Classical recapture and maximality

Abstract

The idea of classical recapture has played a prominent role for non-classical logicians. In the specific case of non-classical theories of truth, although we know that it is not possible to retain classical logic for every statement involving the truth predicate, it is clear that for many such statements this is in principle feasible, and even desirable. What is not entirely obvious or well-known is how far this idea can be pushed. Can the non-classical theorist retain classical logic for every non-paradoxical statement? If not, is she forced to settle for a very weak form of Classical Recapture, or are there robust versions of classical recapture available to her? These are the main questions that I will address in this paper. As a test case I will consider a paracomplete account of the truth-theoretic paradoxes and I will argue for two claims. First, that it is not possible to retain the law of excluded middle for every non-paradoxical statement. Secondly, that there are no robust versions of classical recapture available to the paracomplete logician.