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Contrapositive symmetry of fuzzy implications. (English) Zbl 0845.03007
Summary: 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.

MSC:
03B52 Fuzzy logic; logic of vagueness
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