CONTRAPOSITIVE SYMMETRY OF FUZZY IMPLICATIONS

Contrapositive symmetry of R- and QL-implications defined from t-norms, t-conorms and strong negations is studied. For R-implications, characterizations of contrapositive symmetry are proved when the underlying t-norm satisfies a residuation condition. Contrapositive symmetrization of R-implications not having this property makes it possible to define a conjunction so that the residuation principle is preserved. Cases when this associated conjunction is a t-norm are characterized. As a consequence, a new family of t-norms (called nilpotent minimum) owing several attractive properties is discovered. Concerning QL-implications, contrapositive symmetry is characterized by solving a functional equation. When the underlying t-conorm is continuous and the t-norm is Archimedean, the t-conorm must be isomorphic to the Lukasiewicz one, while the t-norm must be isomorphic to a member from the well-known Frank family of t-norms. Finally, contrapositive symmetry for some new families of fuzzy implications is investigated.