the circle of otters, except where it intersects with the circle of authors. The statement "all authors are fodder," moreover, requires us to shade out the circle of authors, except where it intersects with the circle of fodder. Together, these two shading patterns show that all otters are fodder.

(b) is not correct. As the diagram shows, "all otters are authors" requires us to shade out the circle of otters that does not intersect with the circle of authors. The statement "some authors are fodder," moreover, requires us to place an X where the circle of authors intersects with the circle of fodder. It does not follow, however, that there are any otters who are fodder. If some fodder were also non-otter authors, and if all the fodder were non-otter authors, then the premises of (b) could be true while the conclusion is false. (c) is not correct. As the diagram shows, the fact that some otters are authors, and the fact that some authors are fodder, does not entail that some otters are fodder. If some fodder were also non-otter authors, and if all the fodder were non-otter authors, then the premises of (c) could be true while the conclusion is false. (d) is not correct. As the diagram shows, "no otters are authors" requires us to shade out the circle of otters where it intersects with the circle of authors. The statement "no authors are fodder," moreover, requires us to shade out the circle of authors where it intersects with the circle of fodder. Together, these two shading patterns do not show that no otters are fodder, since the intersection between the circle of otters and the circle of fodder is not shaded. (e) is not correct. One of the options among (a)-(d) is a valid syllogism.

All authors are fodder.

Therefore, all otters are fodder.

(a) is correct. As the diagram shows, "all otters are authors" requires us to shade out the circle of otters, except where it intersects with the circle of authors. The statement "all authors are fodder," moreover, requires us to shade out the circle of authors, except where it intersects with the circle of fodder. Together, these two shading patterns show that all otters are fodder.

(b) is not correct. As the diagram shows, "all otters are authors" requires us to shade out the circle of otters that does not intersect with the circle of authors. The statement "some authors are fodder," moreover, requires us to place an X where the circle of authors intersects with the circle of fodder. It does not follow, however, that there are any otters who are fodder. If some fodder were also non-otter authors, and if all the fodder were non-otter authors, then the premises of (b) could be true while the conclusion is false. (c) is not correct. As the diagram shows, the fact that some otters are authors, and the fact that some authors are fodder, does not entail that some otters are fodder. If some fodder were also non-otter authors, and if all the fodder were non-otter authors, then the premises of (c) could be true while the conclusion is false. (d) is not correct. As the diagram shows, "no otters are authors" requires us to shade out the circle of otters where it intersects with the circle of authors. The statement "no authors are fodder," moreover, requires us to shade out the circle of authors where it intersects with the circle of fodder. Together, these two shading patterns do not show that no otters are fodder, since the intersection between the circle of otters and the circle of fodder is not shaded. (e) is not correct. One of the options among (a)-(d) is a valid syllogism.

Assuming the definition of "GINK" from question 24, which of the following statements are entailed by the conjunction, "GINK sheep are in the field and all of the things in the field are purple"?

Two or more of the above.

(a) is not correct. "GINK sheep are in the field" means the same as "all the sheep are in the field, and some non-sheep is in the field." This claim, when conjoined with "all of the things are purple," does entail the statement, "all sheep are purple." However, it is not the only such statement among (a)-(d).

(b) is not correct. "GINK sheep are in the field" means the same as "all the sheep are in the field, and some non-sheep is in the field." This claim, when conjoined with "all of the things are purple," does not entail the statement, "no sheep is purple."

(c) is not correct. "GINK sheep are in the field" means the same as "all the sheep are in the field, and some non-sheep is in the field." This claim, when conjoined with "all of the things are purple," does entail the statement, "some non-sheep is purple." However, it is not the only such statement among (a)-(d).

(d) is not correct. "GINK sheep are in the field" means the same as "all the sheep are in the field, and some non-sheep is in the field." This claim, when conjoined with "all the things in the field are purple," does not entail the statement, "nothing in the field is purple." In fact, it entails something incompatible with "nothing in the field is purple," namely "something in the field is purple."

(e) is correct. (a) is not correct. "GINK sheep are in the field" means the same as "all the sheep are in the field, and some non-sheep is in the field." This claim, when conjoined with "all of the things are purple," entails the statement "all sheep are purple," as well as the statement "some non-sheep is purple."

(a) is not correct. "GINK sheep are in the field" means the same as "all the sheep are in the field, and some non-sheep is in the field." This claim, when conjoined with "all of the things are purple," does entail the statement, "all sheep are purple." However, it is not the only such statement among (a)-(d).

(b) is not correct. "GINK sheep are in the field" means the same as "all the sheep are in the field, and some non-sheep is in the field." This claim, when conjoined with "all of the things are purple," does not entail the statement, "no sheep is purple."

(c) is not correct. "GINK sheep are in the field" means the same as "all the sheep are in the field, and some non-sheep is in the field." This claim, when conjoined with "all of the things are purple," does entail the statement, "some non-sheep is purple." However, it is not the only such statement among (a)-(d).

(d) is not correct. "GINK sheep are in the field" means the same as "all the sheep are in the field, and some non-sheep is in the field." This claim, when conjoined with "all the things in the field are purple," does not entail the statement, "nothing in the field is purple." In fact, it entails something incompatible with "nothing in the field is purple," namely "something in the field is purple."

(e) is correct. (a) is not correct. "GINK sheep are in the field" means the same as "all the sheep are in the field, and some non-sheep is in the field." This claim, when conjoined with "all of the things are purple," entails the statement "all sheep are purple," as well as the statement "some non-sheep is purple."

Assuming the definition of "GINK" from question (25), which of the following statements are entailed by the conjunction, "GINK sheep are in the field and some of the things in the field are purple"?

None of the above.

(a) is incorrect. "GINK sheep are in the field" means the same thing as "all the sheep are in the field and some non-sheep is also in the field." This conjunction, when combined with the claim "some of the things in the field are purple," does not entail that all sheep are purple. It could be that some but not all of the things in the field are purple, and that some of the non-purple things are the sheep.

(b) is incorrect. Again, "GINK sheep are in the field" means the same thing as "all the sheep are in the field and some non-sheep is also in the field." This conjunction, when combined with the claim "some of the things in the field are purple," does not entail that no sheep is purple. It could be that the sheep are some of the purple things in the field.

(c) is incorrect. Again, "GINK sheep are in the field" means the same thing as "all the sheep are in the field and some non-sheep is also in the field." This conjunction, when combined with the claim "some of the things in the field are purple," does not entail that some non-sheep is purple. It could be that a non-purple non-sheep is in the field with purple sheep.

(d) is incorrect. Again, "GINK sheep are in the field" means the same thing as "all the sheep are in the field and some non-sheep is also in the field." This conjunction, when combined with the claim "some of the things in the field are purple," does not entail that no non-sheep is purple. It could be that the non-sheep, who is in the field with all the sheep, is one of the purple things in the field.

(e) is correct. None of the statements among (a)-(e) is entailed by the conjunction, "GINK sheep are in the field and some of the things in the field are purple."

(a) is incorrect. "GINK sheep are in the field" means the same thing as "all the sheep are in the field and some non-sheep is also in the field." This conjunction, when combined with the claim "some of the things in the field are purple," does not entail that all sheep are purple. It could be that some but not all of the things in the field are purple, and that some of the non-purple things are the sheep.

(b) is incorrect. Again, "GINK sheep are in the field" means the same thing as "all the sheep are in the field and some non-sheep is also in the field." This conjunction, when combined with the claim "some of the things in the field are purple," does not entail that no sheep is purple. It could be that the sheep are some of the purple things in the field.

(c) is incorrect. Again, "GINK sheep are in the field" means the same thing as "all the sheep are in the field and some non-sheep is also in the field." This conjunction, when combined with the claim "some of the things in the field are purple," does not entail that some non-sheep is purple. It could be that a non-purple non-sheep is in the field with purple sheep.

(d) is incorrect. Again, "GINK sheep are in the field" means the same thing as "all the sheep are in the field and some non-sheep is also in the field." This conjunction, when combined with the claim "some of the things in the field are purple," does not entail that no non-sheep is purple. It could be that the non-sheep, who is in the field with all the sheep, is one of the purple things in the field.

(e) is correct. None of the statements among (a)-(e) is entailed by the conjunction, "GINK sheep are in the field and some of the things in the field are purple."

Suppose you had to fill in the rightmost column of the truth table below:

p q p&-q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

p q p&-q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

FTFF

In order for a conjunction to be true, both conjuncts must be true. If one of the conjuncts is a negation, like "-q", then the sentence that is negated (in this case, q) has to be false in order for the conjunct "-q" to be true. So the conjunction "p&-q" is only true when q is false and p is true.

In order for a conjunction to be true, both conjuncts must be true. If one of the conjuncts is a negation, like "-q", then the sentence that is negated (in this case, q) has to be false in order for the conjunct "-q" to be true. So the conjunction "p&-q" is only true when q is false and p is true.

Suppose you had to fill in the rightmost column of the truth table below:

p q p⊃-q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

p q p⊃-q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

FTTT

In order for a conditional to be false, the antecedent must be true and the consequent must be false. If the consequent is a negation, like -q, then the sentence that is negated (in this case, q) has to be true for the consequent to be false. So the conditional "p⊃-q" is only false when p is true and q is true, and it is true otherwise.

In order for a conditional to be false, the antecedent must be true and the consequent must be false. If the consequent is a negation, like -q, then the sentence that is negated (in this case, q) has to be true for the consequent to be false. So the conditional "p⊃-q" is only false when p is true and q is true, and it is true otherwise.

Suppose you had to fill in the rightmost column of the truth table below:

p q r p&-(q&r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill in that column?

p q r p&-(q&r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill in that column?

FTTTFFFF

In order for a conjunction to be true, both conjuncts must be true. In order for a negation to be true, the sentence, which is being negated, must be false. If the sentence being negated is a conjunction (such as "q&r"), then at least one of the conjuncts (either q or r, in this case) must be false. So for the conjunction "p&-(q&r)" to be true, p must be true, and either q or r must be false.

In order for a conjunction to be true, both conjuncts must be true. In order for a negation to be true, the sentence, which is being negated, must be false. If the sentence being negated is a conjunction (such as "q&r"), then at least one of the conjuncts (either q or r, in this case) must be false. So for the conjunction "p&-(q&r)" to be true, p must be true, and either q or r must be false.

Which of these claims follows from p&-q?

none of the above.

Given a conjunction, one can infer either of its conjuncts. So, from the conjunction "p&-q," one can infer "p," and one can infer "-q." Both "p" and "-q" follow from "p&-q." None of the options among (a)-(d) follow, however.

Given a conjunction, one can infer either of its conjuncts. So, from the conjunction "p&-q," one can infer "p," and one can infer "-q." Both "p" and "-q" follow from "p&-q." None of the options among (a)-(d) follow, however.

Which of these claims follows from p⊃-q?

none of the above.

From a conditional, and nothing else, one cannot infer its antecedent, nor can one infer its consequent. One can only infer the consequent if the premises include both the conditional and its antecedent, and one can only infer the negation of the antecedent if the premises include both the conditional and the negation of the consequent. Since all we have is the conditional, none of (a)-(d) follow.

From a conditional, and nothing else, one cannot infer its antecedent, nor can one infer its consequent. One can only infer the consequent if the premises include both the conditional and its antecedent, and one can only infer the negation of the antecedent if the premises include both the conditional and the negation of the consequent. Since all we have is the conditional, none of (a)-(d) follow.

Which of these claims follows from p&-(q&r)?

p

From a conjunction, one can infer any of its conjuncts. One cannot infer the negation of any of its conjuncts. So, from the conjunction "p&-(q&r)," it follows that p, and it also follows that -(q&r). Since "p" is the only one of these options among (a)-(d), it follows that (a) is the correct answer.

From a conjunction, one can infer any of its conjuncts. One cannot infer the negation of any of its conjuncts. So, from the conjunction "p&-(q&r)," it follows that p, and it also follows that -(q&r). Since "p" is the only one of these options among (a)-(d), it follows that (a) is the correct answer.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p&q -(p&q) -r -(pVq)V-r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p&q -(p&q) -r -(pVq)V-r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

FTTTTTTT

(a) is correct. The rightmost proposition is a disjunction. The only way for a disjunction to be false is for both of its disjuncts to be false. The first disjunct, -(p&q), is only false when the conjunction of p&q is true. So for the rightmost proposition to be false, both p and q have to be true. The second disjunct of the rightmost proposition is -r, which is only false when r is true. So the only way for the rightmost proposition to be false is for p, q, and r to all be true. Otherwise, the rightmost proposition is true.

(a) is correct. The rightmost proposition is a disjunction. The only way for a disjunction to be false is for both of its disjuncts to be false. The first disjunct, -(p&q), is only false when the conjunction of p&q is true. So for the rightmost proposition to be false, both p and q have to be true. The second disjunct of the rightmost proposition is -r, which is only false when r is true. So the only way for the rightmost proposition to be false is for p, q, and r to all be true. Otherwise, the rightmost proposition is true.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p&q (p&q)Vr -[(p&q)Vr]

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p&q (p&q)Vr -[(p&q)Vr]

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

FFFTFTFT

(c) is correct. For the rightmost proposition to be true, the following disjunction must be false: (p&q)Vr. For that disjunction to be false, however, both disjuncts have to be false. This means that r must be false. The other disjunct, (p&q), is false when either of its conjuncts are. So there are only three truth-value assignments that make the rightmost proposition true: one where p, q, and r are all false; one where p and r are false but q is true; and one where q and r are false but p is true.

(c) is correct. For the rightmost proposition to be true, the following disjunction must be false: (p&q)Vr. For that disjunction to be false, however, both disjuncts have to be false. This means that r must be false. The other disjunct, (p&q), is false when either of its conjuncts are. So there are only three truth-value assignments that make the rightmost proposition true: one where p, q, and r are all false; one where p and r are false but q is true; and one where q and r are false but p is true.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p⊃q r⊃(q⊃p)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p⊃q r⊃(q⊃p)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

TTTTFTTT

(b) is correct. The rightmost proposition is conditional. So it is only false when its antecedent is true and its consequent is false. So the rightmost proposition is only false when r is true but q⊃p is false. For q⊃p to be false, however, q has to be true while p is false. So there is only one truth-value assignment, which makes the whole rightmost proposition false: it is when r is true, q is true, and p is false.

(b) is correct. The rightmost proposition is conditional. So it is only false when its antecedent is true and its consequent is false. So the rightmost proposition is only false when r is true but q⊃p is false. For q⊃p to be false, however, q has to be true while p is false. So there is only one truth-value assignment, which makes the whole rightmost proposition false: it is when r is true, q is true, and p is false.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p≡q p≡r (p≡q)&(p≡r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p≡q p≡r (p≡q)&(p≡r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

TFFFFFFT

(d) is correct. For the rightmost proposition to be true, both of its conjuncts have to be true. Since the first conjunct is p≡q, and the second conjunct is p≡r, it follows that, for the whole rightmost proposition to be true, p, q, and r must all have the same truth-values. So there are only two truth-value assignments, according to which the whole rightmost proposition comes out as true: one in which p, q, and r are all true; and one in which p, q, and r are all false.

(d) is correct. For the rightmost proposition to be true, both of its conjuncts have to be true. Since the first conjunct is p≡q, and the second conjunct is p≡r, it follows that, for the whole rightmost proposition to be true, p, q, and r must all have the same truth-values. So there are only two truth-value assignments, according to which the whole rightmost proposition comes out as true: one in which p, q, and r are all true; and one in which p, q, and r are all false.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p≡q pVr (p≡q)≡(pVr) -[(p≡q)≡(pVr)]

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p≡q pVr (p≡q)≡(pVr) -[(p≡q)≡(pVr)]

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

none of the above.

(e) is correct. Since the rightmost proposition is the negation of a biconditional, the rightmost proposition will only be true when p≡q has a different truth value than pVr. So the rightmost proposition will only be true under two conditions: either p≡q is true while pVr is false, or pVr is true while p≡q is false. For pVr to be false, however, both p and r have to be false. For p≡q to be true while pVr is false, then, p, q, and r would all have to be false. For p≡q to be false while pVr is true, however, all that has to happen is that either p or r is true while p and q differ in truth values. This could happen on three truth-value assignments: one where p is true, q is false, and r is true; one where p is true, q is false, and r is false; and one where p is false, q is true, and r is true. So there are four truth-value assignments, according tow which the rightmost proposition comes out true. The answer should read, "FFTTTFFT," which is not among (a)-(d).

(e) is correct. Since the rightmost proposition is the negation of a biconditional, the rightmost proposition will only be true when p≡q has a different truth value than pVr. So the rightmost proposition will only be true under two conditions: either p≡q is true while pVr is false, or pVr is true while p≡q is false. For pVr to be false, however, both p and r have to be false. For p≡q to be true while pVr is false, then, p, q, and r would all have to be false. For p≡q to be false while pVr is true, however, all that has to happen is that either p or r is true while p and q differ in truth values. This could happen on three truth-value assignments: one where p is true, q is false, and r is true; one where p is true, q is false, and r is false; and one where p is false, q is true, and r is true. So there are four truth-value assignments, according tow which the rightmost proposition comes out true. The answer should read, "FFTTTFFT," which is not among (a)-(d).

