Adding truth-constants to logics of continuous t-norms

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.