FUZZY LOGICS AND THE GENERALIZED MODUS PONENS REVISITED

Abstract

After a background on the use of triangular norms in multiple-valued logics, a systematic investigation of implication functions based on triangular norms is carried out; five families of implication functions are considered. Then Zadeh's generalized modus ponens is thoroughly reconsidered in the framework of possibility theory. It is pointed out that the choice of the implication function for representing the rule “if X is A, then Y is B” is completely determined by the choice of the conjunctive operation used for aggregating an a priori possibility distribution with a conditional possibility distribution. The chaining of conditional rules is proved to be transitive when we choose a continuous triangular norm as conjunctive operation. The generalized modus tollens, the treatment of truth-qualified rules and the processing of a collection of conditional rules providing an incomplete description of a causal link, are studied in detail. Finally the generalized modus ponens, as a pattern of deductive reasoning taking into account the fuzziness of premises, is contrasted with patterns of plausible reasoning.