Hilbert-style formalism for two-dimensional notions of consequence

Abstract

The present work proposes a two-dimensional Hilbert-style deductive formalism (H-formalism) for B-consequence relations, a class of two-dimensional logics that generalize the usual (Tarskian, one-dimensional) notions of logic. We argue that the two-dimensional environment is appropriate to the study of bilateralism in logic, by allowing the primitive judgments of assertion and denial (or, as we prefer, the cognitive attitudes of acceptance and rejection) to act on independent but interacting dimensions in determining what-follows-from-what. In this perspective, our proposed formalism constitutes an inferential apparatus for reasoning over bilateralist judgments. After a thorough description of the inner workings of the proposed proof formalism, which is inspired by the one-dimensional symmetrical Hilbert-style systems, we provide a proof-search algorithm for finite analytic systems that runs in at most exponential time, in general, and in polynomial time when only rules having at most one formula in the succedent are present in the concerned system. We delve then into the area of two-dimensional non-deterministic semantics via matrix structures containing two sets of distinguished truthvalues, one qualifying some truth-values as accepted and the other as rejected, constituting a semantical path for bilateralism in the two-dimensional environment. We present an algorithm for producing analytic two-dimensional Hilbert-style systems for sufficiently expressive two-dimensional matrices, as well as some streamlining procedures that allow to considerably reduce the size and complexity of the resulting calculi. For finite matrices, we should point out that the procedure results in finite systems. In the end, as a case study, we investigate the logic of formal inconsistency called mCi with respect to its axiomatizability in terms of Hilbert-style systems. We prove that there is no finite one-dimensional Hilbert-style axiomatization for this logic, but that it inhabits a two-dimensional consequence relation that is finitely axiomatizable by a finite two-dimensional Hilbert-style system. The existence of such system follows directly from the proposed axiomatization procedure, in view of the sufficiently expressive 5-valued non-deterministic bidimensional semantics available for the mentioned two-dimensional consequence relation.