Adding truth-constants to logics of continuous t-norms

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Authors:
Francesc Esteva
Joan Gispert
Lluís Godo
Carles Noguera
Title: Adding truth-constants to logics of continuous t-norms: axiomatization and completeness results
Journal: Fuzzy Sets and Systems
Volume 158
Number 6
Pages: 597-618
Year: 2007
Preprint




Abstract

In this paper we study generic expansions of logics of continuous t-norms with truth-constants, taking advantage of previous results for Lukasiewicz logic and more recent results for Gödel and Product logics. Indeed, we consider algebraic semantics for expansions of logics of continuous t-norms with a set of truth-constants $ \{ \overline{r} \mid r \in C \} $, for a suitable countable $ C \subseteq [0, 1] $, and provide a full description of completeness results when (i) the t-norm is a finite ordinal sum of Lukasiewicz, Gödel and Product components, (ii) the set of truth-constants covers all the unit interval in the sense that each component of the t-norm contains at least one value of C different from the bounds of the component, and (iii) the truth-constants in Lukasiewicz components behave as rational numbers.