Lindström theorems in graded model theory

• Published in: Annals of pure and applied logic Vol. 172; no. 3; p. 102916
• Main Authors: Badia, Guillermo, Noguera, Carles
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01-03-2021
• Subjects: Abstract model theory; Lindström theorem; Many-valued predicate logics; Mathematical fuzzy logic; 03C95; 03B52; Mathematical fuzzy logic; 03B50; Many-valued predicate logics; Lindström theorem; 03C90; Abstract model theory
• ISSN: 0168-0072
• DOI: 10.1016/j.apal.2020.102916

Stemming from the works of Petr Hájek on mathematical fuzzy logic, graded model theory has been developed by several authors in the last two decades as an extension of classical model theory that studies the semantics of many-valued predicate logics. In this paper we take the first steps towards an abstract formulation of this model theory. We give a general notion of abstract logic based on many-valued models and prove six Lindström-style characterizations of maximality of first-order logics in terms of metalogical properties such as compactness, abstract completeness, the Löwenheim–Skolem property, the Tarski union property, and the Robinson property, among others. As necessary technical restrictions, we assume that the models are valued on finite MTL-chains and the language has a constant for each truth-value.