The truth-functional connective SKANG has the following truth-table.

p q SKANG(p,q)

T T F

T F T

F T F

F F F

Which of the following has the same truth-table as SKANG?

p q SKANG(p,q)

T T F

T F T

F T F

F F F

Which of the following has the same truth-table as SKANG?

p&-q

(b) is correct. Note that SKANG(p,q) also has the same truth-table as -(p⊃q). This means that it also has the same truth-table as p&-q.

(b) is correct. Note that SKANG(p,q) also has the same truth-table as -(p⊃q). This means that it also has the same truth-table as p&-q.

Which of the following will be true when and only when SKANG(p,q) is true?

two or more of the above.

(a) is not correct. If SKANG(p,q) is true, this requires q to be false.

(b) is not correct. It is correct that, whenever SKANG(p,q) is true, either p or q will also be true. However, (b) is not the only statement among (a)-(d), which is true whenever SKANG(p,q) is.

(c) is not correct. It is correct that, whenever SKANG(p,q) is true, p and q have different truth-values. However, (c) is not the only statement among (a)-(d), which is true whenever SKANG(p,q) is.

(d) is not correct. It is correct that, whenever SKANG(p,q) is true, p and q will not both be true. However, (d) is not the only statement among (a)-(d), which is true whenever SKANG(p,q) is.

(e) is correct. If SKANG(p,q) is true, this means that p is true but q is false. Under such conditions, (b), (c), and (d) would all be true. (b) would be true because p is true, and if p is true then so is either p or q. (c) would be true because, if p is true and q is false, then p and q have different truth-values. Finally, if p is true and q is false, then it is not the case that both p and q are true. This would mean that (d) follows as well.

(a) is not correct. If SKANG(p,q) is true, this requires q to be false.

(b) is not correct. It is correct that, whenever SKANG(p,q) is true, either p or q will also be true. However, (b) is not the only statement among (a)-(d), which is true whenever SKANG(p,q) is.

(c) is not correct. It is correct that, whenever SKANG(p,q) is true, p and q have different truth-values. However, (c) is not the only statement among (a)-(d), which is true whenever SKANG(p,q) is.

(d) is not correct. It is correct that, whenever SKANG(p,q) is true, p and q will not both be true. However, (d) is not the only statement among (a)-(d), which is true whenever SKANG(p,q) is.

(e) is correct. If SKANG(p,q) is true, this means that p is true but q is false. Under such conditions, (b), (c), and (d) would all be true. (b) would be true because p is true, and if p is true then so is either p or q. (c) would be true because, if p is true and q is false, then p and q have different truth-values. Finally, if p is true and q is false, then it is not the case that both p and q are true. This would mean that (d) follows as well.

Which of the following arguments is valid?

none of the above.

(e) is correct. Since SKANG(p,q) is true exactly when p is true and q is false, none of (a)-(d) are valid. Solely from the truth of p, it does not follow that p is true and q is false. Nor can the truth of p and the falsity of q follow from the falsity of p. Nor can the truth of p and the falsity of q follow from p&q. Nor does the truth of p and the falsity of q follow from p⊃q.

(e) is correct. Since SKANG(p,q) is true exactly when p is true and q is false, none of (a)-(d) are valid. Solely from the truth of p, it does not follow that p is true and q is false. Nor can the truth of p and the falsity of q follow from the falsity of p. Nor can the truth of p and the falsity of q follow from p&q. Nor does the truth of p and the falsity of q follow from p⊃q.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r SKANG(p,q) SKANG(r, SKANG (p,q))

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r SKANG(p,q) SKANG(r, SKANG (p,q))

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

none of the above.

(e) is correct. Since SKANG(p,q) is true exactly when p is true and q is false, the rightmost proposition is equivalent to the following conjunction: (r)&-(p&-q). For the rightmost proposition to be true, then, r has to be true while the conjunction p&-q is FALSE. There are two conditions that allow p&-q to be false, however: one where p is false, and one where q is true. So, in order for the rightmost conjunction to be true, r has to be true while either p is false or q is true. So there are only three truth-value assignments, according to which the rightmost proposition comes out true: one in which p is true, q is true, and r is true; one in which p is false, q is true, and r is true; and one in which p is false, q is false, and r is true. The correct answer should read "TFFFTFTF," which is not among (a)-(d).

(e) is correct. Since SKANG(p,q) is true exactly when p is true and q is false, the rightmost proposition is equivalent to the following conjunction: (r)&-(p&-q). For the rightmost proposition to be true, then, r has to be true while the conjunction p&-q is FALSE. There are two conditions that allow p&-q to be false, however: one where p is false, and one where q is true. So, in order for the rightmost conjunction to be true, r has to be true while either p is false or q is true. So there are only three truth-value assignments, according to which the rightmost proposition comes out true: one in which p is true, q is true, and r is true; one in which p is false, q is true, and r is true; and one in which p is false, q is false, and r is true. The correct answer should read "TFFFTFTF," which is not among (a)-(d).

Suppose your premise is [(p⊃q)⊃r]Vq. Which of the following conclusions can be deduced from that premise?

none of the above.

(a) is not correct. The premise [(p⊃q)⊃r]Vq can be true, even if q⊃p is false. To see why, let's suppose that the premise is true, but q⊃p is false. If q⊃p is false, then q must be true, while p is false. Yet if q is true, that means that any disjunction, which has q as a disjunct, is also true. The premise [(p⊃q)⊃r]Vq is such a disjunction. Since the conclusion can be false while the premise is true, the conclusion does not follow from the premise.

(b) is not correct. Suppose that p is true, q is false, and r is true. Then the whole premise, [(p⊃q)⊃r]Vq, is true without p⊃q being true.

(c) is not correct. Suppose that p is false and r is false. Then the whole premise, [(p⊃q)⊃r]Vq, could still be true, so long as q is true.

(d) is not correct. Not all of (a)-(c) can be deduced from the premise [(p⊃q)⊃r]Vq.

(e) is correct; none of (a)-(d) follows from the premise [(p⊃q)⊃r]Vq.

(a) is not correct. The premise [(p⊃q)⊃r]Vq can be true, even if q⊃p is false. To see why, let's suppose that the premise is true, but q⊃p is false. If q⊃p is false, then q must be true, while p is false. Yet if q is true, that means that any disjunction, which has q as a disjunct, is also true. The premise [(p⊃q)⊃r]Vq is such a disjunction. Since the conclusion can be false while the premise is true, the conclusion does not follow from the premise.

(b) is not correct. Suppose that p is true, q is false, and r is true. Then the whole premise, [(p⊃q)⊃r]Vq, is true without p⊃q being true.

(c) is not correct. Suppose that p is false and r is false. Then the whole premise, [(p⊃q)⊃r]Vq, could still be true, so long as q is true.

(d) is not correct. Not all of (a)-(c) can be deduced from the premise [(p⊃q)⊃r]Vq.

(e) is correct; none of (a)-(d) follows from the premise [(p⊃q)⊃r]Vq.

The truth-functional connective BLIM has the following truth-table.

p q r BLIM(p,q,r)

T T T T

T T F F

T F T F

T F F T

F T T T

F T F F

F F T F

F F F T

Which of the following has the same truth-table as BLIM(p,q,r)?

p q r BLIM(p,q,r)

T T T T

T T F F

T F T F

T F F T

F T T T

F T F F

F F T F

F F F T

Which of the following has the same truth-table as BLIM(p,q,r)?

(p≡q)≡(p≡r)

(a) is incorrect because pV(qVr) would be false if p, q, and r were all false. Yet under such a truth-value assignment, the premise would still be true.

(b) is incorrect because (p⊃q)⊃r would be true if p were true, q were false, and r were true, even though BLIM(p,q,r) would be false under that assignment.

(c) is correct. The proposition BLIM(p,q,r) has the same truth-value as (p≡q)≡(p≡r). To see why, note that (p≡q)≡(p≡r) is only true when each of the two biconditionals, (p≡q) and (p≡r), have the same truth value. So there are only two ways for (p≡q)≡(p≡r) to be true: either (p≡q) and (p≡r) are both true, or (p≡q) and (p≡r) are both false. There are only two ways for (p≡q) and (p≡r) to both be true, however: either p, q, and r are all true, or p, q, and r are all false. There are only two ways, moreover, for both (p≡q) and (p≡r) to be false: either p is true while both q and r are false, or else p is false while both q and r are true. So there are exactly four truth-value assignments, which make (p≡q)≡(p≡r) true. They are when p, q, and r are all true; when p is true while q and r are false; when p is false while q and r are true; and when p, q, and r are all false. This is the exact same truth-table for BLIM(p,q,r), however.

(d) is incorrect because, if p were true, q were false, and r were true, the premise BLIM(p,q,r) would be false, even though (p≣q)⊃r would still be true.

(e) is incorrect. One of the options among (a)-(d) has the same truth table as BLIM(p,q,r)

(a) is incorrect because pV(qVr) would be false if p, q, and r were all false. Yet under such a truth-value assignment, the premise would still be true.

(b) is incorrect because (p⊃q)⊃r would be true if p were true, q were false, and r were true, even though BLIM(p,q,r) would be false under that assignment.

(c) is correct. The proposition BLIM(p,q,r) has the same truth-value as (p≡q)≡(p≡r). To see why, note that (p≡q)≡(p≡r) is only true when each of the two biconditionals, (p≡q) and (p≡r), have the same truth value. So there are only two ways for (p≡q)≡(p≡r) to be true: either (p≡q) and (p≡r) are both true, or (p≡q) and (p≡r) are both false. There are only two ways for (p≡q) and (p≡r) to both be true, however: either p, q, and r are all true, or p, q, and r are all false. There are only two ways, moreover, for both (p≡q) and (p≡r) to be false: either p is true while both q and r are false, or else p is false while both q and r are true. So there are exactly four truth-value assignments, which make (p≡q)≡(p≡r) true. They are when p, q, and r are all true; when p is true while q and r are false; when p is false while q and r are true; and when p, q, and r are all false. This is the exact same truth-table for BLIM(p,q,r), however.

(d) is incorrect because, if p were true, q were false, and r were true, the premise BLIM(p,q,r) would be false, even though (p≣q)⊃r would still be true.

(e) is incorrect. One of the options among (a)-(d) has the same truth table as BLIM(p,q,r)

BLIM(p,q,r) is true if which of the following are true?

none of the above.

(a) is not correct. BLIM(p,q,r) is logically equivalent to (p≡q)≡(p≡r). It cannot be true when p and q are true, but r is false.

(b) is not correct. BLIM(p,q,r) is logically equivalent to (p≡q)≡(p≡r). It cannot be true when p and r are true, but q is false..

(c) is not correct. BLIM(p,q,r) is logically equivalent to (p≡q)≡(p≡r). It cannot be true when p and r are false, but q is true.

(d) is not correct. BLIM(p,q,r) is logically equivalent to (p≡q)≡(p≡r). It cannot be true when p and q are false, but r is true.

(e) is correct. None of the options among (a)-(d) is true if BLIM(p,q,r) is true.

(a) is not correct. BLIM(p,q,r) is logically equivalent to (p≡q)≡(p≡r). It cannot be true when p and q are true, but r is false.

(b) is not correct. BLIM(p,q,r) is logically equivalent to (p≡q)≡(p≡r). It cannot be true when p and r are true, but q is false..

(c) is not correct. BLIM(p,q,r) is logically equivalent to (p≡q)≡(p≡r). It cannot be true when p and r are false, but q is true.

(d) is not correct. BLIM(p,q,r) is logically equivalent to (p≡q)≡(p≡r). It cannot be true when p and q are false, but r is true.

(e) is correct. None of the options among (a)-(d) is true if BLIM(p,q,r) is true.

Which of the following arguments is valid?

p&q&r

BLIM(p,q,r)

(a) is correct. BLIM(p,q,r) is logically equivalent to (p≡q)≡(p≡r). So if p, q, and r are all true, then p, q, and r all have the same truth values. It would follow that BLIM(p,q,r). So (a) is true.

BLIM(p,q,r)

(a) is correct. BLIM(p,q,r) is logically equivalent to (p≡q)≡(p≡r). So if p, q, and r are all true, then p, q, and r all have the same truth values. It would follow that BLIM(p,q,r). So (a) is true.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p≡q p≡r q≡r BLIM(p≡q,p≡r,q≡r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p≡q p≡r q≡r BLIM(p≡q,p≡r,q≡r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

TTFFFFTT

(b) is correct. Since the proposition BLIM(p,q,r) is equivalent to (p≡q)≡(p≡r), it follows that BLIM(p≡q,p≡r,q≡r) is equivalent to the following biconditional: ((p≡q)≡(p≡r))≡((p≡q)≡(q≡r)). Such a propositon, however, is a biconditional, which is only true when both of its propositional parts have the same truth-value. There are only four truth-value assignments, according to which the rightmost proposition is true: one according to which all three sentences are true; one according to which both p and q are true while r is false; one according to which both p and q are false while r is true; and one according to which all three sentences are false.

(b) is correct. Since the proposition BLIM(p,q,r) is equivalent to (p≡q)≡(p≡r), it follows that BLIM(p≡q,p≡r,q≡r) is equivalent to the following biconditional: ((p≡q)≡(p≡r))≡((p≡q)≡(q≡r)). Such a propositon, however, is a biconditional, which is only true when both of its propositional parts have the same truth-value. There are only four truth-value assignments, according to which the rightmost proposition is true: one according to which all three sentences are true; one according to which both p and q are true while r is false; one according to which both p and q are false while r is true; and one according to which all three sentences are false.

Which of the following inferences is valid?

none of the above.

(a) is incorrect. Since the proposition BLIM(p,q,r) is equivalent to (p≡q)≡(p≡r), it follows that BLIM(p⊃q,p⊃r,q⊃r) is equivalent to the following biconditional: ((p⊃q)≡(p⊃r))≡((p⊃q)≡(q⊃r)). Since that biconditional can be true while p is true, q is true, and r is false, the biconditional does not require the truth of q⊃r.

(b) is incorrect. Since the proposition BLIM(p,q,r) is equivalent to (p≣q)≣(p≣r), it follows that BLIM(p⊃q,p⊃r,q⊃r) is equivalent to the following biconditional: ((p⊃q)≡(p⊃r))≡((p⊃q)≡(q⊃r)). Since the biconditional can be true while p is false, q is false, and r is true, the biconditional does not require the truth of p⊃r.

(c) is incorrect. Since the proposition BLIM(p,q,r) is equivalent to (p≡q)≡(p≡r), it follows that BLIM(p⊃q,p⊃r,q⊃r) is equivalent to the following biconditional: ((p⊃q)≡(p⊃r))≡((p⊃q)≡(q⊃r)). That whole biconditional can be true while p is false.

(d) is incorrect. Since the proposition BLIM(p,q,r) is equivalent to (p≡q)≡(p≡r), it follows that BLIM(p⊃q,p⊃r,q⊃r) is equivalent to the following biconditional: ((p⊃q)≡(p⊃r))≡((p⊃q)≡(q⊃r)). That whole biconditional can be true while p is true, q is true, and r is false, so it follows that the biconditional can be true even when p≡r is false.

(e) is correct. Since the proposition BLIM(p,q,r) is equivalent to (p≡q)≡(p≡r), it follows that BLIM(p⊃q,p⊃r,q⊃r) is equivalent to the following biconditional: ((p⊃q)≡(p⊃r))≡((p⊃q)≡(q⊃r)). Since that biconditional can be true while p is true, q is true, and r is false, the biconditional does not require the truth of q⊃r. So (a) is incorrect. Since the biconditional can be true while p is false, q is false, and r is true, the biconditional does not require the truth of p⊃r. So (b) is incorrect. Likewise, (c) is incorrect, since the whole biconditional can be true while p is false. Finally, since the biconditional can be true while p is true, q is true, and r is false, it follows that the biconditional can be true even when p≣r is false.

(a) is incorrect. Since the proposition BLIM(p,q,r) is equivalent to (p≡q)≡(p≡r), it follows that BLIM(p⊃q,p⊃r,q⊃r) is equivalent to the following biconditional: ((p⊃q)≡(p⊃r))≡((p⊃q)≡(q⊃r)). Since that biconditional can be true while p is true, q is true, and r is false, the biconditional does not require the truth of q⊃r.

(b) is incorrect. Since the proposition BLIM(p,q,r) is equivalent to (p≣q)≣(p≣r), it follows that BLIM(p⊃q,p⊃r,q⊃r) is equivalent to the following biconditional: ((p⊃q)≡(p⊃r))≡((p⊃q)≡(q⊃r)). Since the biconditional can be true while p is false, q is false, and r is true, the biconditional does not require the truth of p⊃r.

