Consistent Theories in Inconsistent Logics

Abstract

The relationship between logics with sets of theorems including contradictions (“inconsistent logics”) and theories closed under such logics is investigated. It is noted that if we take “theories” to be defined in terms of deductive closure understood in a way somewhat different from the standard, Tarskian, one, inconsistent logics can have consistent theories. That is, we can find some sets of formulas the closure of which under some inconsistent logic need not contain any contradictions. We prove this in a general setting for a family of relevant connexive logics, extract the essential features of the proof in order to obtain a sufficient condition for the consistency of a theory in arbitrary logics, and finally consider some concrete examples of consistent mathematical theories in Abelian logic. The upshot is that on this way of understanding deductive closure, common to relevant logics, there is a rich and interesting kind of interaction between inconsistent logics and their theories. We argue that this suggests an important avenue for investigation of inconsistent logics, from both a technical and a philosophical perspective.