Syntactic characterizations of classes of firstorder structures in mathematical fuzzy logic
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 firstorder 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 MTLalgebra, from which analogues of the Łoś–Tarski and the Chang–Łoś–Suszko preservation theorems follow.
Keywords
Graded model theory Mathematical fuzzy logic Universal classes Universalexistential classes Amalgamation theorems Preservation theorems1 Introduction
Graded model theory is the generalized study, in mathematical fuzzy logic (MFL), of the construction and classification of graded structures. The field was properly started in Hájek and Cintula (2006) and has received renewed attention in recent years (Badia and Noguera 2018a; Bagheri and Moniri 2013; Cintula and Metcalfe 2013; Cintula et al. 2015; Costa and Dellunde 2017; Dellunde 2011, 2014). Part of the programme of graded model theory is to find nonclassical analogues of results from classical model theory (e.g. Hodges 1993; Sacks 1972; Chang and Keisler 1973). This will not only provide generalizations of classical theorems but will also provide insight into what avenues of research are particular to classical firstorder logic and do not make sense in a broader setting.
On the other hand, classical model theory was developed together with the analysis of some very relevant mathematical structures. In consequence, its principal results provided a logical interpretation of such structures. Thus, if we want model theory’s idiosyncratic interaction with other disciplines to be preserved, the redefinition of the fundamental notions of graded model theory cannot be obtained from directly fuzzifying every classical concept. Quite the contrary, the experience acquired in the study of different structures, the results obtained using specific classes of structures, and the potential overlaps with other areas should determine the light the main concepts of graded model theory have to be defined in. It is in this way that several fundamental concepts of the model theory of mathematical fuzzy logic have already appeared in the literature.
The goal of this paper is to give syntactic characterizations of classes of graded structures; more precisely, we want to study which kinds of formulas can be used to axiomatize certain classes of structures based on finite (expansions of) \(\mathrm {MTL}\)chains. Traditional examples of such sort of results are preservation theorems in classical model theory, which, in general, can be obtained as consequences of certain amalgamation properties (cf. Hodges 1993). We provide some amalgamation results using the technique of diagrams which will allow us to establish analogues of the Łoś–Tarski preservation theorem (Hodges 1993, Theorem 6.5.4) and the Chang–Łoś–Suszko theorem (Hodges 1993, Theorem 6.5.9).
This is not the first work that addresses a modeltheoretic study of the preservation and characterization of classes of fuzzy structures. Indeed, Bagheri and Moniri (2013) have obtained results for the particular case of continuous model theory by working over the standard \(\mathrm {MV}\)algebra Open image in new window and with a predicate language enriched with a truthconstant for each element of Open image in new window . In that context, they characterize universal theories in terms of the preservation under substructures (Bagheri and Moniri 2013, Prop. 5.1) and prove versions of the Tarski–Vaught theorem (Bagheri and Moniri 2013, Prop. 4.6) and of the Chang–Łoś–Suszko theorem (Bagheri and Moniri 2013, Prop. 5.5).
The connection between classical model theory and the study of classes of fuzzy structures needs to be clarified. Namely, as explained and developed in previous papers (Cintula et al. 2009; Dellunde et al. 2016, 2018), there is a translation of fuzzy structures into classical manysorted structures, more precisely, twosorted structures with one sort for the firstorder domain and another accounting for truthvalues in the algebra. Such connection certainly allows to directly import to the fuzzy setting several classical results, but, as already noted in the mentioned papers, it does not go a long way. Indeed, the translation does not preserve the syntactical complexity of sentences (regarding quantifiers), and hence, it cannot be used for syntactically sensitive results, such as those studied in the present paper.
The paper is structured as follows: in Sect. 1, we introduce the syntax and semantics of fuzzy predicate logics. In Sect. 2, several fuzzy modeltheoretic notions such as homomorphisms or the method of diagrams are presented. In Sect. 3, we study the preservation of universal formulas, obtain an existential form of amalgamation and derive from it an analogue of the Łoś–Tarski theorem. In Sect. 4, we study classes given by universalexistential sentences by showing that such formulas are preserved under unions of chains, obtaining another corresponding amalgamation result and a version of Chang–Łoś–Suszko preservation theorem. We end with some concluding remarks and suggestions for lines of further research.
2 Preliminaries
In this section, we introduce the syntax and semantics of fuzzy predicate logics, and recall the basic results on diagrams we will use in the paper. We use the notation and definitions of the Handbook of Mathematical Fuzzy Logic (Cintula et al. 2011).
Definition 1
(Syntax of Predicate Languages) A predicate language\(\mathcal {P}\) is a triple \(\left\langle Pred_{\mathcal {P}},Func_{\mathcal {P}},Ar_{\mathcal {P}} \right\rangle \), where \(Pred_{\mathcal {P}}\) is a nonempty set of predicate symbols, \(Func_{\mathcal {P}}\) is a set of function symbols (disjoint from \(Pred_{\mathcal {P}}\)), and \(Ar_{\mathcal {P}}\) represents the arity function, which assigns a natural number to each predicate symbol or function symbol. We call this natural number the arity of the symbol. The predicate symbols with arity zero are called truthconstants, while the function symbols whose arity is zero are named object constants (constants for short).
\(\mathcal {P}\)terms, \(\mathcal {P}\)formulas, \(\forall _n\) and \(\exists _n\)\(\mathcal {P}\)formulas, and the notions of free occurrence of a variable, open formula, substitutability, and sentence are defined as in classical predicate logic. A theory is a set of sentences. When it is clear from the context, we will refer to \(\mathcal {P}\)terms and \(\mathcal {P}\)formulas simply as terms and formulas.
Let MTL stand for the monoidal tnorm based logic introduced by Esteva and Godo (2001). Throughout the paper, we consider the predicate logic MTL\(\forall \) [for a definition of the axiomatic system for MTL\(\forall \) we refer the reader to Cintula et al. (2011, Def. 5.1.2, Ch. I)]. Let us recall that the deduction rules of MTL\(\forall \) are those of MTL and the rule of generalization: from \(\varphi \) infer \((\forall x)\varphi \). The definitions of proof and provability are analogous to the classical ones. We denote by \(\varPhi \vdash _{\text {MTL}\forall }\varphi \) the fact that \(\varphi \) is provable in MTL\(\forall \) from the set of formulas \(\varPhi \). For the sake of clarity, when it is clear from the context we will write \(\vdash \) to refer to \(\vdash _{\text {MTL}\forall }\). The algebraic semantics of MTL\(\forall \) are based on \(\mathrm {MTL}\)algebras (Esteva and Godo 2001).
\({\varvec{A}}\) is called an \(\mathrm {MTL}\)chain if its underlying lattice is linearly ordered. Since it is customary to consider fuzzy logics in languages expanding that of \(\mathrm {MTL}\), henceforth, we will confine our attention to algebras which are expansions of \(\mathrm {MTL}\)chains of such kind and just call them chains.
Definition 2

