My interpretation of this question is metaphysical: 'Is reality classical?' This is shorthand for: 'Is there any fact that is fundamentally, objectively, and in principle...

... inexpressible? ... vague? ... contradictory? ... indeterminate? (I.e., neither the case nor not-the-case.) ... etc.

But this is a strange set of questions, and I suspect most professional philosophers and LessWrongers are instead answering a confused mixture of (mostly trivial) questions like 'which logic do I find most useful?'.