On non-contradictory input/output couples in Zadeh's CRI

Abstract

Deals with the logically unpleasant problem of reaching outputs that are contradictory with the corresponding inputs, using the compositional rule of inference (CRI). This is a situation that is common in some applications; for example, when Mandani's well-known conditional is used. For the goal of presenting some criteria to ensure that non-contradictory outputs are obtained effectively, two definitions of contradiction in fuzzy set theory are introduced and studied, as well as its relationship with the concept of incompatibility. These are two concepts that are equivalent within crisp sets but not always within fuzzy sets. For example, in classical set theory, there is only one self-contradictory object, namely the empty set, but this is not the case for any theory [F(E),N,T,S] of fuzzy sets. It is shown that, under some conditions on the values of the fuzzy conditional relation, the output given by the CRI is not contradictory with the input, provided that this is a normal fuzzy set. In particular, the case in which the conditional relation is a T-fuzzy pre-order is considered. Each section of the paper contains elementary examples.