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. 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 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. "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 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 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 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. 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 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 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. 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 "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. 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. 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. 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. 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. 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. 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. 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. 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? 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. 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. 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. 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. 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. 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. 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.

_____________________________________________

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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 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. 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. 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 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. 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? 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.) 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? 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? 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? 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? 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? 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? 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? 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? 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. 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. 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. 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? 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? 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? 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? 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 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. 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? 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? 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? 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? 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? 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. 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. 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." 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. 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. 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. 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." 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. 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. 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. 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. (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. 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. 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. 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. "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. 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. 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. 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. 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. 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. 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. 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). 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." 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." (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." 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. 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. 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. 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. 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). 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). 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. 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). 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. 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. 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. 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. 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." 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." 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. 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. 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." 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." 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. 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). 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. 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). 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. (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) 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. 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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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. 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. 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. 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. 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. 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. 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. 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." 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." 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. 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. 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). 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." -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. 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. 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. -(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.