\(\Vert x\Vert ^{{\varvec{A}}}_{\mathbf {M},v}=v(x)\);

\(\Vert F(t_1,\ldots ,t_n)\Vert ^{{\varvec{A}}}_{\mathbf {M},v}=F_{\mathbf {M}}(\Vert t_1\Vert ^{{\varvec{A}}}_{\mathbf {M},v},\ldots ,\Vert t_n\Vert ^{{\varvec{A}}}_{\mathbf {M},v})\), for \(F\in Func\);

\(\Vert P(t_1,\ldots ,t_n)\Vert ^{{\varvec{A}}}_{\mathbf {M},v}=P_{\mathbf {M}}(\Vert t_1\Vert ^{{\varvec{A}}}_{\mathbf {M},v},\ldots ,\Vert t_n\Vert ^{{\varvec{A}}}_{\mathbf {M},v})\), for \(P\in Pred\);

\(\Vert c(\varphi _1,\ldots ,\varphi _n)\Vert ^{{\varvec{A}}}_{\mathbf {M},v}=\circ ^{\varvec{A}}(\Vert \varphi _1\Vert ^{{\varvec{A}}}_{\mathbf {M},v},\ldots ,\Vert \varphi _n\Vert ^{{\varvec{A}}}_{\mathbf {M},v})\), for \(\circ \in \mathcal {L}\);

\(\Vert (\forall x)\varphi \Vert ^{{\varvec{A}}}_{\mathbf {M},v}=inf_{\le ^{\varvec{A}}}\{\Vert \varphi \Vert ^{{\varvec{A}}}_{\mathbf {M},v[x\rightarrow d]}\mid d\in M\}\);

