Logic

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.

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It looks to me as if your question is a version of what epistemologists have come to know as "the problem of the criterion"--in this case, with respect to logic. There have come to be three different ways of responding to this problem (which one could also apply to any other sources of information or reasoning, such as sense perception, memory, or induction): reductionism, which provides evidence for the reliability of the source by getting confirmation for that source from some other source (this is more or less the second option you provide in your question, though I am not sure why that seems to you to show that "sometimes logic does not work"); dogmatism, which essentially says that we can be justified in accepting individual samples from a given source without having any (prior) justification for thinking that the source is reliable (this is more or less the first option you present in your question; and holistic coherence theory, which claims that our justification for thinking that the source is reliable ("woks") happens at the same time and for the same reason as our justification for thinking that samples of the source are reliable, and that is when and because we achieve a kind of "coherence" in our thinking about samples and the source itself. My own view is that reductionism simply "kicks the can down the road" by raising the problem of the criterion for the other source appealed to, and that dogmatism allows justification of samples to come too easily (and is thus not plausible).

There are a number of places you can find out more about the problem of the criterion. I did a quick search of the Stanford Encyclopedia of Philosophy and couldn't find a discussion, but there is one on the Internet Encyclopedia of Philosophy. My co-author, Ian Evans and I offer a defense of the holistic coherence response to this problem in our recent book (Knowledge) with Polity Press.

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.

Aristotle gives a nice account of why we must have something "definite in our thinking" and not contradictions in Metaphysics IV. In order to say of something that it is or can be both F and not-F, he writes, we must have successfully identified that thing as the thing that is or can be both F and not-F. But we are in no position to do that if the something both is and is not the something we are talking about, or trying to talk about! So we do not have to abandon the piece of logic, the principle of non-contradiction, in one form, at least, which states that opposite things cannot significantly be said of the same thing. Here, at least, it seems that logic does not break down on the basis of the interesting argument that you gave.

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