Consistent Theories in Inconsistent Logics

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