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Language

There is an infinite number of words - "ONE", "TWO", "THREE"... etc. Every word has a definition. Every definition consists of letters. There is a finite number of arrangement of letters; thus there is a finite number of definitions. Thus there is at least one word that doesn't have a definition. Paradox?

There is a finite number of arrangements of letters; thus there is a finite number of definitions. Is that true if we're allowed to use each letter an increasing number of times? If our stock of letter tokens increases without limit, then can't the number (and length) of our definitions also increase without limit? Certainly the names of the numbers will tend to get longer as the numbers they name increase, and those names will reuse letters to an ever-increasing degree.
Truth

Is truth binary? Can there be degrees of truth? Must we say a statement is either true or false? In everyday conversations we do say things like “The story is only 30% true.”

I'd say yes, truth is binary: there are no degrees of truth in between 0 and 1. For two reasons, I think that every proposition has exactly one of two truth-values, true or else false . (1) I'm not aware of any problem in logic or philosophy whose solution really does require positing fewer or more than two truth-values. (2) To my knowledge, every system of logic or semantics that accommodates other than two truth-values has consequences that are deeply implausible. These systems include three-valued logic, infinite-valued logic, supervaluation semantics, and others. In light of (1), I see no reason to flirt with those implausible consequences. When solving a problem seems to require other than two truth-values, the real trouble lies somewhere else. Or so it seems to me. Now, to say that every proposition is true or false is not to say that every sentence is true or false, not even every grammatically correct declarative sentence. For instance, the self-referential Liar sentence "This...
Logic

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 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.
Logic

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 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.
Logic

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 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).
Language

Could someone help me clear up a paradox? Let’s say there’s a woman who says “I cannot say no to you” and the woman she is speaking with responds “well then, say ‘no’”. Is this really a paradox? In my opinion it isn’t: When the woman says that “she can’t say ‘no’” there are one of two interpretations of that phrase. She either means “I cannot deny your requests” or literally “I cannot utter the word ‘no’”. If it’s the former, then asking her to just “say no” wouldn’t put her in a paradox, since simply uttering the word “no” isn’t denying a request, it’s just making an empty utterance. It’s like how saying “I declare bankruptcy” doesn’t actually do anything, it’s just making noise. If it’s the latter example, and she cannot “say no”, then she technically never said she cannot deny a request—she just can’t use the specific word “no” in her statement. She can still say “I refuse” or “I will not do that” and then not have to say “no”. Am I wrong about this?

If the woman meant (a) "I can't utter the word no in response to any request from you," then she can't abide by her companion's request (to say "no") without falsifying what she has just said. Still, I agree with you that there's no paradox here. The woman can abide by the request to say "no" by saying "no" in response to it. As far as I can see, the appearance of paradox depends on supposing that the woman meant both (a) and also (b) "I can't deny any request from you." But, as you suggest, she can't have meant both (a) and (b). All that follows is that (a) and (b) can't both be true if her companion asks her to say "no." Nothing especially interesting about that.
Freedom

If you were to look back at a certain period (of time) that you did something, can you say with certainty (most likely) you would have done a different action instead of the action that you took that lead you to the present you. Or does it become another probability that you cannot infer that you would have most likely done things differently. Say for example you said that you would have most likely went to school B if you did not go to school A. Are you saying that if you went back to that instance in time, knowing all the stuff that you did you would most likely have chosen school B given your circumstances of you at the past without any of the future biases that you hold from choosing school A? Or does it become another possibility of a what if scenario where you cannot infer that you would have done this or that to certain degree? How can you say for certain you would have done that when you did not (choose school A) therefore there would be no certainty in what you would have done in the past?

If you go back to that choice point exactly as it was -- with your beliefs, desires, and circumstances being exactly as they were originally -- then why think that you would have made a different choice? We can't make sense of your choosing differently unless we imagine that something is different about that choice point. One might claim that your original choice was indeterministic , so it could have been different even in exactly the same circumstances. But calling a choice "indeterministic" is just to say that, beyond a certain point, we can't make sense of it.
Logic
Mathematics

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

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.
Logic

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 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.
Logic

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 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!

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