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 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; tau(ij) = min {max(mu(ik), nu(kj)): all j}, where tau(ij) = m(A)(x(i)) -->m(c)(z(j)), mu(ik) = m(A)(x(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 tau(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 (not equal "<->") fuzzy equivalence relationships. (C) 1998 Elsevier Science B.V.