Löwenheim–Skolem theorems for non-classical first-order algebraizable logics: Table 1.

Abstract

This paper is a contribution to the model theory of non-classical first-order predicate logics. In a wide framework of first-order systems based on algebraizable logics, we study several notions of homomorphisms between models and find suitable definitions of elementary homomorphism, elementary substructure and elementary equivalence. Then we obtain (downward and upward) Löwenheim-Skolem theorems for these non-classical logics, by direct proofs and by describing their models as classical 2-sorted models.

Fuzzy Positive Primitive Formulas

“…In particular, as pointed out in [22], with mathematical fuzzy logic (MFL). Only in recent times, model theory of predicate fuzzy logics has been developed as a subarea of MFL (see for instance [5] or [9]), leaving the important area of fuzzy finite-model theory yet unexplored.…”

Saturated Models in Mathematical Fuzzy Logic

“…Let us start by recalling the notion of (elementary) substructure (see e.g. [17]). A, M is a substructure of B, N if the following conditions are satisfied: 1) M ⊆ N .…”

“…The study of models of first-order fuzzy logics is based on the corresponding strong completeness theorems [10], [21] and has already addressed several crucial topics such as: characterization of completeness properties with respect to models based on particular classes of algebras [7], models of logics with evaluated syntax [26], [27], study of mappings and diagrams [13], ultraproduct constructions [14], [15], characterization of elementary equivalence in terms of elementary mappings [16], characterization of elementary classes as those closed under elementary equivalence and ultraproducts [15], Löwenheim-Skolem theorems [17], and back-and-forth systems for elementary equivalence [18]. A related stream of research is that of continuous model theory [2], [6].…”

Syntactic characterizations of classes of first-order structures in mathematical fuzzy logic

“…Let f, g be a strong homomorphism from A, M to B, N , we say that f, g is an embedding from A, M to B, N if both functions f and g are injective, and we say that f, g is an isomorphism from A, M to B, N if f, g is an embedding and both functions f and g are surjective. For a general study of different kinds of homomorphisms and the formulas they preserve we refer to [13].…”