Adding truth-constants to logics of continuous t-norms: Axiomatization and completeness results

Authors:
Francesc Esteva;Joan Gispert;Lluís Godo;Carles Noguera
Affiliations:
Artificial Intelligence Research Institute (IIIA), Spanish National Research Council (CSIC), Spain;Dept Prob. Logic and Statitistics, University of Barcelona, Spain;Artificial Intelligence Research Institute (IIIA), Spanish National Research Council (CSIC), Spain;Artificial Intelligence Research Institute (IIIA), Spanish National Research Council (CSIC), Spain
Venue:
Fuzzy Sets and Systems
Year:
2007

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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 Godel and Product logics. Indeed, we consider algebraic semantics for expansions of logics of continuous t-norms with a set of truth-constants {r@?|r@?C}, for a suitable countable C@?[0,1], and provide a full description of completeness results when (i) the t-norm is a finite ordinal sum of Lukasiewicz, Godel 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.