The generalized modus ponens considered as a fuzzy relation

Abstract

The generalized modus ponens (GMP) is an inference rule in fuzzy logic, with the following scheme: if S 1 is P 1 then S 2 is P 2, but S 1 is Q 1, hence S 2 is Q 2. The objective is to determine Q 2, given P 1 P 2 and Q 1. Thepredicate P 1 in the antecedent of the if-then rule is fuzzy, hence the condition S 1 is P 1 to derive S 2 is P 2 is not strict. Therefore, it should be possible to render a reasonable approximation of Q 2 when Q 1 diverges slightly from P 1. A number of criteria for this inference rule will be presented. In literature there are three basic assumptions for dealing with the GMP, namely: (1) the if-then rule is implemented by an implication rule from some multivalued logic, i.e. the if-then rule is represented by a fuzzy relation, (2) the premiss P 1 can be strengthened or weakened to obtain Q 1 by taking a power of the membership function of P 1, and (3) the conclusion Q 2 is derived by means of the max-min composition rule. It will be shown by a number of theorems that these three items are incompatible with the criteria. Furthermore, it will be shown that it is arduous to construct an implication rule to meet the conditions (2) and (3). A way to restore this is to alter condition (2). Instead of taking a power of the membership function, it is also possible to change the parameters of this function. But this also causes complications. Hence, requirement (3) should be modified into some functional relation. It will be shown how this can be accomplished in combination with the functions with parameters changed.