Terms in this set (788)

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. "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 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 "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 "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.
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.
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.
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.
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.
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.
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.
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. 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.
All dodos are quotable.
All quotable things are witches.

Therefore, all dodos are witches.

(a) is not valid. As the diagram shows, "all dodos are quotable" requires us to shade out the circle of dodos that does not intersect with the circle of quotable things. The statement "some quotable things are witches," moreover, requires us to place an X where the circle of quotable things intersects with the circle of witches. It does not follow, however, that there are any dodos who are witches. If some witches were also quotable non-dodos, and if all the witches were quotable non-dodos, then the premises of (a) could be true while the conclusion is false.

(b) is not valid. As the diagram shows, the fact that some dodos are quotable, and the fact that some quotable things are witches, does not entail that some dodos are witches. If some witches were also quotable non-dodos, and if all the witches were quotable non-dodos, then the premises of (b) could be true while the conclusion is false
(c) is not valid. As the diagram shows, "no dodos are quotable" requires us to shade out the circle of dodos where it intersects with the circle of quotable things. The statement "no quotable things are witches," moreover, requires us to shade out the circle of quotable things where it intersects with the circle of witches. Together, these two shading patterns do not show that no dodos are witches, since the intersection between the circle of dodos and the circle of witches is not shaded.
(d) is correct. As the diagram shows, "all dodos are quotable" requires us to shade out the circle of dodos, except where it intersects with the circle of quotable things. The statement "all quotable things are witches," moreover, requires us to shade out the circle of quotable things, except where it intersects with the circle of witches. Together, these two shading patterns show that all dodos are witches.
All 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."
(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 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.
All dragons are penguins.
All penguins are fiery.

Therefore, all dragons are fiery.

(a) is not correct. As the diagram shows, "no dragons are penguins" requires us to shade out the circle of dragons where it intersects with the circle of penguins. The statement "no penguins are fiery," moreover, requires us to shade out the circle of penguins where it intersects with the circle of fiery things. Together, these two shading patterns do not show that no dragons are fiery, since the intersection between the circle of dragons and the circle of fiery things is not completely shaded.

(b) is correct. As the diagram shows, "all dragons are penguins" requires us to shade out the circle of dragons, except where it intersects with the circle of penguins. The statement "all penguins are fiery things," moreover, requires us to shade out the circle of penguins, except where it intersects with the circle of fiery things. Together, these two shading patterns show that all dragons are fiery.

(c) is not correct. As the diagram shows, the fact that some dragons are penguins, and the fact that some penguins are fiery, does not entail that some dragons are fiery. If some penguins were also fiery non-dragons, and if all the penguins were fiery non-dragons, then the premises of (b) could be true while the conclusion is false.

(d) is not correct. As the diagram shows, "all dragons are penguins" requires us to shade out the circle of dragons that does not intersect with the circle of penguins. The statement "some penguins are fiery," moreover, requires us to place an X where the circle of fiery things intersects with the circle of penguins. It does not follow, however, that there are any fiery dragons. If some penguins were also fiery non-dragons, and if all the penguins were fiery non-dragons, then the premises of (a) could be true while the conclusion is false.

(e) is not correct. One of the syllogisms among (a)-(d) is valid.
All 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.
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.
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. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." "FRONK trees are deciduous" does not imply that all trees are deciduous. The diagram for "all trees are deciduous" would shade the region of the circle of trees, which is outside the circle of deciduous things. Our diagram does not do this.

(b) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." "FRONK trees are deciduous" does not imply that no trees are deciduous. The diagram for "no trees are deciduous" would shade the intersection between the circle of trees and the circle of deciduous things. Our diagram does not do this.

(c) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." "FRONK trees are deciduous" does not imply that some trees are deciduous. The diagram for "some trees are deciduous" would place an X between the circle of trees and the circle of deciduous things. Our diagram does not do this.

(d) is not correct. As the diagram shows, "FRONK As are Bs" means the same thing as "all Bs are As." So "FRONK trees are deciduous" means the same thing as "all deciduous things are trees." "FRONK trees are deciduous" does not imply that some trees are not deciduous. The diagram for "some trees are not deciduous" would place an X inside the circle of trees but outside the circle of deciduous things. Our diagram does not do this.

(e) is correct. None of the statements among (a)-(d) are implied by "FRONK trees are deciduous."
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.
-(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.