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

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 r1: “ifA, then B” and r2: “if B, then C”is not by the usual max-min composition rule, but by the following min-max rule; τij = min {max(μik, νkj): all j}, where τij = mA(χi) → mc(zj), μik = mA(χi) → mb(yk), and vkj = mB(yk) → mc(zj). The min-max value gives an upper bound on τik. 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.