(c) is incorrect. Since the proposition BLIM(p,q,r) is equivalent to (p≡q)≡(p≡r), it follows that BLIM(p⊃q,p⊃r,q⊃r) is equivalent to the following biconditional: ((p⊃q)≡(p⊃r))≡((p⊃q)≡(q⊃r)). That whole biconditional can be true while p is false.

(d) is incorrect. Since the proposition BLIM(p,q,r) is equivalent to (p≡q)≡(p≡r), it follows that BLIM(p⊃q,p⊃r,q⊃r) is equivalent to the following biconditional: ((p⊃q)≡(p⊃r))≡((p⊃q)≡(q⊃r)). That whole biconditional can be true while p is true, q is true, and r is false, so it follows that the biconditional can be true even when p≡r is false.

(e) is correct. Since the proposition BLIM(p,q,r) is equivalent to (p≡q)≡(p≡r), it follows that BLIM(p⊃q,p⊃r,q⊃r) is equivalent to the following biconditional: ((p⊃q)≡(p⊃r))≡((p⊃q)≡(q⊃r)). Since that biconditional can be true while p is true, q is true, and r is false, the biconditional does not require the truth of q⊃r. So (a) is incorrect. Since the biconditional can be true while p is false, q is false, and r is true, the biconditional does not require the truth of p⊃r. So (b) is incorrect. Likewise, (c) is incorrect, since the whole biconditional can be true while p is false. Finally, since the biconditional can be true while p is true, q is true, and r is false, it follows that the biconditional can be true even when p≣r is false.

Which of the following pairs of statements is consistent?

all of the above.

(a) is incorrect. It is true that "no trees are tall" is consistent with "it's not the case that all trees are tall." However, among (a)-(d), (a) is not the only pair of consistent statements.

(b) is incorrect. It is true that "all trees are not tall" is consistent with "some trees are not tall." However, among (a)-(d), (b) is not the only pair of consistent statements.

(c) is incorrect. It is true that "no trees are not tall" is consistent with "some trees are tall." However, among (a)-(d), (c) is not the only pair of consistent statements.

(d) is correct. The statement "no trees are tall" is consistent with "it's not the case that all trees are tall." The statement "all trees are not tall" is consistent with "some trees are not tall." Finally, the statement "no trees are not tall" is consistent with "some trees are tall." So each of (a)-(c) is a consistent pair of statements.

(e) is not correct. At least one of (a)-(c) is a consistent pair of statements.

(a) is incorrect. It is true that "no trees are tall" is consistent with "it's not the case that all trees are tall." However, among (a)-(d), (a) is not the only pair of consistent statements.

(b) is incorrect. It is true that "all trees are not tall" is consistent with "some trees are not tall." However, among (a)-(d), (b) is not the only pair of consistent statements.

(c) is incorrect. It is true that "no trees are not tall" is consistent with "some trees are tall." However, among (a)-(d), (c) is not the only pair of consistent statements.

(d) is correct. The statement "no trees are tall" is consistent with "it's not the case that all trees are tall." The statement "all trees are not tall" is consistent with "some trees are not tall." Finally, the statement "no trees are not tall" is consistent with "some trees are tall." So each of (a)-(c) is a consistent pair of statements.

(e) is not correct. At least one of (a)-(c) is a consistent pair of statements.

Which of the following Venn diagrams represents the statement that all trees are tall?

B

(a) is incorrect. By shading out the region of the circle of tall things, which is outside the circle of trees, this diagram represents that all tall things are trees, rather than representing that all trees are tall.

(b) is correct. By shading out the region of the circle of trees, which is outside the circle of tall things, this diagram represents that all trees are tall.

(c) is incorrect. By shading out only the region where the circle of trees and the circle of tall things intersect, this diagram represents that no trees are tall.

(d) is incorrect. By placing an X at the intersection of the two circles, this diagram represents that some trees are tall.

(e) is incorrect. One of the diagrams represents that all trees are tall.

(a) is incorrect. By shading out the region of the circle of tall things, which is outside the circle of trees, this diagram represents that all tall things are trees, rather than representing that all trees are tall.

(b) is correct. By shading out the region of the circle of trees, which is outside the circle of tall things, this diagram represents that all trees are tall.

(c) is incorrect. By shading out only the region where the circle of trees and the circle of tall things intersect, this diagram represents that no trees are tall.

(d) is incorrect. By placing an X at the intersection of the two circles, this diagram represents that some trees are tall.

(e) is incorrect. One of the diagrams represents that all trees are tall.

Which of the following Venn diagrams represents the statement that some presidents are not descendants of the Pilgrims?

D

(a) is not correct. By placing an X, which is inside the circle of descendants of Pilgrims but outside the circle of presidents, this diagram represents that some descendants of Pilgrims are not presidents.

(b) is not correct. By shading the intersection between the circle of presidents and the circle of descendants of Pilgrims, this diagram represents that no presidents are descendants of Pilgrims.

(c) is not correct. By shading out the region of the circle of descendants of Pilgrims, this diagram represents that all descendants of Pilgrims are presidents.

(d) is correct. By placing an X, which is inside the circle of presidents but outside the circle of descendants of Pilgrims, this diagram represents that some presidents are not descendants of Pilgrims.

(e) is not correct. One of the diagrams among (a)-(d) correctly represents that some presidents are not descendants of the pilgrims.

(a) is not correct. By placing an X, which is inside the circle of descendants of Pilgrims but outside the circle of presidents, this diagram represents that some descendants of Pilgrims are not presidents.

(b) is not correct. By shading the intersection between the circle of presidents and the circle of descendants of Pilgrims, this diagram represents that no presidents are descendants of Pilgrims.

(c) is not correct. By shading out the region of the circle of descendants of Pilgrims, this diagram represents that all descendants of Pilgrims are presidents.

(d) is correct. By placing an X, which is inside the circle of presidents but outside the circle of descendants of Pilgrims, this diagram represents that some presidents are not descendants of Pilgrims.

(e) is not correct. One of the diagrams among (a)-(d) correctly represents that some presidents are not descendants of the pilgrims.

Which of the statements below represents the information contained in the following Venn diagram?

No moose are chartreuse.

(a) is incorrect. For the diagram to represent "all moose are chartreuse," it would have to shade out the region of the circle of moose, which is outside the circle of chartreuse things.

(b) is correct. By shading out the intersection between the circle of moose and the circle of chartreuse things, the diagram represents that no moose are chartreuse.

(c) is incorrect. For the diagram to represent that some moose are chartreuse, there would have to be in the region where the moose circle and the chartreuse circle intersect. Yet there is no such X in the diagram.

(d) is incorrect. For the diagram to represent that some moose are chartreuse, there would have to be an X in the region of the moose circle, which is outside of the chartreuse circle. Yet there is no such X in the diagram.

(e) is incorrect. One of the options among (a)-(d) correctly states what the diagram represents.

(a) is incorrect. For the diagram to represent "all moose are chartreuse," it would have to shade out the region of the circle of moose, which is outside the circle of chartreuse things.

(b) is correct. By shading out the intersection between the circle of moose and the circle of chartreuse things, the diagram represents that no moose are chartreuse.

(c) is incorrect. For the diagram to represent that some moose are chartreuse, there would have to be in the region where the moose circle and the chartreuse circle intersect. Yet there is no such X in the diagram.

(d) is incorrect. For the diagram to represent that some moose are chartreuse, there would have to be an X in the region of the moose circle, which is outside of the chartreuse circle. Yet there is no such X in the diagram.

(e) is incorrect. One of the options among (a)-(d) correctly states what the diagram represents.

Which of the following statements represents the information contained in the following Venn diagram?

All baboons are maroon.

(a) is correct. By shading out the region in the circle of baboons, which is outside the circle of maroon things, the diagram represents that all baboons are maroon.

(b) is not correct. To represent that no baboons are maroon, the diagram would have to shade out the intersection between the baboon circle and the maroon circle. Yet the intersection between the circles in the diagram is not shaded.

(c) is not correct. To represent that some baboons are maroon, the diagram would have to have an X inside the region where the circle of baboons intersects with the circle of maroon things. Yet there is no such X in the diagram.

(d) is not correct. To represent that some baboons are maroon, the diagram would have to have an X inside the circle of baboons, but outside the circle of maroon things. Yet there is no such X in the diagram.

(e) is not correct. One of the options among (a)-(d) correctly states what the diagram represents.

(a) is correct. By shading out the region in the circle of baboons, which is outside the circle of maroon things, the diagram represents that all baboons are maroon.

(b) is not correct. To represent that no baboons are maroon, the diagram would have to shade out the intersection between the baboon circle and the maroon circle. Yet the intersection between the circles in the diagram is not shaded.

(c) is not correct. To represent that some baboons are maroon, the diagram would have to have an X inside the region where the circle of baboons intersects with the circle of maroon things. Yet there is no such X in the diagram.

(d) is not correct. To represent that some baboons are maroon, the diagram would have to have an X inside the circle of baboons, but outside the circle of maroon things. Yet there is no such X in the diagram.

(e) is not correct. One of the options among (a)-(d) correctly states what the diagram represents.

Which of the following is valid?

Some gerbils are not purple.

Therefore, not all gerbils are purple.

(a) is not correct. If no gerbils are purple, it does not follow that all gerbils are purple. Suppose there are some gerbils, and that none of them are purple. In such a case, the claim "no gerbils are purple" would be true, but "all gerbils are purple" would be false.

(b) is not correct. If some gerbils are purple, it does not follow that not all gerbils are purple. Suppose that not only some but all gerbils are purple. In such a case, the claim "some gerbils are purple" would be true, but "not all gerbils are purple" would be false.

(c) is correct. If some gerbils are purple, then not all gerbils are purple.

(d) is not correct. If all gerbils are purple, it does not follow that some gerbils are purple. Suppose that there are no gerbils. In such a case, since there would be no non-purple gerbils, the claim "all gerbils are purple" would be true. Yet the claim "some gerbils are purple" would still be false, because there would not be any gerbils.

(e) is not correct. One of the immediate categorical inferences stated in (a)-(d) is valid.

Therefore, not all gerbils are purple.

(a) is not correct. If no gerbils are purple, it does not follow that all gerbils are purple. Suppose there are some gerbils, and that none of them are purple. In such a case, the claim "no gerbils are purple" would be true, but "all gerbils are purple" would be false.

(b) is not correct. If some gerbils are purple, it does not follow that not all gerbils are purple. Suppose that not only some but all gerbils are purple. In such a case, the claim "some gerbils are purple" would be true, but "not all gerbils are purple" would be false.

(c) is correct. If some gerbils are purple, then not all gerbils are purple.

(d) is not correct. If all gerbils are purple, it does not follow that some gerbils are purple. Suppose that there are no gerbils. In such a case, since there would be no non-purple gerbils, the claim "all gerbils are purple" would be true. Yet the claim "some gerbils are purple" would still be false, because there would not be any gerbils.

(e) is not correct. One of the immediate categorical inferences stated in (a)-(d) is valid.

Which of the following is valid?

All dragons are penguins.

All penguins are fiery.

Therefore, all dragons are fiery.

(a) is not correct. As the diagram shows, "no dragons are penguins" requires us to shade out the circle of dragons where it intersects with the circle of penguins. The statement "no penguins are fiery," moreover, requires us to shade out the circle of penguins where it intersects with the circle of fiery things. Together, these two shading patterns do not show that no dragons are fiery, since the intersection between the circle of dragons and the circle of fiery things is not completely shaded.

(b) is correct. As the diagram shows, "all dragons are penguins" requires us to shade out the circle of dragons, except where it intersects with the circle of penguins. The statement "all penguins are fiery things," moreover, requires us to shade out the circle of penguins, except where it intersects with the circle of fiery things. Together, these two shading patterns show that all dragons are fiery.

(c) is not correct. As the diagram shows, the fact that some dragons are penguins, and the fact that some penguins are fiery, does not entail that some dragons are fiery. If some penguins were also fiery non-dragons, and if all the penguins were fiery non-dragons, then the premises of (b) could be true while the conclusion is false.

(d) is not correct. As the diagram shows, "all dragons are penguins" requires us to shade out the circle of dragons that does not intersect with the circle of penguins. The statement "some penguins are fiery," moreover, requires us to place an X where the circle of fiery things intersects with the circle of penguins. It does not follow, however, that there are any fiery dragons. If some penguins were also fiery non-dragons, and if all the penguins were fiery non-dragons, then the premises of (a) could be true while the conclusion is false.

(e) is not correct. One of the syllogisms among (a)-(d) is valid.

All penguins are fiery.

Therefore, all dragons are fiery.

(a) is not correct. As the diagram shows, "no dragons are penguins" requires us to shade out the circle of dragons where it intersects with the circle of penguins. The statement "no penguins are fiery," moreover, requires us to shade out the circle of penguins where it intersects with the circle of fiery things. Together, these two shading patterns do not show that no dragons are fiery, since the intersection between the circle of dragons and the circle of fiery things is not completely shaded.

(b) is correct. As the diagram shows, "all dragons are penguins" requires us to shade out the circle of dragons, except where it intersects with the circle of penguins. The statement "all penguins are fiery things," moreover, requires us to shade out the circle of penguins, except where it intersects with the circle of fiery things. Together, these two shading patterns show that all dragons are fiery.

(c) is not correct. As the diagram shows, the fact that some dragons are penguins, and the fact that some penguins are fiery, does not entail that some dragons are fiery. If some penguins were also fiery non-dragons, and if all the penguins were fiery non-dragons, then the premises of (b) could be true while the conclusion is false.

(d) is not correct. As the diagram shows, "all dragons are penguins" requires us to shade out the circle of dragons that does not intersect with the circle of penguins. The statement "some penguins are fiery," moreover, requires us to place an X where the circle of fiery things intersects with the circle of penguins. It does not follow, however, that there are any fiery dragons. If some penguins were also fiery non-dragons, and if all the penguins were fiery non-dragons, then the premises of (a) could be true while the conclusion is false.

(e) is not correct. One of the syllogisms among (a)-(d) is valid.

Which of the following is valid?

Some moose are not chartreuse.

Therefore, not all moose are chartreuse.

(a) is not correct. If all moose are chartreuse, then it cannot be true that some moose are not chartreuse.

(b) is not correct. If no moose are chartreuse, it does not follow that no moose are not chartreuse. Suppose there are some moose, all of whom are non-chartreuse. In such a case, the premise would be true and the conclusion would be false.

(c) is not correct. If some moose are chartreuse, it does not follow that there are any non-chartreuse moose. It could be that there are some moose, all of whom are chartreuse. In such a case, "some moose are chartreuse" would be true, even though "some moose are not chartreuse" would be false.

(d) is correct. If it is some moose are not chartreuse, it immediately follows that not all moose are chartreuse.

(e) is not correct. One of the immediate categorical inferences among (a)-(d) is valid.

Therefore, not all moose are chartreuse.

(a) is not correct. If all moose are chartreuse, then it cannot be true that some moose are not chartreuse.

(b) is not correct. If no moose are chartreuse, it does not follow that no moose are not chartreuse. Suppose there are some moose, all of whom are non-chartreuse. In such a case, the premise would be true and the conclusion would be false.

(c) is not correct. If some moose are chartreuse, it does not follow that there are any non-chartreuse moose. It could be that there are some moose, all of whom are chartreuse. In such a case, "some moose are chartreuse" would be true, even though "some moose are not chartreuse" would be false.

(d) is correct. If it is some moose are not chartreuse, it immediately follows that not all moose are chartreuse.

(e) is not correct. One of the immediate categorical inferences among (a)-(d) is valid.

Which of the following is valid?

All baboons are goons.

All goons are maroon.

Therefore, all baboons are maroon.

(a) is not correct. As the diagram shows, "no baboons are goons" requires us to shade out the circle of baboons where it intersects with the circle of goons. The statement "no goons are maroon," moreover, requires us to shade out the circle of goons where it intersects with the circle of maroon things. Together, these two shading patterns do not show that no baboons are goons, since the intersection between the circle of baboons and the circle of maroon things is not shaded.

(b) is correct. As the diagram shows, "all baboons are goons" requires us to shade out the circle of baboons, except where it intersects with the circle of goons. The statement "all goons are maroon," moreover, requires us to shade out the circle of goons, except where it intersects with the circle of maroon things. Together, these two shading patterns show that all baboons are maroon.

(c) is not correct. As the diagram shows, "all baboons are goons" requires us to shade out the circle of baboons that does not intersect with the circle of goons. The statement "some goons are maroon," moreover, requires us to place an X where the circle of maroon things intersects with the circle of goons. It does not follow, however, that there are any maroon baboons. If some goons were also non-baboon maroon things, and if all the goons were maroon non-baboons, then the premises of (c) could be true while the conclusion is false.

(d) is not correct. As the diagram shows, the fact that some baboons are goons, and the fact that some goons are maroon, does not entail that some baboons are maroon. If some goons were also maroon non-baboons, and if all the goons were maroon non-baboons, then the premises of (d) could be true while the conclusion is false.

e) is not correct. One of the options among (a)-(d) is a valid syllogism.

