Pertinent Reasoning

In this paper we venture beyond one of the fundamental assumptions in the non-monotonic reasoning commu-nity, namely that non-monotonic entailment is supra-classical. We investigate reasoning which uses an infra-classical entailment relation that we call pertinent en-tailment. The notion of pertinence proposed here is induced by a binary accessibility relation on worlds establishing a link (representing some form of perti-nence) between premiss and consequence. We show that this notion can be captured elegantly using a sim-ple modal logic without nested modalities. One road to infra-classicality has been studied extensively, that of substructural logics, which weaken the generating en-gine of axioms and inference rules for producing en-tailment pairs (X,Y). Here we follow an alternative strategy: we first demand that X entails Y classically, and then, with supplementary information provided by an accessibility relation, more, trimming down the set of entailment pairs to infra-classicality. It turns out that pertinent entailment restricts well-known ‘paradoxes’ avoided by relevance/relevant logic in an interesting way. We present its properties, showing that it pos-sesses other non-classical properties, like strong non-explosiveness and non-monotonicity, and we discuss which inference rules traditionally considered in the lit-erature it satisfies