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Adding truth-constants to logics of continuous t-norms: axiomatization and completeness results. (English) Zbl 1117.03030
Summary: We study generic expansions of logics of continuous t-norms with truth-constants, taking advantage of previous results for \({\L}\)ukasiewicz logic and more recent results for Gödel logic and product logic. Indeed, we consider algebraic semantics for expansions of logics of continuous t-norms with a set of truth-constants \(\{\bar r\mid r \in C\}\), 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 Łukasiewicz, 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 Łukasiewicz components behave like rational numbers.

MSC:
03B52 Fuzzy logic; logic of vagueness
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