All goons are maroon.

Therefore, all baboons are maroon.

(a) is not correct. As the diagram shows, "no baboons are goons" requires us to shade out the circle of baboons where it intersects with the circle of goons. The statement "no goons are maroon," moreover, requires us to shade out the circle of goons where it intersects with the circle of maroon things. Together, these two shading patterns do not show that no baboons are goons, since the intersection between the circle of baboons and the circle of maroon things is not shaded.

(b) is correct. As the diagram shows, "all baboons are goons" requires us to shade out the circle of baboons, except where it intersects with the circle of goons. The statement "all goons are maroon," moreover, requires us to shade out the circle of goons, except where it intersects with the circle of maroon things. Together, these two shading patterns show that all baboons are maroon.

(c) is not correct. As the diagram shows, "all baboons are goons" requires us to shade out the circle of baboons that does not intersect with the circle of goons. The statement "some goons are maroon," moreover, requires us to place an X where the circle of maroon things intersects with the circle of goons. It does not follow, however, that there are any maroon baboons. If some goons were also non-baboon maroon things, and if all the goons were maroon non-baboons, then the premises of (c) could be true while the conclusion is false.

(d) is not correct. As the diagram shows, the fact that some baboons are goons, and the fact that some goons are maroon, does not entail that some baboons are maroon. If some goons were also maroon non-baboons, and if all the goons were maroon non-baboons, then the premises of (d) could be true while the conclusion is false.

e) is not correct. One of the options among (a)-(d) is a valid syllogism.

Suppose you had to fill in the rightmost column of the truth table below:

p q -pVq

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

p q -pVq

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

TFTT

In order for a disjunction to be true, at least one of its disjuncts must be true. If one of the disjuncts is a negation, like "-p," then the sentence that is negated (in this case, p) has to be false in order for the conjunct "-p" to be true. So the disjunction "-pVq" is only false when q is false and p is true.

In order for a disjunction to be true, at least one of its disjuncts must be true. If one of the disjuncts is a negation, like "-p," then the sentence that is negated (in this case, p) has to be false in order for the conjunct "-p" to be true. So the disjunction "-pVq" is only false when q is false and p is true.

Suppose you had to fill in the rightmost column of the truth table below:

p q -p≡q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

p q -p≡q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

FTTF

In order for a biconditional to be false, the sentences it connects must have different truth values. If one of the sentences is a negation, like -p, then the sentence that is negated (in this case, p) has to be true for the negation to be false. So the biconditional "-p≡q" is only true when p is false and q is true, or when p is true and q is false. It is false otherwise.

In order for a biconditional to be false, the sentences it connects must have different truth values. If one of the sentences is a negation, like -p, then the sentence that is negated (in this case, p) has to be true for the negation to be false. So the biconditional "-p≡q" is only true when p is false and q is true, or when p is true and q is false. It is false otherwise.

Suppose you had to fill in the rightmost column of the truth table below:

p q r -pV(qVr)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill in that column?

p q r -pV(qVr)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill in that column?

TTTFTTTT

In order for a disjunction to be true, at least one of the disjuncts must be true. In order for a negation to be true, the sentence, which is being negated, must be false. So for the disjunction "-pV(qVr)" to be false, p must be true, and both q and r must be false. The disjunction will be true otherwise.

In order for a disjunction to be true, at least one of the disjuncts must be true. In order for a negation to be true, the sentence, which is being negated, must be false. So for the disjunction "-pV(qVr)" to be false, p must be true, and both q and r must be false. The disjunction will be true otherwise.

Which of these claims follows from -pVq?

none of the above.

From a disjunction, one can not infer either of its disjuncts. None of the options among (a)-(d) follow from the premise in question.

From a disjunction, one can not infer either of its disjuncts. None of the options among (a)-(d) follow from the premise in question.

Which of these claims follows from -p≡q?

none of the above.

From a biconditional, and nothing else, one cannot infer either of the sentences it connects. One cannot discover the truth-values of the sentences connected by a biconditional, in other words, merely from knowing that the biconditional is true. Since all we have is the biconditional, none of (a)-(d) follow.

From a biconditional, and nothing else, one cannot infer either of the sentences it connects. One cannot discover the truth-values of the sentences connected by a biconditional, in other words, merely from knowing that the biconditional is true. Since all we have is the biconditional, none of (a)-(d) follow.

Which of these claims follows from -pV(qVr)?

none of the above.

From a disjunction, one can not infer either of its disjuncts. None of the options among (a)-(d) follow from the premise in question.

From a disjunction, one can not infer either of its disjuncts. None of the options among (a)-(d) follow from the premise in question.

Which of these claims follows from pV-(qVr)?

none of the above.

From a disjunction, one can not infer either of its disjuncts. None of the options among (a)-(d) follow from the premise in question.

From a disjunction, one can not infer either of its disjuncts. None of the options among (a)-(d) follow from the premise in question.

Which of these claims follows from -p⊃(qVr)?

none of the above.

From a conditional, and nothing else, one cannot infer its antecedent, nor can one infer its consequent. One can only infer the consequent if the premises include both the conditional and its antecedent, and one can only infer the negation of the antecedent if the premises include both the conditional and the negation of the consequent. Since all we have is the conditional, none of (a)-(d) follow.

From a conditional, and nothing else, one cannot infer its antecedent, nor can one infer its consequent. One can only infer the consequent if the premises include both the conditional and its antecedent, and one can only infer the negation of the antecedent if the premises include both the conditional and the negation of the consequent. Since all we have is the conditional, none of (a)-(d) follow.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p&q -(p&q) -(p&q)Vr

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p&q -(p&q) -(p&q)Vr

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

TFTTTTTT

(d) is correct. In order for the rightmost disjunction to be true, all that is needed is for either r to be true, or for -(p&q) to be true. For the negation -(p&q) to be true, however, the conjunction p&q has to be false. For p&q to be false, all that is needed is for either p to be false, or q to be false. So for the rightmost disjunction to be true, all that is needed is for either r to be true, or for either p or q to be false. The only way, in other words, for the rightmost disjunction to be false is for r to be false while both p and q are true.

(d) is correct. In order for the rightmost disjunction to be true, all that is needed is for either r to be true, or for -(p&q) to be true. For the negation -(p&q) to be true, however, the conjunction p&q has to be false. For p&q to be false, all that is needed is for either p to be false, or q to be false. So for the rightmost disjunction to be true, all that is needed is for either r to be true, or for either p or q to be false. The only way, in other words, for the rightmost disjunction to be false is for r to be false while both p and q are true.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p&q -r -r⊃(p&q)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p&q -r -r⊃(p&q)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

TTTFTFTF

(b) is correct. In order for the rightmost proposition to be false, the antecedent has to be true and the consequent has to be false. If the the antecedent, -r, is true, this means that r has to be false. If the consequent, p&q, is false, this requires either p or q to be false. So the rightmost proposition is false exactly when both r and either p or q is false. If r is true, or if r is false but both p and q are true, then the rightmost proposition is true.

(b) is correct. In order for the rightmost proposition to be false, the antecedent has to be true and the consequent has to be false. If the the antecedent, -r, is true, this means that r has to be false. If the consequent, p&q, is false, this requires either p or q to be false. So the rightmost proposition is false exactly when both r and either p or q is false. If r is true, or if r is false but both p and q are true, then the rightmost proposition is true.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r q⊃p (q⊃p)⊃r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r q⊃p (q⊃p)⊃r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

none of the above.

(e) is correct. In order for the rightmost proposition to be false, the antecedent must be true and the consequent must be false. The antecedent is q⊃p, and the consequent is r. The antecedent, q⊃p, is only false, however, if q is true and p is false. Otherwise, the antecedent is true. So in order for the rightmost proposition to be false, r has to be false, and, in addition, either q has to be false or p has to be true. If r is true, or if q is true while p is false, then the rightmost proposition is true. Otherwise, the rightmost proposition is false. The correct answer should be "TFTFTTTF," which is not listed in (a)-(d).

(e) is correct. In order for the rightmost proposition to be false, the antecedent must be true and the consequent must be false. The antecedent is q⊃p, and the consequent is r. The antecedent, q⊃p, is only false, however, if q is true and p is false. Otherwise, the antecedent is true. So in order for the rightmost proposition to be false, r has to be false, and, in addition, either q has to be false or p has to be true. If r is true, or if q is true while p is false, then the rightmost proposition is true. Otherwise, the rightmost proposition is false. The correct answer should be "TFTFTTTF," which is not listed in (a)-(d).

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p≡r q≡r (q≡r)⊃(p≡r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p≡r q≡r (q≡r)⊃(p≡r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

TTTFFTTT

(b) is correct. The rightmost proposition is only false when its antecedent is true and its consequent is false. So the rightmost proposition is only false when q≡r is true but p≡r is false. A biconditional is true just in case the propositions on both sides have the same truth-value, and false otherwise. So there are only two truth-value assignments, according to which the whole rightmost proposition is false: one in which p is true, q is false, and r is false; and one in which p is false, q is true, and r is true.

(b) is correct. The rightmost proposition is only false when its antecedent is true and its consequent is false. So the rightmost proposition is only false when q≡r is true but p≡r is false. A biconditional is true just in case the propositions on both sides have the same truth-value, and false otherwise. So there are only two truth-value assignments, according to which the whole rightmost proposition is false: one in which p is true, q is false, and r is false; and one in which p is false, q is true, and r is true.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p&q pVr -r (p&q)≡(pVr)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p&q pVr -r (p&q)≡(pVr)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

TTFFFTFT

(b) is correct. For the rightmost proposition to be false, either the conjunction p&q has to be true while the disjunction pVr is false, or else the conjunction p&q has to be false while the disjunction pVr is true. The first situation cannot obtain, however. If the conjunction p&q is true, this requires p to be true. Yet if p is true, then regardless of the truth of r, pVr is also true. The second situation obtains when either p or q is false while either p or r is true. So if either p or q is false while either p or r is true, then the rightmost proposition is false. Otherwise, the rightmost proposition is true.

(b) is correct. For the rightmost proposition to be false, either the conjunction p&q has to be true while the disjunction pVr is false, or else the conjunction p&q has to be false while the disjunction pVr is true. The first situation cannot obtain, however. If the conjunction p&q is true, this requires p to be true. Yet if p is true, then regardless of the truth of r, pVr is also true. The second situation obtains when either p or q is false while either p or r is true. So if either p or q is false while either p or r is true, then the rightmost proposition is false. Otherwise, the rightmost proposition is true.

Suppose your premise is [(p⊃q)⊃r]Vq. Which of the following conclusions can be deduced from that premise?

none of the above.

(a) is incorrect. The premise [(p⊃q)⊃r]Vq does not imply the conditional q⊃p, since the truth of the premise is consistent with a truth-value assignment, according to which q is true and p is false.

(b) is incorrect. The premise [(p⊃q)⊃r]Vq does not imply the conditional p⊃q, since the premise is consistent with a truth-value assignment, according to which p is true and q is false.

(c) is incorrect. The premise [(p⊃q)⊃r]Vq is also consistent with the falsity of both p and r, so it cannot imply the disjunction pVr.

(d) is incorrect. It is not the case that each of (a)-(c) follows from [(p⊃q)⊃r]Vq.

(e) is correct. The premise [(p⊃q)⊃r]Vq does not imply the conditional q⊃p, since the truth of the premise is consistent with a truth-value assignment, according to which q is true and p is false. The premise does not imply the conditional q⊃p, either, since the premise is consistent with a truth-value assignment, according to which p is true and q is false. Finally, the premise is also consistent with the falsity of both p and r, so it cannot imply the disjunction pVr.

(a) is incorrect. The premise [(p⊃q)⊃r]Vq does not imply the conditional q⊃p, since the truth of the premise is consistent with a truth-value assignment, according to which q is true and p is false.

(b) is incorrect. The premise [(p⊃q)⊃r]Vq does not imply the conditional p⊃q, since the premise is consistent with a truth-value assignment, according to which p is true and q is false.

(c) is incorrect. The premise [(p⊃q)⊃r]Vq is also consistent with the falsity of both p and r, so it cannot imply the disjunction pVr.

(d) is incorrect. It is not the case that each of (a)-(c) follows from [(p⊃q)⊃r]Vq.

(e) is correct. The premise [(p⊃q)⊃r]Vq does not imply the conditional q⊃p, since the truth of the premise is consistent with a truth-value assignment, according to which q is true and p is false. The premise does not imply the conditional q⊃p, either, since the premise is consistent with a truth-value assignment, according to which p is true and q is false. Finally, the premise is also consistent with the falsity of both p and r, so it cannot imply the disjunction pVr.

Which of the following pairs of statements is consistent?

All wizards are tall; no wizards are tall.

(a) is correct. The statements "all wizards are tall" is consistent with the statement "no wizards are tall," provided that there are no wizards. If there are no wizards, then there are no wizards who are not tall. If there are no wizards who are not tall, then all wizards are tall. At the same time, however, if there are no wizards, then there are no tall wizards either. So the statements "all wizards are tall" and "no wizards are tall" can both be true, provided that there are no wizards. The statements are consistent.

(b) is incorrect. If all of the wizards are tall, then it cannot be the case that some wizards are not tall. "All wizards are tall" and "some wizards are not tall" are inconsistent.

(c) is incorrect. If some of the wizards are tall, then it cannot be the case that no wizards are not tall. "Some wizards are tall" and "no wizards are not tall" are inconsistent.

(d) is incorrect. Not all of (a)-(c) are consistent pairs of statements.

(e) is inorrect. At least one of (a)-(c) is a consistent pair of statements.

(a) is correct. The statements "all wizards are tall" is consistent with the statement "no wizards are tall," provided that there are no wizards. If there are no wizards, then there are no wizards who are not tall. If there are no wizards who are not tall, then all wizards are tall. At the same time, however, if there are no wizards, then there are no tall wizards either. So the statements "all wizards are tall" and "no wizards are tall" can both be true, provided that there are no wizards. The statements are consistent.

(b) is incorrect. If all of the wizards are tall, then it cannot be the case that some wizards are not tall. "All wizards are tall" and "some wizards are not tall" are inconsistent.

(c) is incorrect. If some of the wizards are tall, then it cannot be the case that no wizards are not tall. "Some wizards are tall" and "no wizards are not tall" are inconsistent.

(d) is incorrect. Not all of (a)-(c) are consistent pairs of statements.

(e) is inorrect. At least one of (a)-(c) is a consistent pair of statements.

Which of the following Venn Diagrams represents a body of information that includes the information that all living sequoia trees are tall?

D

(a) is not correct. The diagram does correctly represent that all living sequoia trees are tall. It does so by shading out the region of the circle of living sequoia trees, which is outside the circle of tall things. However, the diagram for (a) is not the only one among (a)-(c), which represents that all living sequoia trees are tall.

(b) is not correct. The diagram does correctly represent that all living sequoia trees are tall. It does so by shading out the region of the circle of living sequoia trees, which is outside the circle of tall things. However, the diagram for (b) is not the only one among (a)-(c), which represents that all living sequoia trees are tall.

(c) is not correct. The diagram does correctly represent that all living sequoia trees are tall. It does so by shading out the region of the circle of living sequoia trees, which is outside the circle of tall things. However, the diagram for (c) is not the only one among (a)-(c), which represents that all living sequoia trees are tall.

(d) is correct. The diagrams for each of (a)-(c) each represent that all living sequoia trees are tall. Each one does so by shading out the region of the circle of living sequoia trees, which is outside the circle of tall things. In addition to representing that all living sequoia trees are tall, the diagram for (b) also represents that some living sequoia trees are tall, while the diagram for (c) also represents that all tall things are living sequoia trees.

(e) is not correct. At least one of (a)-(c) represents that all sequoia trees are tall.

(a) is not correct. The diagram does correctly represent that all living sequoia trees are tall. It does so by shading out the region of the circle of living sequoia trees, which is outside the circle of tall things. However, the diagram for (a) is not the only one among (a)-(c), which represents that all living sequoia trees are tall.

(b) is not correct. The diagram does correctly represent that all living sequoia trees are tall. It does so by shading out the region of the circle of living sequoia trees, which is outside the circle of tall things. However, the diagram for (b) is not the only one among (a)-(c), which represents that all living sequoia trees are tall.

(c) is not correct. The diagram does correctly represent that all living sequoia trees are tall. It does so by shading out the region of the circle of living sequoia trees, which is outside the circle of tall things. However, the diagram for (c) is not the only one among (a)-(c), which represents that all living sequoia trees are tall.

(d) is correct. The diagrams for each of (a)-(c) each represent that all living sequoia trees are tall. Each one does so by shading out the region of the circle of living sequoia trees, which is outside the circle of tall things. In addition to representing that all living sequoia trees are tall, the diagram for (b) also represents that some living sequoia trees are tall, while the diagram for (c) also represents that all tall things are living sequoia trees.

(e) is not correct. At least one of (a)-(c) represents that all sequoia trees are tall.

Which of the following Venn Diagrams represents the statement that some living sequoia trees are not tall?

