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. The goal of the present paper is to show the existence of saturated models for first-order graded logics by means of I. INTRODUCTION an elementary construction. Mathematical fuzzy logic studies graded logics as particular The paper is organized as follows: after this introduction, kinds of many-valued inference systems in several formalisms, Section II presents the necessary preliminaries we need by including first-order predicate languages. Models of such first- recalling several semantical notions from mathematical fuzzy order graded logics are variations of classical structures in logic, namely, the algebraic counterpart of extensions of the which predicates are evaluated over wide classes of algebras uninorm logic UL, fuzzy first-order models based on such of truth degrees, beyond the classical two-valued Boolean algebras, and some basic model-theoretic notions. Section III algebra. Such models are relevant for recent computer science introduces the notion of tableaux (necessary for our treatment developments in which they are studied as weighted structures of types) as pairs of sets of formulas and proves that each (see e.g. [23]). consistent tableau has a model. Section IV defines types as The study of models of first-order fuzzy logics is based pairs of sets of formulas in one free variable (roughly speaking, on the corresponding strong completeness theorems [10], [21] expressing the properties that an element should satisfy and and has already addressed several crucial topics such as: falsify) and contains the main results of the paper: a fuzzy characterization of completeness properties with respect to version of the Tarski-Vaught theorem for unions of elementary models based on particular classes of algebras [7], models chains and the existence theorem for saturated models. Finally, of logics with evaluated syntax [26], [27], study of mappings Section V ends the paper with some concluding remarks. arXiv:1810.09742v1 [math.LO] 23 Oct 2018 and diagrams [13], ultraproduct constructions [14], [15], char- acterization of elementary equivalence in terms of elementary II. PRELIMINARIES mappings [16], characterization of elementary classes as those closed under elementary equivalence and ultraproducts [15], In this section we introduce the object of our study, fuzzy L¨owenheim–Skolem theorems [17], and back-and-forth sys- first-order models, and several necessary related notions for tems for elementary equivalence [18]. A related stream of the development of the paper. For comprehensive information research is that of continuous model theory [2], [6]. on the subject, one may consult the handbook of Mathematical Fuzzy Logic [8] (e.g. Chapters 1 and 2). This is a pre-print of a paper published as: Saturated Models in Math- We choose, as the underlying propositional basis for the ematical Fuzzy Logic. Proceedings of the IEEE International Symposium first-order setting, the class of residuated uninorm-based log- on Multiple-Valued Logic 2018, IEEE Computer Society: 150-155. We are indebted to the anonymous referees for their helpful comments. Guillermo ics [25]. This class contains most of the well-studied particular Badia is supported by the project I 1923-N25 of the Austrian Science Fund systems of fuzzy logic that can be found in the literature (FWF). Carles Noguera is supported by the project GA17-04630S of the Czech and has been recently proposed as a suitable framework for Science Foundation (GACR)ˇ and has also received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie reasoning with graded predicates in [11], while it retains Sklodowska-Curie grant agreement No 689176 (SYSMICS project). important properties, such as associativity and commutativity of the residuated conjunction, that will be used to obtain the terms and the truth values of the formulas as (where for ◦ results of this paper. stands for any n-ary connective in L): The algebraic semantics of such logics is based on UL-algebras, that is, algebraic structures in the lan- M kxkv = v(x), guage L = {∧, ∨, &, →, 0, 1, ⊥, ⊤} of the form A = M M M A A A A A A A A kF (t1,...,tn)kv = FM(kt1kv ,..., ktnkv ), hA, ∧ , ∨ , & , → , 0 , 1 , ⊥ , ⊤ i such that M M M kP (t1,...,tn)k = PM(kt1k ,..., ktnk ), A A A A v v v • hA, ∧ , ∨ , ⊥ , ⊤ i is a bounded lattice, M A M M A A k◦(ϕ1,...,ϕn)kv = ◦ (kϕ1kv ,..., kϕnkv ), • hA, & , 1 i is a commutative monoid, M M k(∀x)ϕk = inf A {kϕk | m ∈ M}, • for each a,b,c ∈ A, we have: v ≤ v[x→m] M M k(∃x)ϕk = sup≤A {kϕk | m ∈ M}. A A v v[x→m] a & b ≤ c iff b ≤ a → c, (res) If the infimum or supremum does not exist, the corresponding A A A A A A A M M ((a → b) ∧ 1 ) ∨ ((b → a) ∧ 1 )= 1 (lin) value is undefined. We say that is a safe if kϕkv is defined for each P-formula ϕ and each M-evaluation v. A is called a UL-chain if its underlying lattice is linearly Formulas without free variables are called sentences and a set ordered. Standard UL-chains are those define over the real unit of sentences is called a theory. Observe that if ϕ is a sentence, interval [0, 1] with its usual order; in that case the operation then its value does not depend on a particular M-evaluation; A A & is a residuated uninorm, that is, a left-continuous binary we denote its value as kϕkM. If ϕ has free variables among associative commutative monotonic operation with a neutral {x1,...,xn} we will denote it as ϕ(x1,...,xn); then the A element 1 (which need not coincide with the value 1). value of the formula under a certain evaluation v depends only Let Fm denote the set of propositional formulas written on the values given to the free variables; if v(xi) = di ∈ M L M A M in the language of UL-algebras with a denumerable set of we denote kϕkv as kϕ(d1,...,dn)kM. We say that is a variables and let Fm be the absolutely free algebra de- model of a theory T , in symbols M |= T , if it is safe and for L A A fined on such set. Given a UL-algebra A, we say that an each ϕ ∈ T , kϕkM ≥ 1 . A Fm A -evaluation is a homomorphism from L to . The Observe that we allow arbitrary UL-chains and we do not logic of all UL-algebras is defined by establishing, for each focus in any kind of standard completeness properties. Γ ∪{ϕ}⊆ Fm , Γ |= ϕ if and only if, for each UL-algebra Using the semantics just defined, the notion of semantical L A A and each A-evaluation e, we have e(ϕ) ≥ 1 , whenever consequence is lifted from the propositional to the first-order A e(ψ) ≥ 1 for each ψ ∈ Γ. The logic UL is, hence, defined level in the obvious way. Such first-order logics satisfy two as preservation of truth over all UL-algebras, where the notion important properties that we will use in the paper (see e.g. [9]), of truth is given by the set of designated elements, or filter, for each theory T ∪{ϕ,ψ,χ} (inductively defining for each A 0 n+1 n F A = {a ∈ A | a ≥ 1 }. The standard completeness theorem formula α: α = 1, and for each natural n, α = α & α): of UL proves that the logic is also complete with respect to its 1) Local deduction theorem: T, ϕ |= ψ if, and only if, there intended semantics: the class of UL-chains defined over [0, 1] is a natural number n such that T |= (ϕ ∧ 1)n → ψ. by residuated uninorms (the standard UL-chains); this justifies 2) Proof by cases: If T, ϕ |= χ and T, ψ |= χ, then the name of UL (uninorm logic). T, ϕ ∨ ψ |= χ. Most well-known propositional fuzzy logics can be obtained 3) Consequence compactness: If T ϕ, then for some by extending UL with additional axioms and rules (in a finite T0 ⊆ T , T0 ϕ.