Model-Theoretic Inferentialism and Paraconsistency in Multiple-valued Structures

Abstract

Carnap [5] in 1943 noticed that there exist some non-normal valuations that do not respect the usual truth conditions of classical connectives, yet are sound. This finding reveals an asymmetry between the syntax and semantics of classical logic and might be counted as a challenge to the inferentialist program. While several solutions have been proposed for classical logic, the problem persists for some well-known paraconsistent logics and none of the existing solutions work properly. To address this issue, we propose a method to read off [model-theoretic] semantics from the deductive behavior of these logics in a uniform way. We examine this method for three paraconsistent logics chosen from different traditions: LP, J3 and mbC. We aim to show that the syntax-first approach of inferentialism to logic can still be maintained for these paraconsistent logics, without losing any semantics.