Would someone please clarify the importance of the distinction between a) either A is true, or Not A is true and b) either A is true, or A is not true I've seen answers on this site in which the difference between those two formulations is very important, but I'm not quite sure why. Thank you.

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.

Classical logic says that from a contradiction you can derive anything. I think that depends on how you define a contradiction. If you have two opposing truth values with respect to A, A is true and A is false what can we infer about the truth status of A? Well in one way to look at it you could say that to assert a contradiction means we hold that both statements about A are true regardless of whether they contradict each other. A is true regardless of the contrary position that A is false. Likewise A is false regardless of the contrary position that A is true. If we define a contradiction in this manner then we can separately infer both truth values of A. Given A is true and A is false we can conclude A is true and given A is true and A is false we can conclude that A is false. If you infer A is true from the contradiction then A or B is true. If A or B is true then if A is false then B is true. A is true regardless of whether A is false therefor we can not conclude an explosion occurs. It seems that...

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.

For the given premises P and Not P, is P a valid derivation? Shouldn't the derivation be true for all the premises for it to be valid or is it not sound and yet valid? But aren't we determining its unsoundness by virtue of something other than the content of those premises?

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.

Can paradoxes actually happen?

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...

Is the positing of an infinite regress a legitimate explanation in philosophy respectively are infinite regresses logically possible?

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.

I am confused about how a conditional statement is necessarily true, and not false or unknown, when the antecedent and consequent are both false. According to the truth table, the sentence "If Bill Clinton is Cambodian, then George Bush is Angolan" is true. How can such an absurd sentence be true? It seems initially like the sentence could just as easily, or more easily, be false or unknown.

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 .

I am learning about the principle of noncontradiction ~(p^~p). I can see that this would work if we assume that 'p' can only be true or false. Why should I make this assumption. I can see a lot instances where we need more than 2 truth values (how people feel about the temperature of a room, for instance could have an infinite number of responses, and all would be true because the proposition is based on subjective experiences). What is this type of logic called? If this is a possible logic then can't someone argue that everything is this way?

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 .

I know affirming the consequent is a fallacy, so that any argument with that pattern is invalid. But what what about analytically true premises, or causal premises? Are these not really instances of the fallacy? They seem to take its form, but they don't seem wrong. For example: 1. If John is a bachelor, he is an unmarried man. 2. John’s an unmarried man. 3. Therefore he’s a bachelor. How can 1 and 2 be true, and 3 be false? Yet it looks like affirming the consequent. 1. X is needed to cause Y. 2. We’ve got Y. 3. Therefore there must have been X. Again, it seems like the truth of 1 and 2 guarantee the truth of 3. What am I missing?

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...

How, if at all, is the following paradox resolved? You hand someone a card. On one side is printed "The statement on the other side of this card is true." On the other side is printed, "The statement on the other side of this card is false." Thanks for consideration!

You've asked about one version of an ancient paradox called the "Liar paradox" or the "Epimenides paradox." One good place to start looking, then, is the SEP entry on the Liar paradox, available here . Philosophers are all over the map on how to solve paradoxes of this kind, and their proposed solutions are sometimes awfully complicated! Best of luck.

Here's a quote from Hume: "Nothing, that is distinctly conceivable, implies a contradiction." My question is this: what is the difference between something that is logically a contradiction and something that happens to not be instantiated? For example, ghosts do not exist. Could you explain how the concept of a ghost is not a contradiction? Thanks ^^

What is the difference between something that is logically a contradiction and something that happens to not be instantiated? As I think you already suspect, it's the difference between (1) a concept whose instantiation is contrary to the laws of logic or contrary to the logical relations that obtain among concepts; and (2) a concept whose instantiation isn't contrary to logic but only contrary to fact. Examples of (1) include the concepts colorless red object and quadrilateral triangle . Examples of (2) include the concept child of Elizabeth I of England . Concepts of type (1) are unsatisfiable in the strongest sense; concepts of type (2) are merely unsatisfied. Could you explain how the concept of a ghost is not a contradiction? Good question. I'm not sure the concept isn't internally contradictory. Can ghosts, by their very nature, interact with matter? Some stories seem to want to answer yes and no . If I recall correctly (it's been a while) the movie ...