I would caution against inferring from 'The principle of noncontradiction is an abstraction' to 'The principle of noncontradiction doesn't exist naturally or non-naturally'. A number of philosophers, and maybe an even larger number of mathematicians, think that at least some abstract objects must exist -- and exist non-naturally. It may be that the principle of noncontradiction is among those abstract objects. You may find this SEP entry on the topic helpful.

Translating the argument into symbolic terms quite literally, we get this: 'If F, then R. Not F. Therefore, not R.' That form of argument does indeed commit the fallacy of denying the antecedent: the premises don't logically imply the conclusion; the truth of the premises doesn't logically ensure the truth of the conclusion. The first premise says that F is sufficient for R; it doesn't say that F is necessary for R. In that case, R can obtain even if F fails to obtain. My hunch is that you're interpreting 'If F, then R' as 'R if and only if F': you're interpreting the conditional as a biconditional , i.e., as the claim that F is both necessary and sufficient for R. 'R if and only if F' and 'Not F' together imply 'Not R'. Your interpretation is understandable, because conversationally we often do intend to assert a biconditional when we use conditional language. A parent's 'If you clean your room, you can watch TV' usually means 'You can watch TV if and only if you clean your room...

It would be helpful to know which answers on the site you're referring to, but I'll take a stab at your question anyway. The only difference between your formulations (a) and (b) is the second disjunct in each, so I'll focus on that. I presume A is some statement. What's the difference between "Not A is true" and "A is not true"? I'm not sure there's always a difference. Depending on the system of logic or semantics, "Not A is true" can mean merely (i) "Whatever truth-value (if any) A has, it's not the value true ." Or it can mean (ii) "A is false " in systems in which every statement is true or false. The difference between (i) and (ii) is sometimes important, such as when we're dealing with the classic Liar sentence (L) "This sentence is false." One might say that L is not true and yet not false either: one might say that L is neither true nor false.

You wrote: (i) "It seems that for classical logic to make sense of a contradiction in such a way that it leads to explosion...it must define what it means to hold a contradiction in a particular way" and (ii) "[W]ouldn't it be defined in some arbitrary way that forces us into the 'explosion' scenario?" Regarding (i): If the assertion "A is true and A is false" means anything, then surely it implies that A is true and implies that A is false. I can't think of another way to construe the assertion. Are you suggesting that a conjunction doesn't imply each of its conjuncts? Regarding (ii): How is it arbitrary to infer the truth of A and the falsity of A from the assertion "A is true and A is false"? Again, I can't think of another way to understand the assertion. As far as I know, paraconsistent logicians tend to object to inferring B from (A or B) and not-A: they point out that the inference relies on the implicit assumption that not-A rules out A, an assumption they reject.

Given premises P and not P, it is indeed valid to derive P. I don't know of any logical systems, including non-classical systems, that would deny the validity of that derivation. (A valid derivation needn't use all of its premises: "P; Q; therefore, P" is valid.) The derivation you gave isn't sound, however, because not all of the premises are true: it's guaranteed that one of the premises is false (even if we don't know which one). Yes, we're ascertaining the unsoundness of the derivation without knowing the content of its premises, but that's perfectly fine: If you know that the form of the derivation guarantees that it has a false premise, you don't need to know anything more in order to know that the derivation is unsound.

Yes! But bear in mind that a paradox is an apparent contradiction, an apparent inconsistency, that we're tasked with trying to resolve in a consistent way. For example, a particular argument implies that the Liar sentence ("This sentence is false") is both true and false, and a similar argument implies that the Strengthened Liar sentence ("This sentence is not true") is both true and not true. Usually it's our conviction that those arguments can't be sound that impels us to seek out the flaw in each argument. So too for other famous paradoxes, such as the Paradox of the Heap. Paradoxes abound! But that doesn't mean that contradictory situations do. Now, some philosophers, such as Graham Priest, say it's a mistake to demand a consistent solution to every paradox. Priest says that the Liar Paradox has an inconsistent solution, i.e., the Liar sentence is both true and false: it's both true and a contradiction. So Priest would say that not only do paradoxes actually occur but inconsistent...

Are infinite regresses logically possible? Surely it's logically possible for infinitely many positive or negative integers to exist, and they represent a kind of infinite regress: for every negative integer, there's a smaller one; for every positive integer, there's a larger one. Even those who say that only potentially infinite collections (and not actually infinite collections) are possible must admit the possibility of infinite regresses of this numerical kind. Is the positing of an infinite regress a legitimate explanation in philosophy? I don't see why it couldn't be. It seems to me that the burden rests with whoever denies the acceptability of an infinite regress of explanations. Indeed, I think infinite regresses of explanations are unavoidable given some highly plausible assumptions.

The truth-table for the material conditional says that any material conditional with a false antecedent is true. If we construe the conditional you gave as a material conditional, then (because it has a false antecedent) it comes out true. But the material conditional doesn't come out necessarily true unless it's not just false but impossible that Clinton is Cambodian (or else it's necessarily true that Bush is Angolan) . The material conditional has the advantage of being tidy, and a true material conditional will never let you infer a falsehood from a truth. Still, for the reason you gave (and for other reasons too) many philosophers say that the material conditional does a bad job of translating the conditionals we assert in everyday language. You'll find lots more information in this excellent SEP entry .

Your example about the room temperature doesn't seem to support the idea that we need more than two truth-values, because you classify everyone's responses as true . Instead, the example raises the question of how to interpret the people in the room: as disagreeing with each other because they're making incompatible claims ("It's cold"; "It's not cold") or as only apparently disagreeing with each other because they're making compatible claims ("It feels cold to me"; "OK, but it doesn't feel cold to me "). Standard logic (often called "classical" logic) has just two truth-values. Many-valued logics are nonstandard logics that contain anywhere from three to infinitely many truth-values -- in the latter case, all of the real numbers in the closed interval [0,1], with '0' for 'completely false' and '1' for 'completely true'. You'll find lots of detailed information in this SEP entry .

You asked, "How can 1 and 2 be true, and 3 be false?" Suppose that John is divorced and not remarried; he'd be unmarried but not a bachelor. You can patch up the argument by changing (1) to (1*) "If John is a bachelor, he is a never-married man" and changing (2) to (2*) "John is a never-married man." The argument still wouldn't be formally valid, which is the sense of "valid" that Prof. George uses in his reply. But it would be valid in that the premises couldn't be true unless the conclusion were true, because (2*) by itself implies that John is a bachelor. An argument that isn't formally valid -- i.e., an argument whose form alone doesn't guarantee its validity -- can be valid in the sense that the truth of its premises guarantees the truth of its conclusion. The last sentence of Prof. George's reply suggests that definitions are crucial in enabling conclusions to follow from premises. I think that suggestion is true only if logical implication is a relation holding between items of...