A logical approach to interpolation based on similarity relations

Abstract

One of the possible semantics of fuzzy sets is in terms of similarity; namely, a grade of membership of an item in a fuzzy set can be viewed as the degree of resemblance between this item and prototypes of the fuzzy set. In such a framework, an interesting question is how to devise a logic of similarity, where inference rules can account for the proximity between interpretations. The aim is to capture the notion of interpolation inside a logical setting. In this paper, we investigate how a logic of similarity dedicated to interpolation can be defined, by considering different natural consequence relations induced by the presence of a similarity relation on the set of interpretations. These consequence relations are axiomatically characterized in a way that parallels the characterization of nonmonotonic consequence relationships. It is shown how to reconstruct the similarity relation underlying a given family of consequence relations that obey the axioms. Our approach strikingly differs from the logics of indiscernibility, such as the rough-set logics, because emphasis is put on interpolation capabilities. Potential applications are fuzzy rule-based systems and fuzzy case-based reasoning, where notions of similarity play a crucial role.