A paraconsistent and substructural conditional logic

Abstract

I introduce and motivate a conditional logic based on the substructural system HL from Paoli (Substructural logics: a primer, Kluwer, Dordrecht, 2002). Its hallmark is the presence of three logical levels (each one of which contains its own conditional connective), linked to one another by means of appropriate distribution principles. Such a theory brings about a twofold benefit: on the one hand, it yields a new classification of conditionals where the traditional dichotomies (indicative vs subjunctive, factual vs counterfactual) do not play a decisive role; on the other hand, it allows to retain suitable versions of both substitution of provable equivalents and simplification of disjunctive antecedents, while still keeping out such debatable principles as transitivity, monotonicity, and contraposition.