PRESERVATION THEOREMS IN LUKASIEWICZ MODEL THEORY

Abstract

We present some model theoretic results for Lukasiewicz predicate logic by using the methods of continuous model theory developed by Chang and Keisler. We prove compactness theorem with respect to the class of all structures taking values in the Lukasiewicz BL-algebra. We also prove some appropriate preservation theorems concerning universal and inductive theories. Finally, Skolemization and Morleyization in this framework are discussed and some natural examples of fuzzy theories are presented. ajek and Cintula wrote: \In the last few decades many formal systems of fuzzy logics have been developed. Since the main dierences between fuzzy and classical logics lie at propositional level, the fuzzy predicate logics have developed more slowly (compared to the propositional ones). In this paper we aim to promote interest in fuzzy predicate logics by contributing to the model theory of fuzzy predicate In that paper the authors proved a generalized completeness result for a wide class of fuzzy predicate logics, namely core fuzzy logics. To do this they used a generalized Henkin method for a through survey of the present status of fuzzy predicate logic, see (3). In the present paper we continue the work on fuzzy predicate logics by developing further model theory of these logics. We choose Lukasiewicz predicate logic. The reason is that the Lukasiewicz BL-algebra is equipped with continuous operations and this is essential in proving the compactness theorem which is a fundamental result in model theory. Here, the compactness theorem says that if each nite subset of a set of sentences has a Lukasiewicz model, then the set itself has a Lukasiewicz model. As consequences of compactness, we prove the most basic model theoretic results including the upward and downward Lowenheim-Skolem theorems. We also prove suitable versions of the most well known preservation theorems, including preservation theorems for universal and inductive theories. At the end, we introduce some fuzzy theories and study their model theoretic properties. To prove the compactness theorem, we use ultraproduct and its basic properties which we rst investigate. We follow the old methods developed by Chang and