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 ...

Is it true that anything can be concluded from a contradiction? Can you explain? It's seems like its a tautology if taken figuratively because we can indeed conclude anything if we suspend the rules of reasoning, but there is nothing especially interesting in that fact in my humble opinion.

The topic is controversial (as I indicate below), but the inference rules of standard logic do allow you to derive any conclusion at all from any (formally) contradictory premise. Here's one way (let P and Q be any propositions at all): 1. P & Not-P [Premise: formal contradiction] 2. Therefore: P [From 1, by conjunction elimination] 3. Therefore: P or Q [From 2, by disjunction introduction] 4. Therefore: Not-P [From 1, by conjunction elimination] 5. Therefore: Q [From 3, 4, by disjunctive syllogism] Those who object to such derivations usually call themselves "paraconsistent" logicians; more at this SEP entry . They typically reject step 5 on the grounds that disjunctive syllogism "breaks down" in the presence of contradictions. I confess I've never found their line persuasive.

@William Rapaport: Unless disjunctive syllogism or one of the other two rules used in the derivation fails, the "irrelevance" of the conclusion to the premise is irrelevant to whether the conclusion follows from the premise. Relevance logic has to give up at least one of those rules, none of which is easy to give up.

Is there a way to prove that logic works? It seems that the only two methods for doing this would be to use a logical proof –which would be incorporating an assumed answer into the question– or to use some system other than logic –thus proving that sometimes logic does not work.

Even asking "Is there a way to prove that logic works?" presupposes that logic does work at least at the level of its most basic laws, such as the Law of Noncontradiction, because the question itself has meaning only if the most basic laws of logic hold. To put it a bit differently: No sense at all can be attached to the notion that logic doesn't work (or even sometimes doesn't work). See also my reply to Question 4837 and Question 4884 . So we have what philosophers call a "transcendental" proof of the reliability of logic: If we can so much as ask whether logic is reliable (provably or otherwise), then it follows that the answer to our question is yes . You might say that this proof won't impress someone who doubts the most basic laws of logic in the first place. But I'd reply -- predictably -- that no sense can be attached to the notion of doubting the most basic laws of logic.

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