Adding truth-constants to logics of continuous t-norms
Authors: |
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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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \{ \overline{r} \mid r \in C \}} , for a suitable countable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle 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.