No answer on the question of non-classical logics (i.e., not the Frege-Russell logic often taught by philosophers, mathematicians, computer scientists, etc). This one is tricky — there are literally hundreds of non-classical logics.
People sometimes wonder what principles one uses to literally change how one reasons. This is sometimes how the question is put, but that’s a bit misleading. Formal logic involves hypotheses about how we do and how we ought to reason. Frege and Russell had both noticed (as had others) that the tradition of the syllogism, from Aristotle’s time, did not represent many key inferences recognized by mathematicians.
Both thought that mathematics was as good a field (if any) to have come up with good deductive principles, so they set about to systematize these. This was notoriously hard to do. But it at least allows precedent — we changed our “official” logic once before, we can do it again.
The interesting question then is: Since mathematics is used (as Bunge puts it) for “deductive glue” in many contexts, does that doom us to use classical logic forever? To rewrite all of mathematics seems, well, utopian. Should we even try? (This has not stopped people from trying!)
On to this month’s conundrum. What do the following words have in common? Why?