Saturated Models in Mathematical Fuzzy Logic

TL;DR

This paper develops a method for constructing saturated models in first-order graded logics, extending models to realize as many types as possible, and generalizes classical theorems to fuzzy logic contexts.

Contribution

It introduces a construction for saturated models in fuzzy logic and generalizes key classical model theory results to this setting.

Findings

Constructed saturated models via elementary chains

Proved a generalized Tarski–Vaught theorem for fuzzy logic

Analyzed tableaux consistency and satisfiability in fuzzy models

Abstract

This paper considers the problem of building saturated models for first-order graded logics. We define types as pairs of sets of formulas in one free variable which express properties that an element is expected, respectively, to satisfy and to falsify. We show, by means of an elementary chains construction, that each model can be elementarily extended to a saturated 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.