Reply #124: I want a logic which gives sensible and useful results and which is in some sense "realistic."

[struggle4progress]

As an example of something I don't always consider realistic, I would point to the "all possible universes" explication of "necessity" -- which would be the standard way to interpret your claim "You can only prove that something exists by contradiction when that entity's existence is a necessity" as meaning something like "You can only prove that something exists by contradiction when that entity exists in every possible universe. But such an explication hardly ever makes sense, because usually one has no idea how to examine the purported possible universes.

If one is a Platonist and thinks that every idea reflects some real fact, then there is nothing essentially problematic with "correct" proofs in the math libraries. But even if one is not a Platonist, some of the proofs appear to be telling us something, because some certainly lead to computational rules that are scientifically useful. So some of the arguments seem to involve reasoning techniques that are applicable to real things. Some of the arguments, on the other hand, appear to be perfectly decent arguments, except for fact that there seems no chance of deriving any useful computation from them.

I should avoid attempting much beyond such remarks, since your question suggests that I should define what is "real" -- an exercise almost everyone eventually abandons in favor of individual pragmatic compromises involving experiences, prior beliefs and thought processes, and some (but not all) of what various other people say.