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Residuated fuzzy logics with an involutive negation

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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 \(\overline{0}\), namely \(\neg \varphi\) is $\varphi \to \overline{0}$. However, this negation behaves quite differently depending on the t-norm. For a nilpotent t-norm (a t-norm which is isomorphic to Łukasiewicz t-norm), it turns out that \(\neg\) is an involutive negation. However, for t-norms without non-trivial zero divisors, \(\neg\) 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.

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Received: 14 April 1998

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Esteva, F., Godo, L., Hájek, P. et al. Residuated fuzzy logics with an involutive negation. Arch Math Logic 39, 103–124 (2000). https://doi.org/10.1007/s001530050006

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  • DOI: https://doi.org/10.1007/s001530050006

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