I.: Constrained consequence

Abstract. There are various contexts in which it is not pertinent to generate and attend to all the classical consequences of a given premiss — or to trace all the premisses which classically entail a given con-sequence. Such contexts may involve limited resources of an agent or inferential engine, contextual relevance or irrelevance of certain conse-quences or premisses, modelling everyday human reasoning, the search for plausible abduced hypotheses or potential causes, etc. In this paper we propose and explicate one formal framework for a whole spectrum of consequence relations, flexible enough to be tailored for choices from a variety of contexts. We do so by investigating semantic constraints on classical entailment which give rise to a family of infra-classical logics with appealing properties. More specifically, our infra-classical reason-ing demands (beyond α | = β) that Mod(β) does not run wild, but lies within the scope (whatever that may mean in some specific context) of Mod(α), and which can be described by a sentence •α with β | = •α. Besides being infra-classical, the resulting logic is also non-monotonic and allows for non-trivial reasoning in the presence of inconsistencies