Soft Computing (2019) 23:2177–2186 https://doi.org/10.1007/s00500-019-03850-6 FOCUS Syntactic characterizations of classes of first-order structures in mathematical fuzzy logic Guillermo Badia1,2 · Vicent Costa3 · Pilar Dellunde3,4,5 · Carles Noguera6 Published online: 22 February 2019 © The Author(s) 2019 Abstract This paper is a contribution to graded model theory, in the context of mathematical fuzzy logic. We study characterizations of classes of graded structures in terms of the syntactic form of their first-order axiomatization. We focus on classes given by universal and universal–existential sentences. In particular, we prove two amalgamation results using the technique of diagrams in the setting of structures valued on a finite MTL-algebra, from which analogues of the Ło´s–Tarski and the Chang–Ło´s–Suszko preservation theorems follow. Keywords Graded model theory · Mathematical fuzzy logic · Universal classes · Universal-existential classes · Amalgamation theorems · Preservation theorems 1 Introduction Graded model theory is the generalized study, in mathemati- cal fuzzy logic (MFL), of the construction and classification of graded structures. The field was properly started in Hájek This paper is dedicated to Lluís Godo in the occasion of his 60th birthday. and Cintula (2006) and has received renewed attention in recent years (Badia and Noguera 2018a; Bagheri and Moniri Communicated by L. Spada. 2013; Cintula and Metcalfe 2013; Cintula et al. 2015;Costa and Dellunde 2017; Dellunde 2011, 2014). Part of the pro- B Guillermo Badia [email protected] gramme of graded model theory is to find non-classical analogues of results from classical model theory (e.g. Hodges Vicent Costa [email protected] 1993; Sacks 1972; Chang and Keisler 1973). This will not only provide generalizations of classical theorems but will Pilar Dellunde [email protected] also provide insight into what avenues of research are partic- ular to classical first-order logic and do not make sense in a Carles Noguera [email protected] broader setting. On the other hand, classical model theory was developed 1 Department of Knowledge-Based Mathematical Systems, together with the analysis of some very relevant math- Johannes Kepler University Linz, Linz, Austria ematical structures. In consequence, its principal results 2 School of Historical and Philosophical Inquiry, University provided a logical interpretation of such structures. Thus, if of Queensland, Brisbane, Australia we want model theory’s idiosyncratic interaction with other 3 Department of Philosophy, Universitat Autónoma de disciplines to be preserved, the redefinition of the funda- Barcelona, Bellaterra, Catalonia, Spain mental notions of graded model theory cannot be obtained 4 Artificial Intelligence Research Institute (IIIA – CSIC), from directly fuzzifying every classical concept. Quite the Bellaterra, Catalonia, Spain contrary, the experience acquired in the study of different 5 Barcelona Graduate School of Mathematics, Barcelona, structures, the results obtained using specific classes of struc- Catalonia, Spain tures, and the potential overlaps with other areas should 6 Institute of Information Theory and Automation, Czech determine the light the main concepts of graded model theory Academy of Sciences, Prague, Czech Republic 123 2178 G. Badia et al. have to be defined in. It is in this way that several fundamen- We end with some concluding remarks and suggestions for tal concepts of the model theory of mathematical fuzzy logic lines of further research. have already appeared in the literature. The goal of this paper is to give syntactic character- izations of classes of graded structures; more precisely, 2 Preliminaries we want to study which kinds of formulas can be used to axiomatize certain classes of structures based on finite In this section, we introduce the syntax and semantics (expansions of) MTL-chains. Traditional examples of such of fuzzy predicate logics, and recall the basic results on sort of results are preservation theorems in classical model diagrams we will use in the paper. We use the notation theory, which, in general, can be obtained as consequences and definitions of the Handbook of Mathematical Fuzzy of certain amalgamation properties (cf. Hodges 1993). We Logic (Cintula et al. 2011). provide some amalgamation results using the technique of diagrams which will allow us to establish analogues of the Definition 1 (Syntax of Predicate Languages)Apredicate P , , Ło´s–Tarski preservation theorem (Hodges 1993, Theorem language is a triple PredP FuncP ArP , where PredP 6.5.4) and the Chang–Ło´s–Suszko theorem (Hodges 1993, is a non-empty set of predicate symbols, FuncP is a set Theorem 6.5.9). of function symbols (disjoint from PredP ), and ArP repre- This is not the first work that addresses a model-theoretic sents the arity function, which assigns a