# First: I think every question has a logical answer. is it correct? Second: If the answer to my first question is yes, then what is the logical answer to the question why a cow has four legs?

### If by "a logical answer" you

If by "a logical answer" you mean an answer that is logically consistent , then I agree that every well-posed question has a logical answer. Nevertheless, the logically consistent answer to some question will often rely on information from beyond the subject matter of logic. The answer to why a particular cow has four legs will rely on information about the cow's parentage, genetics, embryology, anatomy, or some such. Logic all by itself cannot answer that question.

# Hello! I'd like to ask about syllogisms. I have a particular problem when understanding this certain syllogism: Some girls are single. Some girls are sad. Therefore, some girls are single and sad. While I think it is valid, I cannot fully make an accurate explanation as to why it is. Hoping somebody could help me. Thanks!

### The syllogism in question is

The syllogism in question is not valid. Nothing logically guarantees that the set of single girls and the set of sad girls overlap. Even if both sets have members, it does not follow that they have any members in common. Compare: Some polygons are squares. Some polygons are triangles. But it is false that some polygons are square triangles.

# I have recently compared two philosophy texts which are very very close in material they present: A Concise Introduction to logic 12th edition by Patrick Hurley and Introduction to Logic by Irving Copi & Carl Cohen 12th edition. I have a question about the logical Equivalence Rule Material Implication which states where ever P imples Q appears one can substitute Not P or Q and vice versa. I noticed if Not P or Q is Implicated the NOT is always on the left hand side. There is no instance of Q or Not P and the rule Material Implication being applied. My question is if I am given "Q or Not P" can I apply Material Implication as written or must I commutate "Q or Not P" to get "Not P or Q" and then use the Material Implication rule? It seems all is done to avoid using material implication with a negative disjunct on the right hand side. What is the deal with that? In other words, Would I get false conclusions if I deduce Q or Not P as Not Q or Not P? I am correct in guessing this may be the case? I...

### Using ">" for material

Using "> " for material implication, (P > Q) is equivalent to each of (~ P v Q) and (Q v ~ P). So you can deduce either of those disjunctions. I think it's just a matter of convention to favor the first of them. The reader is expected to notice the equivalence of the two disjunctions. Now, (Q v ~ P) is certainly not equivalent to (~ Q v ~ P). From Q, you can infer the first of those disjunctions but not the second. The disjunction (Q v ~ P) is equivalent to (P > Q), whereas the disjunction (~ Q v ~ P) is equivalent to (P > ~ Q) and (Q > ~ P).

# Consider two identical sets, A and B; but they're not identical, because they have different names. Paradox?

### If a paradox resulted every

If a paradox resulted whenever one thing had more than one name, then these paradoxes wouldn't be restricted to sets. The names 'Samuel Clemens' and 'Mark Twain' would generate a paradox by referring to the same person. But, of course, there's no paradox here. Everything true of the person named 'Samuel Clemens' is true of the person named 'Mark Twain'. Mark Twain was born in Missouri, and Samuel Clemens wrote The Adventures of Huckleberry Finn . Indeed, all those who know that Mark Twain wrote the novel thereby also know de re (Latin for 'concerning the thing') that Samuel Clemens wrote the novel: they know, concerning the person denoted by 'Samuel Clemens', that he wrote the novel, even if they wouldn't use 'Samuel Clemens' to denote the author.

# Recently I got a question for a philosophy class on how the following set of statements is consistent: "All dragons love Katy Perry. All dragons hate Katy Perry." And we have to think about how to explain that this is consistent. I'm struggling to understand how to make that explanation.

### I don't want to do your

I don't want to do your assigned coursework for you, so I'll say just this: Think about how modern logic translates "All S is P" and "All S is not P" into symbols.

# This is a follow up to a question answered by Dr. Maitzen on December 31 2020. The statement really was “Only if A, then B”. It came up on a test question that asked the following: “If A, then B” and “Only if A, then B” are logically equivalent. True or false? The answer is ‘false’, apparently. I reasoned that “Only if A, then B” is maybe like saying “Necessarily: if A, then B”, and this is clearly different from saying simply “If A, then B”. But I’m not sure. Any chance you might be able to help me see why “If A, then B” and “Only if A, then B” aren’t equivalent? Clearly they say different things, but I’m just not sure how to put my finger on the difference. I really appreciate the help. Thank you again.

### It sounds to me as though

It sounds to me as though your teacher may be using the awkward expression "Only if A, then B" as a way of asserting the biconditional "A if and only if B," which is equivalent to the biconditional "B if and only if A." As I say, the expression is awkward, but in any case I wouldn't read it as adding a modal operator like "Necessarily" to the conditional "If A, then B." Whoever wants to say "necessarily" really needs to use that word. Other than your teacher's decision, I can't think of any reason to treat "Only if A, then B" as the biconditional "A if and only if B." The form "Only if A, then B" isn't something you'll encounter in idiomatic English. Competent speakers wouldn't say, "Only if all humans are mortals, then all nonmortals are nonhuman." Instead, they'd say "All humans are mortals if and only if all nonmortals are nonhumans." But it's probably wise to follow your teacher's decision, at least until you're done with the course!

