n this paper the authors venture beyond one of the fundamental assumptions in the non-monotonic reasoning community, namely that non-monotonic entailment is supra-classical. They investigate reasoning which uses an infra-classical entailment relation that they call pertinent entailment. The notion of pertinence proposed here is induced by a binary accessibility relation on worlds establishing a link (representing some form of pertinence) between premiss and consequence. They show that this notion can be captured elegantly using a simple modal logic without nested modalities. One road to infra-classicality has been studied extensively, that of substructural logics, which weaken the generating engine of axioms and inference rules for producing entailment pairs (X; Y). Here they 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. They present its properties, showing that it possesses other non-classical properties, like strong non-explosiveness and non-monotonicity, and we discuss which inference rules traditionally considered in the literature it satisfies