An Introduction to Fuzzy Implications

Abstract

The implication operator (→) plays a significant role in the classical two-valued logic. Firstly, from the classical implication one can obtain all other basic logical connectives of the binary logic, viz., the binary operators -and (∧), or (∨) -and the unary negation operator (¬). Secondly, the implication operator holds the center stage in the inference mechanisms of any logic, like modus ponens, modus tollens, hypothetical syllogism in classical logic. The truth table for the classical implication is given in Table 1.1.

Two Approaches to Fuzzy Implication

“…Moreover, the operator N is a strong negation, because it is involutive: One can refer here to known facts from the theory of fuzzy implications (cf. [6,7,21]) and write the strong negation N in terms of the standard strong negation N I on the unit interval…”

“…The purpose of this chapter was to present new operators that satisfy the conditions for fuzzy implication in the classical sense [6]. These results emerged from the research conducted by Kosiński, undertaken in his last years of life.…”

Implication Functions in Interval-Valued Fuzzy Set Theory

“…Since implications on the unit interval play an important role in fuzzy set theory [9], several authors have extended this notion to interval-valued fuzzy set theory [4,6,7,8,11,17,26,48]. This chapter gives an overview of the results pertaining to implications in interval-valued fuzzy set theory.…”

Dependencies between fuzzy conjunctions and implications

“…After the contribution of B. Schweizer and A. Sklar [10] the notions of the triangular norm and conorm have played the role of a fuzzy conjunction and disjunction. J. Fodor and M. Roubens [8], M. Baczyński and B. Jayaram [1] examined families of multivalued connectives based on triangular norms and conorms. However, some authors (e.g.…”

“…Implications created in such a way have been considered in the literature in the case when the conjunction is a triangular norm by J. Fodor and M. Roubens [8], M. Baczyński and B. Jayaram [1] and are called R-implications. Also, the case when a conjunction is replaced by an appropriate uninorm was examined by B.…”