The compositional rule of inference in an intuitionistic fuzzy logic setting

The incorporation of imprecise, linguistic information into logical deduction processes, as opposed to the practice of traditional two-valued propositional logic and set theory, continues to be a predominant feature of fuzzy expert systems. Throughout the literature, we can find all sorts of intelligent inference schemes acting under imprecision; common to most approaches is their reliance on if-then rules of the kind "IF X is A THEN Y is B", where A and B are fuzzy sets in given universes U and V. Intuitively, fuzzy sets (FSs) can be used to model elastic constraints on the values a variable may assume. While the theory of FS-based approximate reasoning is surely a well-established and commonly applied one, there is still a demand for further expanding the expressiveness of the formalism. One such improvement can be obtained by using Atanassov's intuitionistic fuzzy sets (IFSs), of which FSs are specific instances, and which highlight the fundamental importance of negation: the degree to which a proposition is false, or equivalently to which an object does not belong to a set, is given an independent status here. In this paper we will contribute to the further development of this relatively young theory, by generalizing the well-known Compositional Rule of Inference (CRI) to IFSs. We also deal with the related problem of checking the validity of the inference, as motivated in "K. T. Atanassov and G. Gargov, Elements of intuitionistic fuzzy logic. Part I, Fuzzy Sets and Systems, vol. 95 (1998), 39-52"