Saturated Models in Mathematical Fuzzy Logic* Guillermo Badia Carles Noguera Department of Knowledge-Based Mathematical Systems Institute of Information Theory and Automation Johannes Kepler University Linz Czech Academy of Sciences Linz, Austria Prague, Czech Republic
[email protected] [email protected] Abstract—This paper considers the problem of building satu- Another important item in the classical agenda is that of rated models for first-order graded logics. We define types as saturated models, that is, the construction of structures rich in pairs of sets of formulas in one free variable which express elements satisfying many expressible properties. In continuous properties that an element is expected, respectively, to satisfy and to falsify. We show, by means of an elementary chains model theory the construction of such models is well known construction, that each model can be elementarily extended to a (cf. [1]). However, the problem has not yet been addressed in saturated model where as many types as possible are realized. mathematical fuzzy logic, but only formulated in [15], where In order to prove this theorem we obtain, as by-products, some Dellunde suggested that saturated models of fuzzy logics could results on tableaux (understood as pairs of sets of formulas) and be built as an application of ultraproduct constructions. This their consistency and satisfiability, and a generalization of the Tarski–Vaught theorem on unions of elementary chains. idea followed the classical tradition found in [5]. However, Index Terms—mathematical fuzzy logic, first-order graded in other classical standard references such as [22], [24], [28] logics, uninorms, residuated lattices, logic UL, types, saturated the construction of saturated structures is obtained by other models, elementary chains methods.