An Inferentialist Interpretation of Classical Logic

Abstract: The aim of this work is to provide an account of the meanings of the standard logical operators meeting the following requirements:1) It should meet Dummettian standards of intelligibility; specifically, it should do without any notion of recognition-transcendent truth.2) It should validate Classical logic.Two theories – essentially variations on a common theme – are offered, the one aiming for formal simplicity, the other aspiring to slightly greater philosophical sophistication.In both theories, the basic idea – adopted from writers such as Prawitz and Tennant – is to regard inferential relations among logically complex sentences as supervenient on, and motivated by, a base of conceptually and/or empirically grounded inferential relations among the logical atoms. In the simple version, such a base is taken to be any (single-conclusion) consequence relation closed under Cut and the other standard structural rules. The more complex theory takes bases to be finite sets of inference rules for atomic sentences, notably including some rules embodying disjunctive judgments. The object-language syntax is quite liberal, capable of accommodating, but not limited to, second-order logic.Surprisingly, in both of its variants, this Intuitionistically flavoured semantics turns out, by constructively valid meta-reasoning, to validate Excluded Middle in the object language – despite failure, from the semantic perspective, of Bivalence. That is to say, a base neither supporting A nor not-A will nevertheless support A-or-not-A.For the simpler semantics, Classical logic is shown to be both sound and complete. By contrast, the more complex version turns out to render valid, in addition to Classical logic, a limited, predicative principle of Comprehension, as well as the Axiom of Choice.