Residuated fuzzy logics with an involutive negation

• DOI: 10.1007/s001530050006
• Publication Date: 2003-11-24T14:09:53Z
• Authors: Francesc Esteva; Lluís Godo; Petr Hájek; Mirko Navara
• Keywords: Negation; T-norm; Residuated lattice; Negation as failure; T-norm fuzzy logics; Zero (linguistics); Truth function

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.<br/>Peer Reviewed<br/>