Saturated models of first-order many-valued logics

• Published in: Logic journal of the IGPL Vol. 30; no. 1; pp. 1 - 20
• Main Authors: Badia, Guillermo, Noguera, Carles
• Format: Journal Article
• Language: English
• Published: Oxford University Press 01-02-2022
• Subjects: residuated lattices; uninorms; types; first-order graded logics; logic UL; Mathematical fuzzy logic; elementary chains; saturated models
• ISSN: 1367-0751; 1368-9894
• DOI: 10.1093/jigpal/jzaa027

Abstract
This paper is devoted to the problem of existence of saturated models for first-order many-valued logics. We consider a general notion of type as pairs of sets of formulas in one free variable that express properties that an element of a model should, respectively, satisfy and falsify. By means of an elementary chains construction, we prove that each model can be elementarily extended to a $\kappa $-saturated model, i.e. a model where as many types as possible are realized. In order to prove this theorem we obtain, as by-products, some results on tableaux (understood as pairs of sets of formulas) and their consistency and satisfiability and a generalization of the Tarski–Vaught theorem on unions of elementary chains. Finally, we provide a structural characterization of $\kappa $-saturation in terms of the completion of a diagram representing a certain configuration of models and mappings.