Classical Logic, Uniformity, and Weak Excluded Middle in Non-Monotonic Proof-Theoretic Semantics

Abstract

Non-monotonic base-extension semantics (nB-eS), a kind of non-monotonic proof-theoretic semantics (nPTS), is known to validate classical logic when its meta-logic is classical. Schroeder-Heister has remarked that classical meta-logic is as problematic for the project of modelling intuitionistic logic, as an intuitionistic proof of incompleteness would be. It may be unclear, though, whether Schroeder-Heister’s remark holds for non-monotonic proof-theoretic validity (nP-tV) as well, i.e., for Prawitz’s original version of nPTS. We only know that, with classical meta-logic again, classical logic is sound over a variant of nP-tV, which I shall call liberal non-monotonic proof-theoretic validity (LnP-tV). The latter, in turn, differs from nP-tV in that reductions for the rewriting of proof-structures are not required to be uniform. After drawing attention to a number of divergences between nB-eS, nP-tV and LnP-tV, I show that Schroeder-Heister’s remark might after all apply to nP-tV too. In particular, Weak Excluded Middle (WEM) is logically valid via uniform reductions (with a meta-logic which is non-intuitionistic, but non-classical either).