Is there a way to prove that logic works? It seems that the only two methods for
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.