On the chaining syllogism in fuzzy logic

In this paper we deal with an extension of the fuzzy chaining syllogism in fuzzy logic, which was initiated by Zadeh [L.A. Zadeh, Syllogistic reasoning in fuzzy logic and its application to reasoning with dispositions, IEEE Trans, on Systems, Man and Cybernetics, 15, 754-763, 1985]. We introduce the extended fuzzy chaining syllogism, and investigate it with three important classes of fuzzy implications against the validity of their deduction scheme in fuzzy logic. We show that the fuzzy chaining syllogism holds for an arbitrary fuzzy implication and a continuous extended triangular norm as long as the fuzzy syllogism holds. For an arbitrary triangular norm and an S- or QL-implication we prove that the generalised Kleene-Dienes implication constitutes a lower bound for the inference result of the chaining syllogism. Similarly for Zadeh's triangular norm (minimum) and an R-implication I we obtain this I as a lower bound for the conclusion of the chaining syllogism.