A new class of fuzzy implications. Axioms of fuzzy implication revisited

Many diierent fuzzy implication operators have been proposed; most of them t into one of the two classes: implication operations that are based on an explicit representation of implication A ! B in terms of &, _, and : (e.g., S-implications that are based on the formula B _ :A), and R-implications that are based on an implicit representation of implication A ! B as the weakest C for which C&B implies A. However, some fuzzy implication operations (such as b a) cannot be naturally represented in this form. To describe such operations, we propose a new (third) class of implication operations called A-implications whose relation to &, _, and : is described by (implicit) axioms. 1. Formulation of the problem 1:1: It is necessary to represent implication in expert systems Our knowledge of complex systems is often incomplete, and therefore, we have to rely on the expert's statements, These statements are usually formulated not in mathematical terms, but in words of natural languages. Fuzzy logic is a methodology for formalizing such statements. Since a large part of expert knowledge consists of if{then statements, it is therefore important to formalize implication (i.e., if{then statements) in fuzzy logic. 1 1:2: How implication is represented now : two general ideas There has been a lot of research on representing &, _, and : in fuzzy logic. So, let us assume that we already have operations f & ; f _ : 0; 1] 0; 1] ! 0; 1] and f : : 0; 1] ! 0; 1] that correspond to &, _, and :. In this situation, there are two basic ways to describe an implication operation f !