# I'm confused about the nature of antecedents and conditionals like: (i) "Only if A, then B". I was told in my logic class that antecedents are always sufficient conditions and consequents are always necessary conditions. But if that's the case, then the antecedent in (i) "Only if A" is a sufficient condition. Particularly a sufficient condition for B. But saying "Only if A, then B" means that A is a necessary condition for B as well. So it appears that the antecedent in (i) is both a sufficient and necessary condition. But that doesn’t seem right, given that (i) is equivalent to (ii) If B, then A. And this means A is only a necessary but not a sufficient condition for B. Option 1: Maybe antecedents only are sufficient conditions in simple conditionals like (iii) “If A, then B”; but they aren’t sufficient conditions in conditionals like "Only if A, then B". That might be right. Option 2: On the other hand, we might say "Only if A" just seems to be an antecedent but isn't really. That would...

### Like you, I'm puzzled by the

Like you, I'm puzzled by the form of the conditional "Only if A, then B." It doesn't seem to be idiomatic English. One might say "Only if you go to the party will I go," but one wouldn't say "Only if you go to the party, then I will go." That would be unidiomatic. So I presume that the conditional form you're learning is "Only if A, B" rather than "Only if A, then B." I would interpret "Only if A, B" as stating that A is a necessary condition for B, and therefore implying that B is a sufficient condition for A. If one wants to say that A is both necessary and sufficient for B, then one can say "If and only if A, B" -- although "A if and only if B" would be a smoother way of saying it. In any case, make sure that your logic teacher really did say "Only if A, then B" and, if so, ask if he/she meant to say "Only if A, B."

# Let ‘B’= to be; let ‘~B’=not to be. P1: B v ~B P2: ~B C: ~B P2 is the negation of the left disjunct in P1, not the affirmation of the right disjunct in P1. P1: To be or not to be. P2: Not to be. C: Not to be. It seems to me that, argumentatively, there’s a difference between affirming ‘not to be’, the right disjunct, and negating ‘to be’, the left disjunct. It just happens that, in this case, what’s affirmed and what’s negated are logically equivalent. Is there a convention for conveying that argumentative difference? Also, can you recommend any articles or books where I can learn more about issues like this? Thank you very much :)

### Interesting question! I think

Interesting question! I think you're right that there's something peculiar about this disjunctive syllogism: (1) B v ~ B (2) ~ B (3) ~ B You say that (2) must be the negation of (1)'s left disjunct rather than the assertion of (1)'s right disjunct, even though both of those are syntactically the same. You may find allies in those who distinguish between (i) denying or rejecting a proposition and (ii) asserting the proposition's negation. See Section 2.5 of this SEP entry . But here's a different diagnosis. Although (1)-(3) is a valid argument, and even a valid instance of disjunctive syllogism, the argument is informally defective because premise (1) is superfluous: (1) isn't needed for the argument's validity. Furthermore, anyone justified in asserting (2) is thereby justified in asserting (3) without need of (1). This argument is similar: (4) ~ B v B (5) ~ ~ B (6) B The claim that (5) is the negation of (4)'s left disjunct is at least as plausible as the claim that (2) is the negation of (1...

# I wonder about the nature of modal concepts such as necessity and possibility. When I say "It is possible that this page is white" or "it is necessary that two plus two equals four" I use modal words in my speech. Where do these concepts belong to? Are they in my mind or I receive them from the objects themselves?

### It's a good idea to

It's a good idea to distinguish between epistemic uses of modal language (which have to do with our knowledge) and alethic uses (which have to do with truth independently of our knowledge). When you say, "It is possible that this page is white," you might be wearing tinted glasses and simply admitting that, for all you know, the page that looks amber to you is in fact white (i.e., it looks white to normal observers in normal conditions). That use of "possible" would be epistemic. Or, instead, you might be saying that the page, which in fact emerged a mottled gray from the unreliable paper mill, could have been white had the mill done a better job. Or you might simply infer from the fact that the page is white that it's possible that the page is white: what is true is of course also possible. Those uses of "possible" would be alethic. Where do alethic modal concepts belong? I'd say that they belong to logic, in the sense that they are at the foundation of the concept of logical consequence. To...

# I am reading a by book by the great logician Raymond Smullyan. In this book he says that any statement of the form, "All As are Bs" are true if there are no "As". That is, these statements are vacuously true. He gives the following example, "All Unicorns have 5 legs" is true since there are no unicorns. So is "All unicorns have 6 legs", and "All unicorns are purple", etc. But this strikes me as obviously false. For example, "All unicorns have two horns" and "All unicorns are necessarily existing" are false statements. The first is false in virtue of the fact that unicorns are by definition one-horned. The second is false in virtue by the fact that it is impossible for something to be both necessarily existing and nonexistent. Am I missing something here or misreading Smullyan? Or are these counterexamples sufficient in refuting the claim that any statement of the form "All As are Bs" is vacuously true if there are no "As"? For reference the book is, "Logical Labyrinths" from pages 99-101. Thanks...

### I don't know that book in

I don't know that book in particular, but I can give you a standard explanation that at least makes sense of the view you find puzzling. In Aristotle's logic, any statement of the form "All S are P" implies that at least one S is P, so the statement comes out false (rather than vacuously true) if nothing is S. By contrast, in contemporary logic, "All S are P" is interpreted as saying "For anything at all, if it is S, then it is P": it is interpreted as a universal quantification applied to a conditional statement. Crucially, the conditional statement "If it is S, then it is P" is standardly treated as a truth-functional conditional that is equivalent to the disjunction "It is not S, or it is P." Now suppose that nothing is S, so that "It is not S" is true of everything. Then the disjunction "It is not S, or it is P" will come out true no matter what we substitute for "it," because a true disjunction needs only one true disjunct. In that case, the truth-functional conditionals "If it is S, then it...