Abstract

In Orlowska and Weingartner an approach to relevance was proposed based on the notion of sematic relevance criterion which is understood as a binary relation in the set of predicates of a language under consideration. Some general properties were investigated of sets of formulas relevant with respect to relevance criteria or sets of relevance criteria. Informally, a predicate P is relevant for a predicate Q whenever some information about instances of Q can be derived from information about instances of P. Relevance criteria given by means of binary relations reflect semantic relationship between predicates. We assume that relevance criteria are reflexive, that is every predicate is relevant for itself. In the present note we propose to consider propositional operations of implication which are relative to a semantic relevance criterion. The intuition for relevant implication is, that a necessary condition for the validity of an inference from F to G is that F be relevant to G. We define a logic with implications → , for a relevance criterion R, reflecting this intuition. We treat relevance criterion in the set of predicates as a primary notion, and next, we define its extension to arbitrary formulas. Our view is, that relevance relation between predicates reflects a background knowledge of a user of the language. This knowledge is relative to an investigator and to a theory within which he makes inferences, and in general it cannot be expressed explicitly as a formula of the language. Our semantics for relevant implication includes both validity of inference and its relativity with respect to some background information about relevant predicates