The min-max composition rule and its superiority over the usual max-min composition rule

Abstract

A close analysis of the Syllogism inference rule shows that if one uses Zadeh's notion of fuzzy if-then, then the proper way of combining the membership values of two fuzzy rules r 1: “if A, then B” and r 2: “if B, then C”is not by the usual max-min composition rule, but by the following min-max rule; τ ij = min {max( μ ik , ν kj ): all j}, where τ ij = m A ( χ i ) → m c ( z j ), μ ik = m A ( χ i ) → m b ( y k ), and v kj = m B ( y k ) → m c ( z j ). The min-max value gives an upper bound on τ ik . The min-max rule results in a new notion of transitivity and a corresponding notion of a fuzzy equivalence relation. We demonstrate the superiority of the min-max rule in terms of the properties of this equivalence relation. In particular, we argue that the new form of transitivity is particularly suitable for studying non-logical ( ≠ “↔”) fuzzy equivalence relationships.