C

(a) is not correct. By shading out the region of the circle of tall things, which does not intersect with the circle of living sequoia trees, the diagram represents that all tall things are living sequoia trees, not that some living sequoia trees are not tall.

(b) is not correct. By shading out the intersection between the circle of tall things and the circle of living sequoia trees, the diagram represents that no living sequoia trees are tall, not that some living sequoia trees are not tall.

(c) is correct. By placing an X in the region of the circle of sequoia trees, which is outside the circle of tall things, the diagram represents that some living sequoia trees are not tall.

(d) is not correct. Not all of (a)-(c) represent that some living sequoia trees are not tall.

(e) is not correct. One of (a)-(c) represents that some living sequoia trees are not tall.

(a) is not correct. By shading out the region of the circle of tall things, which does not intersect with the circle of living sequoia trees, the diagram represents that all tall things are living sequoia trees, not that some living sequoia trees are not tall.

(b) is not correct. By shading out the intersection between the circle of tall things and the circle of living sequoia trees, the diagram represents that no living sequoia trees are tall, not that some living sequoia trees are not tall.

(c) is correct. By placing an X in the region of the circle of sequoia trees, which is outside the circle of tall things, the diagram represents that some living sequoia trees are not tall.

(d) is not correct. Not all of (a)-(c) represent that some living sequoia trees are not tall.

(e) is not correct. One of (a)-(c) represents that some living sequoia trees are not tall.

Which of the following statements represents the information contained in the following Venn Diagram?

Some ferrets are not sheriffs.

(a) is not correct. To represent "all ferrets are sheriffs," the diagram would have to shade out the circle of ferrets, which is outside the circle of sheriffs. Yet the diagram has no such shading.

(b) is not correct. To represent "no ferrets are sheriffs," the diagram would have to shade out the circle of ferrets where it intersects with the circle of sheriffs. Yet the diagram has no such shading.

(c) is not correct. To represent "some ferrets are sheriffs," the diagram would have to place an X in the circle of ferrets where it intersects with the circle of sheriffs. Yet the diagram has no such X.

(d) is correct. To represent "some ferrets are not sheriffs," the diagram would have to place an X in the circle of sheriffs, which is outside the circle of ferrets. The diagram does exactly this.

(e) is not correct. One of (a)-(d) correctly states what the diagram shows.

(a) is not correct. To represent "all ferrets are sheriffs," the diagram would have to shade out the circle of ferrets, which is outside the circle of sheriffs. Yet the diagram has no such shading.

(b) is not correct. To represent "no ferrets are sheriffs," the diagram would have to shade out the circle of ferrets where it intersects with the circle of sheriffs. Yet the diagram has no such shading.

(c) is not correct. To represent "some ferrets are sheriffs," the diagram would have to place an X in the circle of ferrets where it intersects with the circle of sheriffs. Yet the diagram has no such X.

(d) is correct. To represent "some ferrets are not sheriffs," the diagram would have to place an X in the circle of sheriffs, which is outside the circle of ferrets. The diagram does exactly this.

(e) is not correct. One of (a)-(d) correctly states what the diagram shows.

none of the above.

(a) is not correct. If "no ferrets are sheriffs" were true, then the intersection between the circle of ferrets and the circle of sheriffs would be shaded. Yet the intersection is not shaded.

(b) is not correct. If "some ferrets are sheriffs" were true, then the intersection between the circle of ferrets and the circle of sheriffs would have an X in it. Yet there is no such X.

(c) is not correct. If "no ferrets are sheriffs" were true, then the intersection between the circle of ferrets and the circle of sheriffs would be shaded. Yet the intersection is not shaded.

(d) is not correct. If "some ferrets are not sheriffs" were true, then the region of the circle of ferrets, which is outside the circle of sheriffs, would have an X in it. Likewise, if some sheriffs were not ferrets, then the region of the circle of sheriffs, which is outside the circle of ferrets, would also have an X in it. Yet there are no such Xs.

(e) is correct. By shading out the circle of ferrets and the circle of sheriffs, except where they intersect, the diagram represents that all ferrets are sheriffs and that all sheriffs are ferrets.

(a) is not correct. If "no ferrets are sheriffs" were true, then the intersection between the circle of ferrets and the circle of sheriffs would be shaded. Yet the intersection is not shaded.

(b) is not correct. If "some ferrets are sheriffs" were true, then the intersection between the circle of ferrets and the circle of sheriffs would have an X in it. Yet there is no such X.

(c) is not correct. If "no ferrets are sheriffs" were true, then the intersection between the circle of ferrets and the circle of sheriffs would be shaded. Yet the intersection is not shaded.

(d) is not correct. If "some ferrets are not sheriffs" were true, then the region of the circle of ferrets, which is outside the circle of sheriffs, would have an X in it. Likewise, if some sheriffs were not ferrets, then the region of the circle of sheriffs, which is outside the circle of ferrets, would also have an X in it. Yet there are no such Xs.

(e) is correct. By shading out the circle of ferrets and the circle of sheriffs, except where they intersect, the diagram represents that all ferrets are sheriffs and that all sheriffs are ferrets.

Which of the following is valid?

Some marsupials are not brutal.

Therefore, not all marsupials are brutal.

(a) is correct. If some marsupials are not brutal, then it cannot be the case that all marsupials are brutal.

(b) is not correct. If all marsupials are brutal, it does not follow that some marsupials are brutal. Suppose there are no marsupials. In such a case, there would not be any non-brutal marsupials. If there are no non-brutal marsupials, however, could make it true that all marsupials are brutal, even when there are no marsupials.

(c) is not correct. If no marsupials are brutal, it does not follow that some marsupials are not brutal. Suppose there are no marsupials. In such a case it can be true that no marsupials are brutal, without it also being true that some marsupials are not brutal.

(d) is not correct. If some marsupials are brutal, it does not follow that not all marsupials are brutal. Suppose that all marsupials are brutal, and that there are some marsupials. In such a case "some marsupials are brutal" would be true, but "not all marsupials are brutal" would be false.

(e) is not correct. One of the immediate categorical inferences among (a)-(d) is valid.

Therefore, not all marsupials are brutal.

(a) is correct. If some marsupials are not brutal, then it cannot be the case that all marsupials are brutal.

(b) is not correct. If all marsupials are brutal, it does not follow that some marsupials are brutal. Suppose there are no marsupials. In such a case, there would not be any non-brutal marsupials. If there are no non-brutal marsupials, however, could make it true that all marsupials are brutal, even when there are no marsupials.

(c) is not correct. If no marsupials are brutal, it does not follow that some marsupials are not brutal. Suppose there are no marsupials. In such a case it can be true that no marsupials are brutal, without it also being true that some marsupials are not brutal.

(d) is not correct. If some marsupials are brutal, it does not follow that not all marsupials are brutal. Suppose that all marsupials are brutal, and that there are some marsupials. In such a case "some marsupials are brutal" would be true, but "not all marsupials are brutal" would be false.

(e) is not correct. One of the immediate categorical inferences among (a)-(d) is valid.

Which of the following is valid?

All ferrets are sheriffs.

All sheriffs pay tariffs.

Therefore, all ferrets pay tariffs

(a) is not correct. As the diagram shows, "some ferrets are sheriffs" requires an X to be placed at the intersection between the circle of ferrets and the circle of sheriffs, and "some sheriffs pay tariffs" requires an X to be placed at the intersection between the circle of sheriffs and the circle of things that pay tariffs. However, it does not follow that some ferrets pay tariffs. It could be that the ferret sheriffs are the ones who don't pay tariffs. (b) is not correct. As the diagram shows, "no ferrets are sheriffs" requires the intersection between the circle of ferrets and the circle of sheriffs to be shaded, and "no sheriffs pay tariffs" requires the intersection between the circle of sheriffs and the circle of things that pay tariffs to be shaded. However, it does not follow that no ferrets pay tariffs, since the intersection between the circle of ferrets and the circle of things that pay tariffs is not completely shaded. (c) is correct. As the diagram shows, "all ferrets are sheriffs" requires the region of the circle of ferrets, which is outside the circle of sheriffs, to be shaded; and "all sheriffs pay tariffs" requires the region of the circle of sheriffs, which is outside the circle of things that pay tariffs, to be shaded. From these two claims it follows that all ferrets pay tariffs, since every region of the circle of ferrets, which is outside the circle of things that pay tariffs, is shaded.

(d) is not correct. As the diagram shows, "all ferrets are sheriffs" requires the region of the circle of ferrets, which is outside the circle of sheriffs, to be shaded; and "some sheriffs pay tariffs" requires an X to be placed at the intersection between the circle of sheriffs and the circle of things that pay tariffs. However, it does not follow that some ferrets pay tariffs. It could be that the ferret sheriffs are the ones who don't pay tariffs.

(e) is not correct. One of the syllogisms among (a)-(d) is valid.

All sheriffs pay tariffs.

Therefore, all ferrets pay tariffs

(a) is not correct. As the diagram shows, "some ferrets are sheriffs" requires an X to be placed at the intersection between the circle of ferrets and the circle of sheriffs, and "some sheriffs pay tariffs" requires an X to be placed at the intersection between the circle of sheriffs and the circle of things that pay tariffs. However, it does not follow that some ferrets pay tariffs. It could be that the ferret sheriffs are the ones who don't pay tariffs. (b) is not correct. As the diagram shows, "no ferrets are sheriffs" requires the intersection between the circle of ferrets and the circle of sheriffs to be shaded, and "no sheriffs pay tariffs" requires the intersection between the circle of sheriffs and the circle of things that pay tariffs to be shaded. However, it does not follow that no ferrets pay tariffs, since the intersection between the circle of ferrets and the circle of things that pay tariffs is not completely shaded. (c) is correct. As the diagram shows, "all ferrets are sheriffs" requires the region of the circle of ferrets, which is outside the circle of sheriffs, to be shaded; and "all sheriffs pay tariffs" requires the region of the circle of sheriffs, which is outside the circle of things that pay tariffs, to be shaded. From these two claims it follows that all ferrets pay tariffs, since every region of the circle of ferrets, which is outside the circle of things that pay tariffs, is shaded.

(d) is not correct. As the diagram shows, "all ferrets are sheriffs" requires the region of the circle of ferrets, which is outside the circle of sheriffs, to be shaded; and "some sheriffs pay tariffs" requires an X to be placed at the intersection between the circle of sheriffs and the circle of things that pay tariffs. However, it does not follow that some ferrets pay tariffs. It could be that the ferret sheriffs are the ones who don't pay tariffs.

(e) is not correct. One of the syllogisms among (a)-(d) is valid.

Statements of the form "FRONK As are Bs" are represented by means of Venn Diagrams that look like the one below. Which of the following statements follows from "FRONK trees are deciduous"?

None of the above.

(a) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." "FRONK trees are deciduous" does not imply that all trees are deciduous. The diagram for "all trees are deciduous" would shade the region of the circle of trees, which is outside the circle of deciduous things. Our diagram does not do this.

(b) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." "FRONK trees are deciduous" does not imply that no trees are deciduous. The diagram for "no trees are deciduous" would shade the intersection between the circle of trees and the circle of deciduous things. Our diagram does not do this.

(c) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." "FRONK trees are deciduous" does not imply that some trees are deciduous. The diagram for "some trees are deciduous" would place an X between the circle of trees and the circle of deciduous things. Our diagram does not do this.

(d) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." "FRONK trees are deciduous" does not imply that some trees are not deciduous. The diagram for "some trees are not deciduous" would place an X inside the circle of trees but outside the circle of deciduous things. Our diagram does not do this.

(e) is correct. None of the statements among (a)-(d) are implied by "FRONK trees are deciduous."

(a) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." "FRONK trees are deciduous" does not imply that all trees are deciduous. The diagram for "all trees are deciduous" would shade the region of the circle of trees, which is outside the circle of deciduous things. Our diagram does not do this.

(b) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." "FRONK trees are deciduous" does not imply that no trees are deciduous. The diagram for "no trees are deciduous" would shade the intersection between the circle of trees and the circle of deciduous things. Our diagram does not do this.

(c) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." "FRONK trees are deciduous" does not imply that some trees are deciduous. The diagram for "some trees are deciduous" would place an X between the circle of trees and the circle of deciduous things. Our diagram does not do this.

(d) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." "FRONK trees are deciduous" does not imply that some trees are not deciduous. The diagram for "some trees are not deciduous" would place an X inside the circle of trees but outside the circle of deciduous things. Our diagram does not do this.

(e) is correct. None of the statements among (a)-(d) are implied by "FRONK trees are deciduous."

Which of the following is a valid inference?

None of the above.

(a) is not correct. If no ghosts are ghoulish, it does not follow that some ghosts are not ghoulish. Suppose there are no ghosts. In such a case it can be true that no ghosts are ghoulish, without it also being true that some ghosts are not ghoulish.

(b) is not correct. If all ghosts are ghoulish, it does not follow that some ghosts are ghoulish. Suppose there are no ghosts. In such a case, there would not be any non-ghoulish ghosts. If there are no non-ghoulish ghosts, however, could make it true that all ghosts are ghoulish, even when there are no ghosts.

(c) is not correct. If some ghosts are not ghoulish, it does not follow that all ghosts are not ghoulish. Suppose there are some ghoulish and some non-ghoulish ghosts. In such a case, "some ghosts are not ghoulish" would be true, even though "all ghosts are not ghoulish" would be false.

(d) is not correct. If some ghosts are ghoulish, it does not follow that not all ghosts are ghoulish. Suppose that all ghosts are ghoulish, and that there are some ghosts. In such a case "some ghosts are ghoulish" would be true, but "not all ghosts are ghoulish" would be false.

(e) is correct. None of the immediate categorical inferences among (a)-(d) is valid.

(a) is not correct. If no ghosts are ghoulish, it does not follow that some ghosts are not ghoulish. Suppose there are no ghosts. In such a case it can be true that no ghosts are ghoulish, without it also being true that some ghosts are not ghoulish.

(b) is not correct. If all ghosts are ghoulish, it does not follow that some ghosts are ghoulish. Suppose there are no ghosts. In such a case, there would not be any non-ghoulish ghosts. If there are no non-ghoulish ghosts, however, could make it true that all ghosts are ghoulish, even when there are no ghosts.

(c) is not correct. If some ghosts are not ghoulish, it does not follow that all ghosts are not ghoulish. Suppose there are some ghoulish and some non-ghoulish ghosts. In such a case, "some ghosts are not ghoulish" would be true, even though "all ghosts are not ghoulish" would be false.

(d) is not correct. If some ghosts are ghoulish, it does not follow that not all ghosts are ghoulish. Suppose that all ghosts are ghoulish, and that there are some ghosts. In such a case "some ghosts are ghoulish" would be true, but "not all ghosts are ghoulish" would be false.

(e) is correct. None of the immediate categorical inferences among (a)-(d) is valid.

Which of the following statements is consistent with "FRONK trees are deciduous"?

Two or more of the above.

e) is correct. Each of (a)-(d) is consistent with "FRONK trees are deciduous." As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim is consistent with the claim, "all trees are deciduous," since it is possible to shade out the region of the circle of trees, which is outside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees. Such a claim is consistent with the claim, "no trees are deciduous," since it is possible to shade out the circle of deciduous things where it intersects with the circle of trees, while also shading out the region of the circle of deciduous things that is outside the circle of trees. (In such a case there would not be any deciduous things.) Such a claim is consistent with the claim, "some trees are deciduous," since it is possible to place an X in the circle of deciduous things where it intersects with the circle of trees, while also shading out the region of the circle of deciduous things that is outside the circle of trees. Finally, such a claim is consistent with the claim, "some trees are not deciduous," since it is possible to place an X in the region of the circle of trees, which is outside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees.

e) is correct. Each of (a)-(d) is consistent with "FRONK trees are deciduous." As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim is consistent with the claim, "all trees are deciduous," since it is possible to shade out the region of the circle of trees, which is outside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees. Such a claim is consistent with the claim, "no trees are deciduous," since it is possible to shade out the circle of deciduous things where it intersects with the circle of trees, while also shading out the region of the circle of deciduous things that is outside the circle of trees. (In such a case there would not be any deciduous things.) Such a claim is consistent with the claim, "some trees are deciduous," since it is possible to place an X in the circle of deciduous things where it intersects with the circle of trees, while also shading out the region of the circle of deciduous things that is outside the circle of trees. Finally, such a claim is consistent with the claim, "some trees are not deciduous," since it is possible to place an X in the region of the circle of trees, which is outside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees.

Which of the following is valid?

All marsupials are brutal.

All brutal things know judo.

Therefore, all marsupials know judo.

(a) is not correct. As the diagram shows, "all marsupials are brutal" requires the region of the circle of marsupials, which is outside the circle of brutal things, to be shaded; and "some brutal things know judo" requires an X to be placed at the intersection between the circle of brutal things and the circle of things that know judo. However, it does not follow that some marsupials know judo. It could be that the brutal marsupials are the ones who don't know judo.

