A Logical Study of Local and Global Graded Similarities

Abstract

Abstract. In this paper we study the relationship between global and local similarities in the graded framework of Fuzzy Class Theory (FCT) in which already exists a graded notion of similarity. In FCT we can express the fact that a fuzzy relation is reflexive, symmetric or transitive up to a certain degree, and similarity is defined as a first-order sentence which is the fusion of three sentences corresponding to the graded notions of reflexivity, symmetry and transitivity. This allows to speak in a natural way of the degree of similarity of a relation. We consider global similarities defined from local similarities using t-norms as aggregation operators and we obtain some results in the framework of FCT that adequately interpreted allow us to say that, when we take a t-norm as aggregation operator the properties of reflexivity, symmetry and transitivity of fuzzy binary relations at the global level inherit the properties of the fuzzy binary relations at the local level when we fusion them, and that the global similarity is a congruence if some of the local similarities are congruences.