non-contradiction
Gone down a non-classical logic rabbit-hole. Here are two things I've come across:
(1) the standard logical system of Buddhist India was four-valued.
(2) there's this great paper by Łukasiewicz, the inventor of the first many-valued logic in the Western math-logic tradition, taking apart Aristotle's arguments for the law of non-contradiction (LNC). Some nice bits:
(2a) a cute theme that keeps popping up is that many arguments for the LNC are reductios, and thus assume the LNC.
(2b) Aristotle states the LNC not just as a logical claim (a statement and its negation can't both be true) but also as an ontological one (a thing can't both have and not have a property) and, most troublingly, a psychological one (a person can't believe two contradictory propositions). There's a great quote about this last part from Husserl:
In the same individual, or still better, in the same consciousness, contrary acts of believing could never persist during even the smallest interval of time. But is this really a law? May we really state it with unlimited generality? Where are the psychological inductions which justify its adoption? Might there not have been and might there not be men, who confused by fallacies for instance, occasionally held opposites to be true simultaneously? Has scientific research been conducted as to whether something like this does not occur among the insane and perhaps even in plain contradictions? How does the hypothesis fare with the conditions of fever delirium, etc.? Is the law also valid for animals?
(2c) My untutored belief about Aristotle was that he had this ridiculously huge influence on Western metaphysics by conceptualizing the world as made of objects which have essences which have attributes etc., the whole shebang drawing on the discrete and static over the continuous and changing. This ur-idea is at the heart of some of the big East-West contrasts, and worked its way into our logic and math as well. (Lakoff and Nuñez argue that the mathematical conceptualization of the line as a set of points, each of which corresponds to a real number, was another example of this "discretization" phenomenon.)
But it looks like he has interesting things to say about change as well! In fact, the whole category of "the potential" was developed for this purpose, and in Brother Łuke's interpretation, at least, the LNC needn't apply there. (Thus things are able to change, at least in appearance, from A to not-A.) The LNC is supposed to apply to actual things/essences/noumena, so figuring out how to feel about the LNC should be linked to figuring out how to feel about noumena.
(3) The weird thing about all this is that, as a mathematician, I use classical logic all the time, and without thinking about it. Even if I were to come down in favor of some transconsistent or relevant logic, I wouldn't have the faintest idea how to start using it in my mathematical life. Not to mention it would get me ostracized from the math community -- not in the sense that people would think I was a crank, though they might, but in the sense that I'd be unable to prove anything in homotopy theory because I'd have to draw on a century of classical-logic-using papers, and conversely, no non-logician would use my work in their own.