\(\Vert (\exists x)\varphi \Vert ^{{\varvec{A}}}_{\mathbf {M},v}=sup_{\le ^{\varvec{A}}}\{\Vert \varphi \Vert ^{{\varvec{A}}}_{\mathbf {M},v[x\rightarrow d]}\mid d\in M\}\).
For a set of formulas \(\varPhi \), we write \(\Vert \varPhi \Vert ^{\varvec{A}}_{\mathbf {M},v}=1\), if \(\Vert \varphi \Vert ^{\varvec{A}}_{\mathbf {M},v}=1\) for every \(\varphi \in \varPhi \). We denote by \(\Vert \varphi \Vert ^{\varvec{A}}_{\mathbf {M}}=1\) the fact that \(\Vert \varphi \Vert ^{\varvec{A}}_{\mathbf {M},v}=1\) for all \(\mathbf {M}\)evaluations v. We say that \(\langle \varvec{A},\mathbf {M}\rangle \) is a model of a set of formulas\(\varPhi \), if \(\Vert \varphi \Vert ^{\varvec{A}}_{\mathbf {M}}=1\) for any \(\varphi \in \varPhi \). Sometimes, we will denote by \(\overrightarrow{x}\) a sequence of variables \(x_1,\ldots ,x_n\) (and the same with sequences \(\overrightarrow{d}\) of elements of the domain). Given a structure \(\langle \varvec{A},\mathbf {M}\rangle \) and a formula \(\varphi (\overrightarrow{x})\), we say that \(\overrightarrow{d}\subseteq M\)satisfies\(\varphi (\overrightarrow{x})\) (or that \(\varphi (\overrightarrow{x})\) is satisfied by \(\overrightarrow{d}\)) if \(\left\ {\varphi (\overrightarrow{x})}\right\ ^{{{{\varvec{A}}}}}_{\mathbf{M },v[\overrightarrow{x}\rightarrow \overrightarrow{d}]}=\overline{1}^{\varvec{A}}\) for any \(\mathbf M \)evaluation v (also written \(\left\ {\varphi [\overrightarrow{d}]}\right\ ^{{{{\varvec{A}}}}}_\mathbf{M }=\overline{1}^{\varvec{A}}\)); for the sake of clarity, we will use also the notation \(\langle {{\varvec{A}}, \mathbf M}\rangle \models \varphi [\overrightarrow{d}]\) when is needed. Two theories T and U are said to be 1equivalent if a structure is a model of T if it is also a model of U (in the case where T and U are singletons of formulas, we say that these formulas are 1equivalent).
Given a set of sentences \(\varSigma \), and a sentence \(\phi \), we denote by \(\varSigma \models _{\varvec{A}}\phi \) the fact that every \(\varvec{A}\)model of \(\varSigma \) is also an \(\varvec{A}\)model of \(\phi \). We focus on classes of structures over a fixed finite chain A whose set of elements is denoted by \(\{a_1,\ldots ,a_k\}\). Such restriction is due to the fact that dropping finiteness can cause to lose compactness, which is an essential element of our proofs. However, the results will still be quite encompassing in practice. Indeed, for instance, prominent examples of weighted structures in computer science are valued over finite chains. Structures over a fixed finite chain A have two important properties: they are witnessed (the values of the quantifiers are maxima and minima achieved in particular instances) and have the compactness property, both for satisfiability and for consequence (see e.g. Dellunde 2014).
Proposition 1
 1.
If every finite subset \(\varSigma _0\subseteq \varSigma \) has a model \(\langle {{\varvec{A}}},\mathbf{M }_{\varSigma _0}\rangle \), then \(\varSigma \) has a model \(\langle {{\varvec{A}}},\mathbf{N }\rangle \).
 2.
If \(\varSigma \models _{\varvec{A}} \alpha \), then there is a finite subset \(\varSigma _0\subseteq \varSigma \) such that \(\varSigma _0 \models _{\varvec{A}} \alpha \).
From now on, we refer to Astructures simply as structures (or as \(\mathcal {P}\)structures if we need to specify the language). For the remainder of the article, let us assume that we have a crisp identity \(\approx \) in the language.
