Residuated fuzzy logics with an involutive negation

Abstract

Residuated fuzzy logic calculi are related to continuous t-norms, which are used as truth functions for conjunction, and their residua as truth functions for implication. In these logics, a negation is also definable from the implication and the truth constant 0̄, namely ¬φ is φ → 0̄. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to LŁukasiewicz t-norm), it turns out that ¬ is an involutive negation. However, for t-norms without non-trivial zero divisors, ¬ is Gödel negation. In this paper we investigate the residuated fuzzy logics arising from continuous t-norms without non-trivial zero divisors and extended with an involutive negation.