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 .

Is philosophy about the world or is it just about our concepts and the way we use them? Or both?

I agree: both. There seems to me to be a false dichotomy between "the world" and "our concepts and the way we use them": our concepts and the way we use them are surely part of the world.

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...

I've recently read that some mathematician's believe that there are "no necessary truths" in mathematics. Is this true? And if it is, what implications would it have on deductive logic, it being the case that deductive logical forms depend on mathematical arguments to some degree. Would in this case, mathematical truths be "contingently-necessary"?

Your question is tantalizing. I wish it had included a citation to mathematicians who say what you report them as saying. On the face of it, their claim looks implausible. Are there no necessary truths at all? If there are necessary truths, how could the mathematical truth that 1 = 1 not be among them? One way to hold that mathematicians seek only contingent truths might be as follows. If some philosophers are correct that propositions are to be identified with sets of possible worlds, then there's only one necessarily true proposition, because there's only one set whose members are all the possible worlds there are. That single necessarily true proposition (call it "T") will be expressed by indefinitely many different sentences , including the sentences "1 = 1" and "No red things are colorless," and it will be contingent just which sentences express T. On this view, mathematicians don't try to discover various necessary truths, since there's just one necessary truth, T. ...

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 ...