Definition 3
Let \(\langle f,g\rangle \) be a strong homomorphism from \(\langle {{{\varvec{A}}}},{\mathbf {M}}\rangle \) to \(\langle {{\varvec{B}}},{\mathbf {N}}\rangle \), we say that \(\langle f,g\rangle \) is an embedding from \(\langle {{{\varvec{A}}}},{\mathbf {M}}\rangle \) to \(\langle {{\varvec{B}}},{\mathbf {N}}\rangle \) if both functions f and g are injective, and we say that \(\langle f,g\rangle \) is an isomorphism from \(\langle {{{\varvec{A}}}},{\mathbf {M}}\rangle \) to \(\langle {{\varvec{B}}},{\mathbf {N}}\rangle \) if \(\langle f,g\rangle \) 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 Dellunde et al. (2016).
Later in the article, we will use diagram techniques. We present here some corollaries of the results obtained in Dellunde (2011). Given a language \(\mathcal {P}\), we start by introducing three different expansions adding either a new truthconstant for each element of the algebra, or new object constants. For any element a of \({\varvec{A}}\), we will use the truthconstant \(\overline{a}\) to denote it. When \(a=\overline{1}^{\varvec{A}}\) or \(a=\overline{0}^{\varvec{A}}\), then \(\overline{a}=\overline{1}\) or \(\overline{a}=\overline{0}\), respectively.
Definition 4
Given a predicate language \(\mathcal {P}\), we expand it by adding an individual constant symbol \(c_m\) for every \(m\in M\), and denote it by \(\mathcal {P}^{\mathbf {M}}\). If \(\langle {{\varvec{A}}},\mathbf {M}\rangle \) is a \(\mathcal {P}^{\mathbf {M}}\)structure, we denote by \(\langle {{\varvec{A}}},\mathbf {M}^\sharp \rangle \) the expansion of the structure \(\langle {{\varvec{A}}},\mathbf {M}\rangle \) to \(\mathcal {P}^{\mathbf {M}}\), where for every \(m\in M\), \((c_m)_\mathbf{M ^\sharp }=m\).
Definition 5
Given a predicate language \(\mathcal {P}\), we expand it by adding a truthconstant symbol \(\overline{a}\) for every \(a \in A\), and denote it by \(\mathcal {P}^{{{\varvec{A}}}}\). When we expand the language \(\mathcal {P}^{{{\varvec{A}}}}\) further by adding an individual constant symbol \(c_m\) for every \(m\in M\), we will denote it by \(\mathcal {P^{\langle {{\varvec{A}}},{\mathbf {M}}\rangle }}\).
Definition 6
Following the same lines of the proof of Dellunde (2011, Prop. 32), we can obtain a characterization of strong and elementary embeddings between two \(\mathcal {P}\)structures over a chain \({\varvec{A}}\).
Corollary 2
 1.
There is an expansion of \(\langle {{\varvec{A}}},{\mathbf {N}}\rangle \) that is a model of Diag\(({{\varvec{A}}},\mathbf{M })\) (ElDiag\(({{\varvec{A}}},\mathbf{M })\), respectively).
 2.
There is a mapping \(g:M \rightarrow N\) such that \(\langle Id_{{{\varvec{A}}}},g \rangle \) is a strong (elementary, respectively) embedding from \(\langle {{\varvec{A}}},{\mathbf {M}}\rangle \) into \(\langle {{\varvec{A}}},{\mathbf {N}}\rangle \).
3 Universal classes
In this section, we prove a result on existential amalgamation (Proposition 4) from which we extract a Łoś–Tarski preservation theorem for universal theories (Theorem 5) and a characterization of universal classes of structures (Theorem 6). Relevant structures in computer science are axiomatized by sets of universal formulas; one prominent example is the class of weighted graphs. Particular versions of the abovementioned results appeared for Ł\(\forall \) in Spada (2009). In the context of fuzzy logic programming, Gerla (2005) studied universal formulas with relation to Herbrand interpretations.
