A logical approach to interpolation based on similarity relations

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 sets 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. 2