(b) is correct. As the diagram shows, "all marsupials are brutal" requires the region of the circle of marsupials, which is outside the circle of brutal things, to be shaded; and "all brutal things know judo" requires the region of the circle of brutal things, which is outside the circle of things that know judo, to be shaded. From these two claims it follows that all marsupials know judo, since every region of the circle of marsupials, which is outside the circle of things that know judo, is shaded.

(c) is not correct. As the diagram shows, "no marsupials are brutal" requires the intersection between the circle of marsupials and the circle of brutal things to be shaded, and "no brutal things know judo" requires the intersection between the circle of brutal things and the circle of things that know judo to be shaded. However, it does not follow that no marsupials know judo, since the intersection between the circle of marsupials and the circle of things that know judo is not completely shaded.

(d) is not correct. As the diagram shows, "some marsupials are brutal" requires an X to be placed at the intersection between the circle of marsupials and the circle of brutal things, and "some brutal things know judo" requires an X to be placed at the intersection between the circle of brutal things and the circle of things that know judo. However, it does not follow that some marsupials know judo. It could be that the brutal marsupials are the ones who don't know judo.

(e) is not correct. One of the syllogisms among (a)-(d) is valid.

All brutal things know judo.

Therefore, all marsupials know judo.

(a) is not correct. As the diagram shows, "all marsupials are brutal" requires the region of the circle of marsupials, which is outside the circle of brutal things, to be shaded; and "some brutal things know judo" requires an X to be placed at the intersection between the circle of brutal things and the circle of things that know judo. However, it does not follow that some marsupials know judo. It could be that the brutal marsupials are the ones who don't know judo.

(b) is correct. As the diagram shows, "all marsupials are brutal" requires the region of the circle of marsupials, which is outside the circle of brutal things, to be shaded; and "all brutal things know judo" requires the region of the circle of brutal things, which is outside the circle of things that know judo, to be shaded. From these two claims it follows that all marsupials know judo, since every region of the circle of marsupials, which is outside the circle of things that know judo, is shaded.

(c) is not correct. As the diagram shows, "no marsupials are brutal" requires the intersection between the circle of marsupials and the circle of brutal things to be shaded, and "no brutal things know judo" requires the intersection between the circle of brutal things and the circle of things that know judo to be shaded. However, it does not follow that no marsupials know judo, since the intersection between the circle of marsupials and the circle of things that know judo is not completely shaded.

(d) is not correct. As the diagram shows, "some marsupials are brutal" requires an X to be placed at the intersection between the circle of marsupials and the circle of brutal things, and "some brutal things know judo" requires an X to be placed at the intersection between the circle of brutal things and the circle of things that know judo. However, it does not follow that some marsupials know judo. It could be that the brutal marsupials are the ones who don't know judo.

(e) is not correct. One of the syllogisms among (a)-(d) is valid.

Which of the following statements is entailed by the conjunction, "FRONK trees are deciduous and all deciduous things are organisms"?

None of the above.

(a) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "all deciduous things are organisms," does not entail that some trees are organisms. In the above diagram, there is no X in any region where the circle of trees and the circle of organisms intersect.

(b) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "all deciduous things are organisms," does not entail that all trees are organisms. In the above diagram, there is a region in the circle of trees, which is outside the circle of organisms, which is not shaded.

(c) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "all deciduous things are organisms," does not entail that some trees are not organisms. In the above diagram, there is no X in any region, which is inside the circle of trees but outside the circle of organisms.

(d) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "all deciduous things are organisms," does not entail that no trees are organisms. In the above diagram, there is a region in the circle of trees, which intersects with the circle of organisms, which is not shaded.

(e) is correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "all deciduous things are organisms," does not entail any of the following: that some trees are organisms, that all trees are organisms, that some trees are not organisms, or that no trees are organisms. All the two claims imply is that, if there are any deciduous things, they are both trees and organisms.

(a) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "all deciduous things are organisms," does not entail that some trees are organisms. In the above diagram, there is no X in any region where the circle of trees and the circle of organisms intersect.

(b) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "all deciduous things are organisms," does not entail that all trees are organisms. In the above diagram, there is a region in the circle of trees, which is outside the circle of organisms, which is not shaded.

(c) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "all deciduous things are organisms," does not entail that some trees are not organisms. In the above diagram, there is no X in any region, which is inside the circle of trees but outside the circle of organisms.

(d) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "all deciduous things are organisms," does not entail that no trees are organisms. In the above diagram, there is a region in the circle of trees, which intersects with the circle of organisms, which is not shaded.

(e) is correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "all deciduous things are organisms," does not entail any of the following: that some trees are organisms, that all trees are organisms, that some trees are not organisms, or that no trees are organisms. All the two claims imply is that, if there are any deciduous things, they are both trees and organisms.

Which of the following statements is entailed by the conjunction, "FRONK trees are deciduous and some deciduous things are organisms"?

Some trees are organisms.

(a) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are organisms," does not entail that all trees are organisms. In the above diagram, there is a region in the circle of trees, which is outside the circle of organisms, which is not shaded.

(b) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are organisms," does not entail that no trees are organisms. In the above diagram, there is a region in the circle of trees, which intersects with the circle of organisms, which is not shaded.

(c) is correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are organisms," requires putting an X in a region, which is inside the circle of trees and the circle of organisms. In the above diagram, there is an X in the region where all three circles intersect.

(d) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are organisms," does not entail that some trees are not organisms. In the above diagram, there is no X in any region, which is inside the circle of trees but outside the circle of organisms.

(e) is not correct. One of the statements among (a)-(d) is entailed by the conjunction, "FRONK trees are deciduous and some deciduous things are organisms."

(a) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are organisms," does not entail that all trees are organisms. In the above diagram, there is a region in the circle of trees, which is outside the circle of organisms, which is not shaded.

(b) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are organisms," does not entail that no trees are organisms. In the above diagram, there is a region in the circle of trees, which intersects with the circle of organisms, which is not shaded.

(c) is correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are organisms," requires putting an X in a region, which is inside the circle of trees and the circle of organisms. In the above diagram, there is an X in the region where all three circles intersect.

(d) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are organisms," does not entail that some trees are not organisms. In the above diagram, there is no X in any region, which is inside the circle of trees but outside the circle of organisms.

(e) is not correct. One of the statements among (a)-(d) is entailed by the conjunction, "FRONK trees are deciduous and some deciduous things are organisms."

Which of the following statements is entailed by the conjunction, "FRONK trees are deciduous, and some deciduous things are not organisms"?

Some trees are not organisms.

(a) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are not organisms," does not entail that all trees are organisms. In the above diagram, there is a region in the circle of trees, which is outside the circle of organisms, which is not shaded.

(b) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are not organisms," does not entail that no trees are organisms. In the above diagram, there is a region in the circle of trees, which intersects with the circle of organisms, which is not shaded.

(c) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are not organisms," does not require putting an X in a region, which is inside both the circle of trees and the circle of organisms.

(d) is correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are not organisms," requires putting an X in a region, which is inside the circle of trees but outside the circle of organisms.

(e) is not correct. One of the statements among (a)-(d) is entailed by the conjunction, "FRONK trees are deciduous and some deciduous things are not organisms."

(a) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are not organisms," does not entail that all trees are organisms. In the above diagram, there is a region in the circle of trees, which is outside the circle of organisms, which is not shaded.

(b) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are not organisms," does not entail that no trees are organisms. In the above diagram, there is a region in the circle of trees, which intersects with the circle of organisms, which is not shaded.

(c) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are not organisms," does not require putting an X in a region, which is inside both the circle of trees and the circle of organisms.

(d) is correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim, when combined with "some deciduous things are not organisms," requires putting an X in a region, which is inside the circle of trees but outside the circle of organisms.

(e) is not correct. One of the statements among (a)-(d) is entailed by the conjunction, "FRONK trees are deciduous and some deciduous things are not organisms."

Which of the following statements is inconsistent with "FRONK trees are deciduous"?

None of the above.

a) is not correct. "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "alldeciduous things are trees." Such a claim is consistent with the claim, "all trees are deciduous," since it is possible to shade out the region of the circle of trees, which is outside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees.

(b) is not correct. "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim is consistent with the claim, "no trees are deciduous," since it is possible to shade out the region of the circle of trees, which intersects with the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees. In such a case, there would be no deciduous things.

(c) is not correct. "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim is consistent with the claim, "some trees are deciduous," since it is possible to place an X in the circle of trees, which is inside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees.

(d) is not correct. "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim is consistent with the claim, "some trees are not deciduous," since it is possible to place an X in the circle of trees, which is outside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees.

(e) is correct. "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim is consistent with the claim, "all trees are deciduous," since it is possible to shade out the region of the circle of trees, which is outside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees. Such a claim is consistent with the claim, "no trees are deciduous," since it is possible to shade out the region of the circle of trees, which intersects with the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees. In such a case, there would be no deciduous things. Such a claim is consistent with the claim, "some trees are not deciduous," since it is possible to place an X in the circle of trees, which is outside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees. Finally, such a claim is also consistent with the claim, "some trees are not deciduous," since it is possible to place an X in the circle of trees, which is outside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees.

a) is not correct. "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "alldeciduous things are trees." Such a claim is consistent with the claim, "all trees are deciduous," since it is possible to shade out the region of the circle of trees, which is outside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees.

(b) is not correct. "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim is consistent with the claim, "no trees are deciduous," since it is possible to shade out the region of the circle of trees, which intersects with the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees. In such a case, there would be no deciduous things.

(c) is not correct. "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim is consistent with the claim, "some trees are deciduous," since it is possible to place an X in the circle of trees, which is inside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees.

(d) is not correct. "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim is consistent with the claim, "some trees are not deciduous," since it is possible to place an X in the circle of trees, which is outside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees.

(e) is correct. "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." Such a claim is consistent with the claim, "all trees are deciduous," since it is possible to shade out the region of the circle of trees, which is outside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees. Such a claim is consistent with the claim, "no trees are deciduous," since it is possible to shade out the region of the circle of trees, which intersects with the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees. In such a case, there would be no deciduous things. Such a claim is consistent with the claim, "some trees are not deciduous," since it is possible to place an X in the circle of trees, which is outside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees. Finally, such a claim is also consistent with the claim, "some trees are not deciduous," since it is possible to place an X in the circle of trees, which is outside the circle of deciduous things, while also shading out the region of the circle of deciduous things that is outside the circle of trees.

Which of the following is valid?

Some goblins are not maze-dwellers.

Therefore, not all goblins are maze-dwellers.

(a) is not correct. If some goblins are maze-dwellers, it does not follow that not all goblins are maze-dwellers. Suppose that all goblins are maze-dwellers, and that there are some goblins. In such a case "some goblins are maze-dwellers" would be true, but "not all goblins are maze-dwellers" would be false.

(b) is not correct. If no goblins are maze-dwellers, it does not follow that some goblins are not maze-dwellers. Suppose there are no goblins. In such a case it can be true that no goblins are maze-dwellers, without it also being true that some goblins are not maze-dwellers.

(c) is not correct. If all goblins are maze-dwellers, it does not follow that some goblins are maze-dwellers. Suppose there are no goblins. In such a case, there would not be any non-maze-dwelling goblins. If there are no non-maze-dwelling goblins, however, could make it true that all goblins are maze-dwellers, even when there are no goblins.

(d) is correct. If some goblins are not maze-dwellers, then it cannot be the case that all goblins are maze-dwellers.

(e) is not correct. One of the immediate categorical inferences among (a)-(d) is valid.

Therefore, not all goblins are maze-dwellers.

(a) is not correct. If some goblins are maze-dwellers, it does not follow that not all goblins are maze-dwellers. Suppose that all goblins are maze-dwellers, and that there are some goblins. In such a case "some goblins are maze-dwellers" would be true, but "not all goblins are maze-dwellers" would be false.

(b) is not correct. If no goblins are maze-dwellers, it does not follow that some goblins are not maze-dwellers. Suppose there are no goblins. In such a case it can be true that no goblins are maze-dwellers, without it also being true that some goblins are not maze-dwellers.

(c) is not correct. If all goblins are maze-dwellers, it does not follow that some goblins are maze-dwellers. Suppose there are no goblins. In such a case, there would not be any non-maze-dwelling goblins. If there are no non-maze-dwelling goblins, however, could make it true that all goblins are maze-dwellers, even when there are no goblins.

(d) is correct. If some goblins are not maze-dwellers, then it cannot be the case that all goblins are maze-dwellers.

(e) is not correct. One of the immediate categorical inferences among (a)-(d) is valid.

Suppose you had to fill in the rightmost column of the truth table below:

p q pV-q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

p q pV-q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

TTFT

In order for a disjuntion to be true, at least one of the disjuncts must be true. If one of the disjuncts is a negation, like " -q" , then the sentence that is negated (in this case, q) has to be false in order for the conjunct " -q" to be true. So the disjunction " pV-q" is only false when p is false and q is true. The disjunction is true otherwise.

In order for a disjuntion to be true, at least one of the disjuncts must be true. If one of the disjuncts is a negation, like " -q" , then the sentence that is negated (in this case, q) has to be false in order for the conjunct " -q" to be true. So the disjunction " pV-q" is only false when p is false and q is true. The disjunction is true otherwise.

Suppose you had to fill in the rightmost column of the truth table below:

p q p≡-q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

p q p≡-q

T T

T F

F T

F F

Going from top to bottom, how would you fill in that column?

FTTF

In order for a biconditional to be false, the sentences it connects must have different truth values. If one of the sentences is a negation, like -q, then the sentence that is negated (in this case, q) has to be true for the negation to be false. So the biconditional " p≡-q" is only true when q is false and p is true, or when q is true and p is false. It is false otherwise.

In order for a biconditional to be false, the sentences it connects must have different truth values. If one of the sentences is a negation, like -q, then the sentence that is negated (in this case, q) has to be true for the negation to be false. So the biconditional " p≡-q" is only true when q is false and p is true, or when q is true and p is false. It is false otherwise.

Suppose you had to fill in the rightmost column of the truth table below:

p q r pV-(qVr)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill in that column?

p q r pV-(qVr)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill in that column?

TTTTFFFT

D is correct. In order for a disjunction to be true, at least one of the disjuncts must be true. In order for a negation to be true, the sentence, which is being negated, must be false. If the sentence being negated is a disjunction, this means that both disjuncts must be false for the whole negation to be true. So for the disjunction "pV-(qVr)" to be false, p must be false, and either q or r must be true. The disjunction will be true otherwise.

D is correct. In order for a disjunction to be true, at least one of the disjuncts must be true. In order for a negation to be true, the sentence, which is being negated, must be false. If the sentence being negated is a disjunction, this means that both disjuncts must be false for the whole negation to be true. So for the disjunction "pV-(qVr)" to be false, p must be false, and either q or r must be true. The disjunction will be true otherwise.

Which of these claims follows from pV-q?

none of the above

From a disjunction, one can not infer either of its disjuncts. None of the options among (a)-(d) follow from the premise in question.

From a disjunction, one can not infer either of its disjuncts. None of the options among (a)-(d) follow from the premise in question.

Which of these claims follows from p≡-q?

none of the above.

From a biconditional, and nothing else, one cannot infer either of the sentences it connects. One cannot discover the truth-values of the sentences connected by a biconditional, in other words, merely from knowing that the biconditional is true. Since all we have is the biconditional, none of (a)-(d) follow.

From a biconditional, and nothing else, one cannot infer either of the sentences it connects. One cannot discover the truth-values of the sentences connected by a biconditional, in other words, merely from knowing that the biconditional is true. Since all we have is the biconditional, none of (a)-(d) follow.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r pVq -(pVq) -(pVq)&r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r pVq -(pVq) -(pVq)&r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

FFFFFFTF

(c) is correct. The rightmost proposition is a conjunction. The only way for a conjunction to be true is if both of its conjuncts are true. The first conjunct, however, is the negation of the disjunction pVq. So the first conjunct of the rightmost proposition is only true when the disjunction, pVq, is FALSE. For the disjunction of pVq to be false, however, both p and q have to be false. The second conjunct of the rightmost proposition is r. So the only way for the rightmost proposition to be true is if p and q are both false, but r is true.

(c) is correct. The rightmost proposition is a conjunction. The only way for a conjunction to be true is if both of its conjuncts are true. The first conjunct, however, is the negation of the disjunction pVq. So the first conjunct of the rightmost proposition is only true when the disjunction, pVq, is FALSE. For the disjunction of pVq to be false, however, both p and q have to be false. The second conjunct of the rightmost proposition is r. So the only way for the rightmost proposition to be true is if p and q are both false, but r is true.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r pVq -r (pVq)⊃-r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r pVq -r (pVq)⊃-r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

FTFTFTTT

(D) is correct. The rightmost proposition is a conditional. So it is only false when its antecedent is true and its consequent is false. So the rightmost proposition is only false when (pVq) is true but -r is false. So as long as r is true, and either p or q is true as well, the whole rightmost proposition is false. Otherwise, it is true.