For the upcoming results, we need to recall the notion of substructure.
Definition 7
 (1)
\({{\varvec{A}}}\) is a subalgebra of \({\varvec{B}};\)
 (2)
\(M\subseteq N;\)
 (3)for any nary function symbol \(F\in \mathcal {P}\) and elements \(d_1,\ldots ,d_n\in M\), we have$$\begin{aligned} F_{\mathbf{M }}(d_1,\ldots ,d_n)=F_{\mathbf{N }}(d_1,\ldots ,d_n); \end{aligned}$$
 (4)for any nary predicate symbol \(P\in \mathcal {P}\) and elements \(d_1,\ldots ,d_n\in M\), we have$$\begin{aligned} P_{\mathbf{M }}(d_1,\ldots ,d_n)=P_{\mathbf{N }}(d_1,\ldots ,d_n). \end{aligned}$$
Definition 8
Since our characterizations will be based on axiomatizability of classes, we need to recall the definition of elementary class of structures.
Definition 9
Definition 10
Let \(\mathcal {P}\) be a predicate language. We say that a \(\mathcal {P}\)formula \(\varphi (x_1,\ldots ,x_n)\) is preserved under substructures if for any \(\mathcal {P}\)structure \(\langle {{\varvec{A}}},\mathbf{M }\rangle \) and any substructure \(\langle {\varvec{B}},\mathbf{N }\rangle \), if \(\left\ {\varphi (d_1,\ldots ,d_n)}\right\ ^{{{\varvec{A}}}}_{\mathbf{M }}=\overline{1}^{\varvec{A}}\) for some \(d_1,\ldots ,d_n\in N\), then \(\left\ {\varphi (d_1,\ldots ,d_n)}\right\ ^{{\varvec{B}}}_{\mathbf{N }}=\overline{1}^{\varvec{B}}\).
The following lemma can be easily proved by induction on the complexity of universal formulas.
Lemma 3
Let \(\varphi (x_1,\ldots ,x_n)\) be a universal formula. Then, \(\varphi (x_1,\ldots ,x_n)\) is preserved under substructures.
In classical model theory, amalgamation properties are often related in elegant ways to preservation theorems (see e.g. Hodges 1993). We will try an analogous approach to obtain our desired preservation result. The importance of this idea is that the problem of proving a preservation result reduces then to finding a suitable amalgamation counterpart. This provides us with proofs that have a neat common structure (such as those of the main results in this section and the next one).
We will write \( \langle {\mathbf{A }, \mathbf{M}_2, \overrightarrow{d}}\rangle \Rrightarrow _{\exists _n} \langle {\mathbf{A }, \mathbf{M}_1, \overrightarrow{d}}\rangle \) if for any \(\exists _n\)formula \(\varphi \), \( \langle {\mathbf{A }, \mathbf{M}_2}\rangle \models \varphi [\overrightarrow{d}] \) only if \( \langle {\mathbf{A }, \mathbf{M}_1}\rangle \models \varphi [\overrightarrow{d}]\).
Proposition 4
Proof
Observe that the proof can be similarly carried out, mutatis mutandi, when \(\langle {{{{\varvec{A}}}}, \mathbf{M}_1}\rangle \) and \(\langle {{{{\varvec{A}}}}, \mathbf{M}_2}\rangle \) have no common part as well. \(\square \)
Now, we have the elements to establish an exact analogue of Theorem 5 from Łoś (1955), Łoś–Tarski preservation theorem.
Theorem 5
 (i)
For any models of T, \(\langle {{\varvec{A}}, \mathbf M}\rangle \subseteq \langle {{\varvec{A}}, \mathbf N}\rangle \), we have: if \(\langle {{\varvec{A}}, \mathbf N}\rangle \models \varPhi \), then \(\langle {{\varvec{A}}, \mathbf M}\rangle \models \varPhi \).
 (ii)
There is a set of universal \({{\mathcal {P}}}^{{\varvec{A}}}\)formulas \(\varTheta (\overrightarrow{x})\) such that: \(T, \varPhi \vDash \varTheta \) and \(T, \varTheta \vDash \varPhi \).
Proof
