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

A new (third) class of implication operations are proposed called A-implications whose relation to &, ∨, and ¬ is described by (implicit) axioms.

Abstract

Many different fuzzy implication operators have been proposed; most of them fit 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 ba) 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.