(D) is correct. The rightmost proposition is a conditional. So it is only false when its antecedent is true and its consequent is false. So the rightmost proposition is only false when (pVq) is true but -r is false. So as long as r is true, and either p or q is true as well, the whole rightmost proposition is false. Otherwise, it is true.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p⊃q (p⊃q)⊃r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p⊃q (p⊃q)⊃r

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

none of the above.

(e) is correct. The rightmost proposition is only false when its antecedent is true and its consequent is false. So the rightmost proposition is only false when p⊃q is true but r is false. The conditional p⊃q is true under EVERY truth-value assignment, however, except the one according to which p is true and q is false. So there are only three truth-value assignments, according to which the whole rightmost proposition is false: one in which p, q, and r are all false; one in which p is false, q is true, and r is false, and one in which p is true, q is true, and r is false. The correct answer should read " TFTTTFTF," which is not among (a)-(d).

(e) is correct. The rightmost proposition is only false when its antecedent is true and its consequent is false. So the rightmost proposition is only false when p⊃q is true but r is false. The conditional p⊃q is true under EVERY truth-value assignment, however, except the one according to which p is true and q is false. So there are only three truth-value assignments, according to which the whole rightmost proposition is false: one in which p, q, and r are all false; one in which p is false, q is true, and r is false, and one in which p is true, q is true, and r is false. The correct answer should read " TFTTTFTF," which is not among (a)-(d).

Suppose you had to fill in the rightmost column of the following truth-table:

p q r q⊃r p⊃(q⊃r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r q⊃r p⊃(q⊃r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

TFTTTTTT

(d) is correct. The rightmost proposition is only false when its antecedent is true and its consequent is false. So the rightmost proposition is only false when p is true but q⊃r is false. The conditional q⊃r is true under EVERY truth-value assignment, however, except the one according to which q is true and r is false. So there is only one truth-value assignment, according to which the whole rightmost proposition is false: one in which p, is true and both q and r are false.

(d) is correct. The rightmost proposition is only false when its antecedent is true and its consequent is false. So the rightmost proposition is only false when p is true but q⊃r is false. The conditional q⊃r is true under EVERY truth-value assignment, however, except the one according to which q is true and r is false. So there is only one truth-value assignment, according to which the whole rightmost proposition is false: one in which p, is true and both q and r are false.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r pVq p&r (pVq)≡(p&r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r pVq p&r (pVq)≡(p&r)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

TFTFFFTT

(d) is correct. Since the rightmost proposition is a biconditional, it can only be true on two conditions: when pVq and p&r are both true, and when pVq and p&r are both false. Since the truth of p&r requires the truth of both p and r, it follows that there are only two truth-value assignments, according to which both of the biconditional's parts come out as true: one in which p, q, and r are all true; and one in which p is true, q is false, and r is true. Likewise, because the falsity of pVq requires the falsity of both p and q, it follows that there are only two truth-value assignments, according to which both of the biconditional's parts come out as false: one in which p, q, and r are all false; and one in which p is false, q is false, and r is true. So the whole rightmost proposition is true under four truth-value assignments. The answer should read, " TFTFFFTT."

(d) is correct. Since the rightmost proposition is a biconditional, it can only be true on two conditions: when pVq and p&r are both true, and when pVq and p&r are both false. Since the truth of p&r requires the truth of both p and r, it follows that there are only two truth-value assignments, according to which both of the biconditional's parts come out as true: one in which p, q, and r are all true; and one in which p is true, q is false, and r is true. Likewise, because the falsity of pVq requires the falsity of both p and q, it follows that there are only two truth-value assignments, according to which both of the biconditional's parts come out as false: one in which p, q, and r are all false; and one in which p is false, q is false, and r is true. So the whole rightmost proposition is true under four truth-value assignments. The answer should read, " TFTFFFTT."

The truth-functional connective SNERG has the following truth-table:

p q r SNERG(p,q,r)

T T T F

T T F F

T F T F

T F F T

F T T T

F T F F

F F T F

F F F F

Which of the following has the same truth-table as SNERG?

p q r SNERG(p,q,r)

T T T F

T T F F

T F T F

T F F T

F T T T

F T F F

F F T F

F F F F

Which of the following has the same truth-table as SNERG?

none of the above.

(e) is correct. In the above truth-table for SNERG, SNERG(p,q,r) is false on every assignment in which p has the same truth-value as either q or r. SNERG(p,q,r) would have the same truth table as the following conjunction: -(p≡q)&-(p≡r).

(e) is correct. In the above truth-table for SNERG, SNERG(p,q,r) is false on every assignment in which p has the same truth-value as either q or r. SNERG(p,q,r) would have the same truth table as the following conjunction: -(p≡q)&-(p≡r).

Which of the following would be true if and only if SNERG(p,q,r) is true?

Either p is false while q and r are true, or else p is true while q and r are false.

(c) is correct. As the truth table for SNERG shows, there are two cases where SNERG(p,q,r) is true. Answer (a) describes one of these cases, and answer (d) describes the other. Answer (c) is the only one that covers both cases.

(c) is correct. As the truth table for SNERG shows, there are two cases where SNERG(p,q,r) is true. Answer (a) describes one of these cases, and answer (d) describes the other. Answer (c) is the only one that covers both cases.

Which of the following arguments is valid?

-p

q&r

SNERG(p,q,r)

(a) is not correct. SNERG(p,q,r) is true exactly when p differs in its truth-values from both q and r. If p and q are true, it cannot follow that SNERG(p,q,r).

(b) is not correct. SNERG(p,q,r) is true exactly when p differs in its truth-values from both q and r. If -pVr is true, and if -q is true, it does not follow that SNERG(p,q,r). If p, q, and r are all false, then the premises of (b) can be true without SNERG(p,q,r) being true.

(c) is correct. SNERG(p,q,r) is true exactly when p differs in its truth-values from both q and r. If p is false, and if q and r are both true, then it follows that SNERG(p,q,r).

(d) is not correct. SNERG(p,q,r) is true exactly when p differs in its truth-values from both q and r. If pV-r is true, and if q is true, it does not follow that SNERG(p,q,r). If p, q, and r are all true, then the premises of (d) can be true without SNERG(p,q,r) being true.

(e) is not correct. One of the arguments among (a)-(d) is valid.

q&r

SNERG(p,q,r)

(a) is not correct. SNERG(p,q,r) is true exactly when p differs in its truth-values from both q and r. If p and q are true, it cannot follow that SNERG(p,q,r).

(b) is not correct. SNERG(p,q,r) is true exactly when p differs in its truth-values from both q and r. If -pVr is true, and if -q is true, it does not follow that SNERG(p,q,r). If p, q, and r are all false, then the premises of (b) can be true without SNERG(p,q,r) being true.

(c) is correct. SNERG(p,q,r) is true exactly when p differs in its truth-values from both q and r. If p is false, and if q and r are both true, then it follows that SNERG(p,q,r).

(d) is not correct. SNERG(p,q,r) is true exactly when p differs in its truth-values from both q and r. If pV-r is true, and if q is true, it does not follow that SNERG(p,q,r). If p, q, and r are all true, then the premises of (d) can be true without SNERG(p,q,r) being true.

(e) is not correct. One of the arguments among (a)-(d) is valid.

Which of the following arguments is valid?

none of the above.

a) is not correct. SNERG(p,q,r) is true exactly when the truth-value of p is different from both the truth-value of q and the truth-value of r. In (a), p has the same truth value as q.

(b) is not correct. SNERG(p,q,r) is true exactly when the truth-value of p is different from both the truth-value of q and the truth-value of r. In (b), only the truth-value of q is established. If p and r were both false, the premises of (b) could be true without SNERG(p,q,r) being true.

(c) is not correct. SNERG(p,q,r) is true exactly when the truth-value of p is different from both the truth-value of q and the truth-value of r. In (c), p has the same truth-value as r.

(d) is not correct. SNERG(p,q,r) is true exactly when the truth-value of p is different from both the truth-value of q and the truth-value of r. In (d), p has the same truth-value as r.

(e) is correct. SNERG(p,q,r) is true exactly when the truth-value of p is different from both the truth-value of q and the truth-value of r. In (a), p has the same truth value as q. In (b), only the truth-value of q is established. Finally, in (c) and (d), p has the same truth-value as r.

a) is not correct. SNERG(p,q,r) is true exactly when the truth-value of p is different from both the truth-value of q and the truth-value of r. In (a), p has the same truth value as q.

(b) is not correct. SNERG(p,q,r) is true exactly when the truth-value of p is different from both the truth-value of q and the truth-value of r. In (b), only the truth-value of q is established. If p and r were both false, the premises of (b) could be true without SNERG(p,q,r) being true.

(c) is not correct. SNERG(p,q,r) is true exactly when the truth-value of p is different from both the truth-value of q and the truth-value of r. In (c), p has the same truth-value as r.

(d) is not correct. SNERG(p,q,r) is true exactly when the truth-value of p is different from both the truth-value of q and the truth-value of r. In (d), p has the same truth-value as r.

(e) is correct. SNERG(p,q,r) is true exactly when the truth-value of p is different from both the truth-value of q and the truth-value of r. In (a), p has the same truth value as q. In (b), only the truth-value of q is established. Finally, in (c) and (d), p has the same truth-value as r.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p⊃r p⊃q r⊃q SNERG(p⊃r,p⊃q,r⊃q)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p⊃r p⊃q r⊃q SNERG(p⊃r,p⊃q,r⊃q)

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

FTTFFFFF

(b) is correct. Because SNERG(p,q,r) is logically equivalent to -(p≣q)&-(p≣r), the rightmost proposition is logically equivalent to the following conjunction: -((p⊃r)≡(p⊃q))&-((p⊃r)≡(r⊃q)). This, in turn, is equivalent to ((-(p⊃r)&(p⊃q))V((p⊃r)&-(p⊃q))) & ((-(p⊃r)&(r⊃q))V((r⊃q)&-(p⊃r))). In other words, one of (p⊃r) and (p⊃q) has to be true while the other is false, and one of (p⊃r) and (r⊃q) has to be true while the other is false. Since -(p⊃q) is equivalent to (p&-q), the rightmost proposition is equivalent to (((p&-r)&(p⊃q))V((p⊃r)&(p&-q))) & (((p&-r)&(r⊃q))V((p⊃r)&(r&-q))). Since (p⊃q) is equivalent to (-pVq), the rightmost proposition is equivalent to the following: (((p&-r)&(-pVq))V((-pVr)&(p&-q))) & (((p&-r)&(-rVq))V((-pVr)&(r&-q))). So the rightmost proposition, ultimately, is a conjunction of two disjunctions, where each of the disjuncts is itself a conjunction of one conjunction and one disjunction. Since the overall proposition is a conjunction, both conjuncts (i.e. both disjunctions) must be true. So, if we discover a truth-value assignment, which makes at least one of the conjuncts false, then the whole rightmost proposition is false. By carefully going through the truth table, we can see the answer will be FTTFFFFF. So (b) is the correct answer.

(b) is correct. Because SNERG(p,q,r) is logically equivalent to -(p≣q)&-(p≣r), the rightmost proposition is logically equivalent to the following conjunction: -((p⊃r)≡(p⊃q))&-((p⊃r)≡(r⊃q)). This, in turn, is equivalent to ((-(p⊃r)&(p⊃q))V((p⊃r)&-(p⊃q))) & ((-(p⊃r)&(r⊃q))V((r⊃q)&-(p⊃r))). In other words, one of (p⊃r) and (p⊃q) has to be true while the other is false, and one of (p⊃r) and (r⊃q) has to be true while the other is false. Since -(p⊃q) is equivalent to (p&-q), the rightmost proposition is equivalent to (((p&-r)&(p⊃q))V((p⊃r)&(p&-q))) & (((p&-r)&(r⊃q))V((p⊃r)&(r&-q))). Since (p⊃q) is equivalent to (-pVq), the rightmost proposition is equivalent to the following: (((p&-r)&(-pVq))V((-pVr)&(p&-q))) & (((p&-r)&(-rVq))V((-pVr)&(r&-q))). So the rightmost proposition, ultimately, is a conjunction of two disjunctions, where each of the disjuncts is itself a conjunction of one conjunction and one disjunction. Since the overall proposition is a conjunction, both conjuncts (i.e. both disjunctions) must be true. So, if we discover a truth-value assignment, which makes at least one of the conjuncts false, then the whole rightmost proposition is false. By carefully going through the truth table, we can see the answer will be FTTFFFFF. So (b) is the correct answer.

Which of the following arguments is valid?

-(p⊃q)&(p⊃r)&-(q⊃r)

-SNERG(p⊃q, p⊃r, q⊃r)

(a) is not correct. SNERG(p⊃q,p⊃r,q⊃r) is equivalent to the following conjunction: -((p⊃q)≡(p⊃r))&-((p⊃q)≡(q⊃r)). In other words, the conditional p⊃q must either be true while p⊃r and q⊃r are false, or the conditional p⊃q must be false while p⊃r and q⊃r are true. Yet the premise of (a) states that p, q, and r are all true, which means that the three conditionals are all true as well. So the premise of (a) is inconsistent with SNERG(p⊃q,p⊃r,q⊃r).

(b) is not correct. SNERG(p⊃q,p⊃r,q⊃r) is equivalent to the following conjunction: -((p⊃q)≡(p⊃r))&-((p⊃q)≡(q⊃r)). In other words, the conditional p⊃q must either be true while p⊃r and q⊃r are false, or the conditional p⊃q must be false while p⊃r and q⊃r are true. Yet the premise of (b) states that p is true while q and r are false, which means that p⊃q and p⊃r are both false. So the premise of (b) is inconsistent with SNERG(p⊃q,p⊃r,q⊃r).

(c) is not correct. SNERG(p⊃q,p⊃r,q⊃r) is equivalent to the following conjunction: -((p⊃q)≡(p⊃r))&-((p⊃q)≡(q⊃r)). In other words, the conditional p⊃q must either be true while p⊃r and q⊃r are false, or the conditional p⊃q must be false while p⊃r and q⊃r are true. Yet the premise of (c) states that p⊃q has the same truth-value as q⊃r, which is inconsistent with SNERG(p⊃q,p⊃r,q⊃r).

(d) is correct. The negation of SNERG(p⊃q,p⊃r,q⊃r) is equivalent to the following disjunction: ((p⊃q)≡(p⊃r))V((p⊃q)≡(q⊃r)). In other words, the conditional p⊃q must either have the same truth-value as p⊃r, or else p⊃q must have the same truth value as q⊃r. In (d), however, p⊃q has the same truth-value as q⊃r, which implies that -SNERG(p⊃q,p⊃r,q⊃r).

(e) is not correct. One of the arguments among (a)-(d) is valid.

-SNERG(p⊃q, p⊃r, q⊃r)

(a) is not correct. SNERG(p⊃q,p⊃r,q⊃r) is equivalent to the following conjunction: -((p⊃q)≡(p⊃r))&-((p⊃q)≡(q⊃r)). In other words, the conditional p⊃q must either be true while p⊃r and q⊃r are false, or the conditional p⊃q must be false while p⊃r and q⊃r are true. Yet the premise of (a) states that p, q, and r are all true, which means that the three conditionals are all true as well. So the premise of (a) is inconsistent with SNERG(p⊃q,p⊃r,q⊃r).

(b) is not correct. SNERG(p⊃q,p⊃r,q⊃r) is equivalent to the following conjunction: -((p⊃q)≡(p⊃r))&-((p⊃q)≡(q⊃r)). In other words, the conditional p⊃q must either be true while p⊃r and q⊃r are false, or the conditional p⊃q must be false while p⊃r and q⊃r are true. Yet the premise of (b) states that p is true while q and r are false, which means that p⊃q and p⊃r are both false. So the premise of (b) is inconsistent with SNERG(p⊃q,p⊃r,q⊃r).

(c) is not correct. SNERG(p⊃q,p⊃r,q⊃r) is equivalent to the following conjunction: -((p⊃q)≡(p⊃r))&-((p⊃q)≡(q⊃r)). In other words, the conditional p⊃q must either be true while p⊃r and q⊃r are false, or the conditional p⊃q must be false while p⊃r and q⊃r are true. Yet the premise of (c) states that p⊃q has the same truth-value as q⊃r, which is inconsistent with SNERG(p⊃q,p⊃r,q⊃r).

(d) is correct. The negation of SNERG(p⊃q,p⊃r,q⊃r) is equivalent to the following disjunction: ((p⊃q)≡(p⊃r))V((p⊃q)≡(q⊃r)). In other words, the conditional p⊃q must either have the same truth-value as p⊃r, or else p⊃q must have the same truth value as q⊃r. In (d), however, p⊃q has the same truth-value as q⊃r, which implies that -SNERG(p⊃q,p⊃r,q⊃r).

(e) is not correct. One of the arguments among (a)-(d) is valid.