natural number to study of the preservation and characterization of classes of each predicate symbol or function symbol. We call this nat- fuzzy structures. Indeed, Bagheri and Moniri (2013)have ural number the arity of the symbol. The predicate symbols obtained results for the particular case of continuous model with arity zero are called truth-constants, while the func- tion symbols whose arity is zero are named object constants theory by working over the standard MV-algebra [0, 1]Ł and with a predicate language enriched with a truth-constant for (constants for short). each element of [0, 1] . In that context, they characterize Ł P-terms, P-formulas, ∀n and ∃n P-formulas, and the universal theories in terms of the preservation under sub- notions of free occurrence of a variable, open formula, sub- structures (Bagheri and Moniri 2013, Prop. 5.1) and prove stitutability, and sentence are defined as in classical predicate versions of the Tarski–Vaught theorem (Bagheri and Moniri logic. A theory is a set of sentences. When it is clear from 2013, Prop. 4.6) and of the Chang–Ło´s–Suszko theorem the context, we will refer to P-terms and P-formulas simply (Bagheri and Moniri 2013, Prop. 5.5). as terms and formulas. The connection between classical model theory and the Let MTL stand for the monoidal t-norm based logic intro- study of classes of fuzzy structures needs to be clarified. duced by Esteva and Godo (2001). Throughout the paper, we Namely, as explained and developed in previous papers (Cin- consider the predicate logic MTL∀ [for a definition of the tula et al. 2009; Dellunde et al. 2016, 2018), there is a axiomatic system for MTL∀ we refer the reader to Cintula et translation of fuzzy structures into classical many-sorted al. (2011, Def. 5.1.2, Ch. I)]. Let us recall that the deduction structures, more precisely, two-sorted structures with one sort rules of MTL∀ are those of MTL and the rule of general- for the first-order domain and another accounting for truth- ization: from ϕ infer (∀x)ϕ. The definitions of proof and values in the algebra. Such connection certainly allows to provability are analogous to the classical ones. We denote by directly import to the fuzzy setting several classical results, Φ MTL∀ ϕ the fact that ϕ is provable in MTL∀ from the set but, as already noted in the mentioned papers, it does not go of formulas Φ. For the sake of clarity, when it is clear from a long way. Indeed, the translation does not preserve the syn- the context we will write to refer to MTL∀. The algebraic tactical complexity of sentences (regarding quantifiers), and semanticsofMTL∀ are based on MTL-algebras (Esteva and hence, it cannot be used for syntactically sensitive results, Godo 2001). such as those studied in the present paper. A is called an MTL-chain if its underlying lattice is lin- The paper is structured as follows: in Sect. 1,weintro- early ordered. Since it is customary to consider fuzzy logics duce the syntax and semantics of fuzzy predicate logics. in languages expanding that of MTL, henceforth, we will In Sect. 2, several fuzzy model-theoretic notions such as confine our attention to algebras which are expansions of homomorphisms or the method of diagrams are presented. MTL-chains of such kind and just call them chains. In Sect. 3, we study the preservation of universal formu- las, obtain an existential form of amalgamation and derive Definition 2 (Semantics of Predicate Fuzzy Logics [Cintula from it an analogue of the Ło´s–Tarski theorem. In Sect. 4, et al. 2011, Def. 5.2.1, Ch. I]) Consider a predicate language we study classes given by universal-existential sentences by P =PredP , FuncP , ArP and let A be a chain. We define showing that such formulas are preserved under unions of an A-structure M for P as a pair M =A, M where chains, obtaining another corresponding amalgamation result and a version of Chang–Ło´s–Suszko preservation theorem. M =M,(PM )P∈Pred,(FM )F∈Func, 123 Syntactic characterizations of classes of first-order structures… 2179 where M is a non-empty domain, PM is an n-ary fuzzy rela- important properties: they are witnessed (the values of the tion for each n-ary predicate symbol, i.e. a function from quantifiers are maxima and minima achieved in particular Mn to A, identified with an element of A if n = 0; and instances) and have the compactness property, both for sat- n FM is a function from M to M, identified with an ele- isfiability and for consequence (see e.g. Dellunde 2014). ment of M if n = 0. As usual, if M is an A-structure for P,anM-evaluation of the object variables is a mapping Proposition 1 Let A be a fixed finite chain. For every set of Σ ∪{α} v assigning to each object variable an element of M.The sentences , the following holds: set of all object variables is denoted by Var.Ifv is an M- Σ ⊆ Σ , Σ evaluation, x is an object variable and d ∈ M, we denote 1.