A Bit More About Fuzzy Modus Ponens

Abstract

Since Lotfi A. Zadeh formulated the compositional rule of inference so called “extension and projection,” various kinds of fuzzy reasoning methods have been proposed with their mathematical properties. Mizumoto studied about many cases based on Max-⊗ composition. The cases lead to T-norm family of fuzzy reasoning methods. In the fuzzy modus ponens, ⊗ is used twice, when defining fuzzy implication and the process of composition. Under the normative criteria required for fuzzy modus ponens, we discuss which T-norms noted by ⊗ have good properties. In the usual modus ponens, when both “P implies Q” and the antecedent are valid, the consequent is right, but when the antecedent is false, we can say nothing about the truth of the conclusion. Also like the above case, unknown should be assigned to the conclusion inferred by fuzzy modus ponens, which is one of the five criteria.