Suppose you had to fill in the rightmost column of the following truth-table:

p q r p≡q (p≡q)⊃r SNERG[p≡q,(p≡q)⊃r,SNERG(p,q,r)]

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

p q r p≡q (p≡q)⊃r SNERG[p≡q,(p≡q)⊃r,SNERG(p,q,r)]

T T T

T T F

T F T

T F F

F T T

F T F

F F T

F F F

Going from top to bottom, how would you fill it in?

none of the above.

(e) is correct. Because SNERG(p,q,r) is logically equivalent to -(p≡q)&-(p≡r), it follows that SNERG[p≡q,(p≡q)⊃r,SNERG(p,q,r)] is logically equivalent to the following massive conjunction: -((p≡q)≡((p≡q)⊃r))&-((p≡q)≡(-(p≡q)&-(p≡r)). For such a conjunction to be true, both its conjuncts must be true. It is false otherwise. For the first conjunct to be true, however, the biconditional (p≡q)≡((p≡q)⊃r) must be FALSE. This means that the biconditional p≡q must have a different truth-value from the conditional (p≡q)⊃r. So, for the first conjunct of the rightmost proposition to be true, either p≡q must be true while (p≡q)⊃r is false, or p≡q must be false while (p≡q)⊃r is true. So there are four truth-value assignments, according to which the first conjunct of the rightmost proposition is true: one according to which p and q are true while r is false; one according to which p and q are both false while r is false; one according to which p is true, q is false, and r is true; and one according to which p is false and both q and r are true. The second conjunct of the rightmost proposition, however, is the following biconditional: (p≡q)≡(-(p≡q)&-(p≡r)). For the second conjunct to be true, either (p≡q) must be true while (-(p≡q)&-(p≡r)) is true; or (p≡q) must be false while (-(p≡q)&-(p≡r)) is false. The first case is logically impossible, however: p and q cannot have the same truth value while a conjunction, one of whose conjuncts is -(p≡q) is true. The second case obtains under two truth-value assignments: one according to which p is true, q is false, and r is false; and one according to which p is false, q is true, and r is true. Since the truth of the whole rightmost proposition requires the truth of both conjuncts, it follows that there is only one truth-value assignment under which the premise is true: when p is false, q is true, and r is true.

(e) is correct. Because SNERG(p,q,r) is logically equivalent to -(p≡q)&-(p≡r), it follows that SNERG[p≡q,(p≡q)⊃r,SNERG(p,q,r)] is logically equivalent to the following massive conjunction: -((p≡q)≡((p≡q)⊃r))&-((p≡q)≡(-(p≡q)&-(p≡r)). For such a conjunction to be true, both its conjuncts must be true. It is false otherwise. For the first conjunct to be true, however, the biconditional (p≡q)≡((p≡q)⊃r) must be FALSE. This means that the biconditional p≡q must have a different truth-value from the conditional (p≡q)⊃r. So, for the first conjunct of the rightmost proposition to be true, either p≡q must be true while (p≡q)⊃r is false, or p≡q must be false while (p≡q)⊃r is true. So there are four truth-value assignments, according to which the first conjunct of the rightmost proposition is true: one according to which p and q are true while r is false; one according to which p and q are both false while r is false; one according to which p is true, q is false, and r is true; and one according to which p is false and both q and r are true. The second conjunct of the rightmost proposition, however, is the following biconditional: (p≡q)≡(-(p≡q)&-(p≡r)). For the second conjunct to be true, either (p≡q) must be true while (-(p≡q)&-(p≡r)) is true; or (p≡q) must be false while (-(p≡q)&-(p≡r)) is false. The first case is logically impossible, however: p and q cannot have the same truth value while a conjunction, one of whose conjuncts is -(p≡q) is true. The second case obtains under two truth-value assignments: one according to which p is true, q is false, and r is false; and one according to which p is false, q is true, and r is true. Since the truth of the whole rightmost proposition requires the truth of both conjuncts, it follows that there is only one truth-value assignment under which the premise is true: when p is false, q is true, and r is true.

Which of the following arguments is valid?

None of the above.

Given the truth table for SNERG, there are only two ways for the premise to be true: either p⊃q is true, while both (p⊃q)⊃r and SNERG(p,q,r) are false, or else p⊃q is false, and both (p⊃q)⊃r and SNERG(p,q,r) are true. In the former case, r must be false. In the latter case, p must be true while q is false. In no case, however, do any of the conclusions among (a)-(d) follow from the premise. (e) is correct.

Given the truth table for SNERG, there are only two ways for the premise to be true: either p⊃q is true, while both (p⊃q)⊃r and SNERG(p,q,r) are false, or else p⊃q is false, and both (p⊃q)⊃r and SNERG(p,q,r) are true. In the former case, r must be false. In the latter case, p must be true while q is false. In no case, however, do any of the conclusions among (a)-(d) follow from the premise. (e) is correct.

Suppose your premise is (pVq)&[(p⊃q)⊃r]. Which of the following conclusions can be deduced from that premise?

none of the above

(a) is incorrect. Suppose that p is true, q is false, and r is false. Then the premise (pVq)&[(p⊃q)⊃r] would be true, even though r is false.

(b) is incorrect. Suppose that p is false, q is true, and r is true. Then the premise (pVq)&[(p⊃q)⊃r] is true without p being true.

(c) is incorrect. Suppose that p is true and q is false. Then p⊃q is false, but, if r is true, then the premise (pVq)&[(p⊃q)⊃r] can still be true.

(d) is incorrect. Suppose that p is true and q is false. Then p⊃q is false, but, if r is true, then the premise (pVq)&[(p⊃q)⊃r] can still be true.

(e) is correct. For the premise to be true, it is not necessary that r is true. Suppose that p is true, q is false, and r is false. Then the premise (pVq)&[(p⊃q)⊃r] would be true, even though r is false. So (a) is wrong. Likewise, (b) is wrong, too. Suppose that p is false, q is true, and r is true. Then the whole premise is true without p being true. Likewise, (c) is wrong. Suppose that p is true and q is false. Then p⊃q is false, but, if r is true, then the premise can still be true. For the same reason, (d) is wrong, too.

(a) is incorrect. Suppose that p is true, q is false, and r is false. Then the premise (pVq)&[(p⊃q)⊃r] would be true, even though r is false.

(b) is incorrect. Suppose that p is false, q is true, and r is true. Then the premise (pVq)&[(p⊃q)⊃r] is true without p being true.

(c) is incorrect. Suppose that p is true and q is false. Then p⊃q is false, but, if r is true, then the premise (pVq)&[(p⊃q)⊃r] can still be true.

(d) is incorrect. Suppose that p is true and q is false. Then p⊃q is false, but, if r is true, then the premise (pVq)&[(p⊃q)⊃r] can still be true.

(e) is correct. For the premise to be true, it is not necessary that r is true. Suppose that p is true, q is false, and r is false. Then the premise (pVq)&[(p⊃q)⊃r] would be true, even though r is false. So (a) is wrong. Likewise, (b) is wrong, too. Suppose that p is false, q is true, and r is true. Then the whole premise is true without p being true. Likewise, (c) is wrong. Suppose that p is true and q is false. Then p⊃q is false, but, if r is true, then the premise can still be true. For the same reason, (d) is wrong, too.

Which of the following pairs of statements is consistent?

All odd numbers are primes; no odd numbers are primes.

(a) is correct. Suppose that there are no odd numbers. In such a case, because there would not be any non-prime odd numbers, the statement " all odd numbers are primes" would be true. Yet the statement " no odd numbers are primes" would also be true, since there would not be any prime odd numbers either. So the statements " all odd numbers are primes" and " no odd numbers are primes" can both be true, provided that there are no odd numbers.

(b) is incorrect. If all odd numbers are primes, then there cannot be any odd numbers that are not primes. So the statement " all odd numbers are primes" is inconsistent with the statement " some odd numbers are not primes."

(c) is incorrect. If some odd numbers are primes, then it cannot be that there are no prime odd numbers. So the statement "some odd numbers are primes" is inconsistent with the statement " no odd numbers are primes."

(d) is incorrect. Not all of the pairs of statements among (a)-(c) are consistent.

(e) is incorrect. At least one of the pairs of statements among (a)-(c) is consistent.

(a) is correct. Suppose that there are no odd numbers. In such a case, because there would not be any non-prime odd numbers, the statement " all odd numbers are primes" would be true. Yet the statement " no odd numbers are primes" would also be true, since there would not be any prime odd numbers either. So the statements " all odd numbers are primes" and " no odd numbers are primes" can both be true, provided that there are no odd numbers.

(b) is incorrect. If all odd numbers are primes, then there cannot be any odd numbers that are not primes. So the statement " all odd numbers are primes" is inconsistent with the statement " some odd numbers are not primes."

(c) is incorrect. If some odd numbers are primes, then it cannot be that there are no prime odd numbers. So the statement "some odd numbers are primes" is inconsistent with the statement " no odd numbers are primes."

(d) is incorrect. Not all of the pairs of statements among (a)-(c) are consistent.

(e) is incorrect. At least one of the pairs of statements among (a)-(c) is consistent.

Which of the following Venn Diagrams represents the statement that all odd numbers are primes?

none of the above.

(a) is not correct. The Venn diagram for " all odd numbers are primes" shades out the region of the circle of odd numbers, which is outside the circle of prime numbers. This diagram, however, does the opposite. What (a)'s diagram represents is that all prime numbers are odd, rather than representing that all odd numbers are primes.

(b) is not correct. The Venn diagram for " all odd numbers are primes" shades out the region of the circle of odd numbers, which is outside the circle of prime numbers. This diagram, however, shades out the region where they intersect. What (b)'s diagram represents is that no odd numbers are primes, rather than representing that all odd numbers are primes.

(c) is not correct. The Venn diagram for " all odd numbers are primes" shades out the region of the circle of odd numbers, which is outside the circle of prime numbers. This diagram, however, places an X in that region instead of shading it. What (c)'s diagram represents is that some odd numbers are not prime, rather than representing that all odd numbers are primes.

(d) is not correct. Not all of (a)-(c) correctly represent that all odd numbers are primes.

(e) is correct. None of (a)-(c) correctly represent that all odd numbers are primes.

(a) is not correct. The Venn diagram for " all odd numbers are primes" shades out the region of the circle of odd numbers, which is outside the circle of prime numbers. This diagram, however, does the opposite. What (a)'s diagram represents is that all prime numbers are odd, rather than representing that all odd numbers are primes.

(b) is not correct. The Venn diagram for " all odd numbers are primes" shades out the region of the circle of odd numbers, which is outside the circle of prime numbers. This diagram, however, shades out the region where they intersect. What (b)'s diagram represents is that no odd numbers are primes, rather than representing that all odd numbers are primes.

(c) is not correct. The Venn diagram for " all odd numbers are primes" shades out the region of the circle of odd numbers, which is outside the circle of prime numbers. This diagram, however, places an X in that region instead of shading it. What (c)'s diagram represents is that some odd numbers are not prime, rather than representing that all odd numbers are primes.

(d) is not correct. Not all of (a)-(c) correctly represent that all odd numbers are primes.

(e) is correct. None of (a)-(c) correctly represent that all odd numbers are primes.

Which of the following Venn Diagrams represents the statement that some odd numbers are not primes?

None of the above.

(a) is not correct. The correct Venn diagram for " some odd numbers are not primes" has an X in the region of the circle of odd numbers, which is outside the circle of prime numbers. This diagram, however, shades that region instead of putting an X in it. This diagram represents that all odd numbers are prime.

(b) is not correct. The correct Venn diagram for " some odd numbers are not primes" has an X in the region of the circle of odd numbers, which is outside the circle of prime numbers. This diagram, however, shades that region instead of putting an X in it, and places an X at the intersection of the two circles instead. This diagram represents that all odd numbers are prime, and that some odd numbers are prime, too.

(c) is not correct. The correct Venn diagram for " some odd numbers are not primes" has an X in the region of the circle of odd numbers, which is outside the circle of prime numbers. This diagram, however, shades that region instead of putting an X in it, and shades the non-intersecting region in the circle of prime numbers, too. This diagram represents that all odd numbers are prime, and that all prime numbers are odd.

(d) is not correct. Not all of the Venn diagrams among (a)-(d) represent that some odd numbers are not prime.

(e) is correct. None of the Venn diagrams among (a)-(d) represent that some odd numbers are not prime.

(a) is not correct. The correct Venn diagram for " some odd numbers are not primes" has an X in the region of the circle of odd numbers, which is outside the circle of prime numbers. This diagram, however, shades that region instead of putting an X in it. This diagram represents that all odd numbers are prime.

(b) is not correct. The correct Venn diagram for " some odd numbers are not primes" has an X in the region of the circle of odd numbers, which is outside the circle of prime numbers. This diagram, however, shades that region instead of putting an X in it, and places an X at the intersection of the two circles instead. This diagram represents that all odd numbers are prime, and that some odd numbers are prime, too.

(c) is not correct. The correct Venn diagram for " some odd numbers are not primes" has an X in the region of the circle of odd numbers, which is outside the circle of prime numbers. This diagram, however, shades that region instead of putting an X in it, and shades the non-intersecting region in the circle of prime numbers, too. This diagram represents that all odd numbers are prime, and that all prime numbers are odd.

(d) is not correct. Not all of the Venn diagrams among (a)-(d) represent that some odd numbers are not prime.

(e) is correct. None of the Venn diagrams among (a)-(d) represent that some odd numbers are not prime.

Some elephants are not elegant.

(d) is correct. By placing an X in the region of the circle of elephants, which is outside the circle of elegant things, this Venn diagram represents that some elephants are not elegant.

(d) is correct. By placing an X in the region of the circle of elephants, which is outside the circle of elegant things, this Venn diagram represents that some elephants are not elegant.

All shepherds are leopards.

(a) is not correct. The correct Venn diagram for " all leopards are shepherds" shades out the region of the circle of leopards, which is outside the circle of shepherds. The diagram above, however, does not do this.

(b) is correct. The correct Venn diagram for " all shepherds are leopards" shades out the region of the circle of shepherds, which is outside the circle of leopards. The diagram above does exactly this.

(c) is not correct. The correct Venn diagram for " no leopards are shepherds" shades out the region of the circle of leopards, which intersects with the circle of shepherds. The diagram above, however, does not do this.

(d) is not correct. The correct Venn diagram for " some shepherds are not leopards" has an X in the region of the circle of shepherds, which is outside the circle of leopards. The diagram above, however, does not do this.

(e) is not correct. One of the options among (a)-(d) correctly states what the diagram represents.

(a) is not correct. The correct Venn diagram for " all leopards are shepherds" shades out the region of the circle of leopards, which is outside the circle of shepherds. The diagram above, however, does not do this.

(b) is correct. The correct Venn diagram for " all shepherds are leopards" shades out the region of the circle of shepherds, which is outside the circle of leopards. The diagram above does exactly this.

(c) is not correct. The correct Venn diagram for " no leopards are shepherds" shades out the region of the circle of leopards, which intersects with the circle of shepherds. The diagram above, however, does not do this.

(d) is not correct. The correct Venn diagram for " some shepherds are not leopards" has an X in the region of the circle of shepherds, which is outside the circle of leopards. The diagram above, however, does not do this.

(e) is not correct. One of the options among (a)-(d) correctly states what the diagram represents.

Which of the following is valid?

Some leopards are not shepherds.

Therefore not all leopards are shepherds.

(a) is not correct. " No leopards are shepherds" does not imply " some leopards are not shepherds" . Suppose there are no leopards. In such a case, the first statement would be true and the second would be false.

(b) is not correct. " Not all leopards are shepherds" does not imply " no leopards are shepherds" . Suppose there are some non-shepherd leopards, and that there are some shepherd leopards, too. In such a case, the first statement would be true and the second would be false.

(c) is correct. " Some leopards are not shepherds" implies " not all leopards are shepherds" . If there are some non-shepherd leopards, then it cannot be true that all the leopards are shepherds.

(d) is not correct. " Some leopards are not shepherds" does not imply " some leopards are shepherds" . Suppose there are some non-shepherd leopards, and that all the leopards are non-shepherds. In such a case, the first statement would be true and the second would be false.

(e) is not correct. One of the immediate categorical inferences among (a)-(c) is valid.

Therefore not all leopards are shepherds.

(a) is not correct. " No leopards are shepherds" does not imply " some leopards are not shepherds" . Suppose there are no leopards. In such a case, the first statement would be true and the second would be false.

(b) is not correct. " Not all leopards are shepherds" does not imply " no leopards are shepherds" . Suppose there are some non-shepherd leopards, and that there are some shepherd leopards, too. In such a case, the first statement would be true and the second would be false.

(c) is correct. " Some leopards are not shepherds" implies " not all leopards are shepherds" . If there are some non-shepherd leopards, then it cannot be true that all the leopards are shepherds.

(d) is not correct. " Some leopards are not shepherds" does not imply " some leopards are shepherds" . Suppose there are some non-shepherd leopards, and that all the leopards are non-shepherds. In such a case, the first statement would be true and the second would be false.

(e) is not correct. One of the immediate categorical inferences among (a)-(c) is valid.