First-order t-norm based fuzzy logics with truth-constants: distinguished semantics and completeness properties

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Authors:
Francesc Esteva
Lluís Godo
Carles Noguera
Title: First-order t-norm based fuzzy logics with truth-constants: distinguished semantics and completeness properties
Journal: Annals of Pure and Applied Logic
Volume 161
Number 2
Pages: 185-202
Year: 2009
Preprint




Abstract

This paper aims at being a systematic investigation of different completeness properties of first-order predicate logics with truth-constants based on a large class of left-continuous t-norms (mainly continuous and weak nilpotent minimum t-norms). We consider standard semantics over the real unit interval but also we explore alternative semantics based on the rational unit interval and on finite chains. We prove that expansions with truth-constants are conservative and we study their real, rational and finite chain completeness properties. Particularly interesting is the case of considering canonical real and rational semantics provided by the algebras where the truth-constants are interpreted as the numbers they actually name. Finally, we study completeness properties restricted to evaluated formulae of the kind $ \overline{r} \to \varphi $, where $ \varphi $ has not additional truth-constants.