Let us prove the difficult direction (the converse direction is clear by Lemma 3). Consider \((T \cup \varPhi (\overrightarrow{x}))_{\forall _1}\), the collection of all \(\forall _1\) logical consequences of \(T \cup \varPhi (\overrightarrow{x})\). We need to establish that the only models of \((T \cup \varPhi (\overrightarrow{x}))_{\forall _1}\) among the models of T are the substructures of models of \(\varPhi (\overrightarrow{x})\). Let \(\langle {{\varvec{A}}, \mathbf M}\rangle \) be a model of \((T \cup \varPhi (\overrightarrow{x}))_{\forall _1}\). All we need to do is find a model \(\langle {{\varvec{A}}, \mathbf N}\rangle \) of the theory \(T \cup \varPhi (\overrightarrow{x})\) such that \(\langle {{\varvec{A}}, \mathbf M}\rangle \Rrightarrow _{\exists _1} \langle {{\varvec{A}}, \mathbf N}\rangle \) and then quote the existential amalgamation theorem.
Following a similar proof, we can obtain an algebraic characterization equivalent to Theorem 5 (as long as we have truthconstants around).
Theorem 6
 (i)
\({\mathbb {K}}\) is closed under isomorphisms, substructures, and ultraproducts.
 (ii)
\({\mathbb {K}}\) is axiomatized by a set of universal \(\mathcal {P}^{{{\varvec{A}}}}\)sentences.
The following corollary can be obtained because in our setting, two forms of compactness (that are generally distinct, in, say, Łukasiewicz logic) collapse, namely (1) the compactness of the consequence relation and (2) the compactness of the satisfiability relation. (1) clearly implies (2) in the presence of \(\overline{0}\) in our language. To see the converse, say that \(T \vDash \varphi \), which amounts to say that \(T \cup \{\varphi \rightarrow \overline{a}\}\) (where a is the predecessor of \(\overline{1}^{\varvec{A}}\)) does not have a model. Hence, by (2), there is a finite \(T_0 \subseteq T\) such that \(T_0 \cup \{\varphi \rightarrow \overline{a}\}\) has no model, so, in fact, \(T_0 \vDash \varphi \).
Corollary 7
Let \(T \cup \{\varphi \}\) be a set of \({{\mathcal {P}}}^{{\varvec{A}}}\)sentences. Then, \(\varphi \) is preserved under substructures of models of T if, and only if, \(\varphi \) is 1equivalent to a universal \({{\mathcal {P}}}^{{\varvec{A}}}\)sentence modulo T.
Proof
Apply Theorem 5 for \(\varPhi = \{\varphi \}\). Consequently, \(\varphi \) is axiomatized by a set of universal \({{\mathcal {P}}}^{{\varvec{A}}}\)sentences. Then, bring it down to a single such formula using Acompactness for consequence. \(\square \)
Needless to say, the previous results, in particular, allow to conclude that a class of \(\mathcal {P}\)structures (that is, structures for a language without additional truthconstants) closed under substructures can be axiomatized by universal \(\mathcal {P}^{{{\varvec{A}}}}\)sentences. One might wonder, of course, if it is really necessary to resort a universal axiomatization in the expanded language.
Whether constants are necessary for Theorem 5 in general is an open question, we conjecture that they are.
4 Universal–existential classes
This section runs quite parallel to the previous one. We recall the notion of elementary chain of structures and its corresponding Tarski–Vaught theorem and prove that universal–existential formulas are preserved under unions of chains (Lemma 9). After that, we prove a result on existential–universal amalgamation (Proposition 10) and derive from it a Chang–Łoś–Suszko preservation theorem (Theorem 11).

\((\forall x) (\exists y) (y^n \approx x)\) for each \(n \geqslant 2\).

\((\forall x, y) ((x\cdot y)\cdot z \approx x\cdot (y \cdot z))\).

\((\forall x) (x \cdot 1 \approx x)\).

\((\forall x) (x\cdot x^{1} \approx 1)\).

\((\forall x, y)(x\cdot y \approx y\cdot x)\).

\((\forall x,y) ((Gx \wedge Gy) \rightarrow G(xy))\).

\((\forall x) (Gx \rightarrow G(x^{1}))\).
Given an ordinal \(\gamma \), a sequence \(\{\langle {{\varvec{ A}}, \mathbf{M}_i}\rangle \mid i < \gamma \}\) of models is called a chain when for all \(i<j<\gamma \) we have that \(\langle {{\varvec{ A}} , \mathbf{M}_i}\rangle \) is a substructure of \(\langle {{\varvec{ A}}, \mathbf{M}_j}\rangle \). If, moreover, these substructures are elementary, we speak of an elementary chain. The union of the chain \(\{\langle {{\varvec{ A}}, \mathbf{M}_i}\rangle \mid i < \gamma \}\) is the structure \(\langle {{\varvec{ A}}, \mathbf{M}}\rangle \) where \(\mathbf M \) is defined by taking as its domain \(\bigcup _{i<\gamma }{ M}_i\), interpreting the constants of the language as they were interpreted in each \(\mathbf{M}_i\) and similarly with the relational symbols of the language. Observe as well that \(\mathbf{M}\) is well defined given that \(\{\langle {{\varvec{ A}}, \mathbf{M}_i}\rangle \mid i < \gamma \}\) is a chain.
Next, we recall a useful theorem that has been established and used to construct saturated models in the context of mathematical fuzzy logic in Badia and Noguera (2018b).
Theorem 8
(Badia and Noguera 2018b) (Tarski–Vaught) Let \(\langle {{\varvec{A}}, \mathbf M}\rangle \) be the union of the elementary chain \(\{\langle {{\varvec{A}},\mathbf M_i}\rangle \mid i < \gamma \}\). Then, for every sequence \(\overrightarrow{d}\) of elements of \(\mathbf{M}_i\) and formula \(\varphi \), \( \left\ {\varphi (\overrightarrow{d})}\right\ ^{\varvec{A}}_{\mathbf M} = \left\ {\varphi (\overrightarrow{d})}\right\ ^{{\varvec{A}}}_{\mathbf M_i}\). Moreover, if the chain is not elementary, we still have that \( \left\ {\varphi (\overrightarrow{d})}\right\ ^{{\varvec{A}}}_{\mathbf M} = \left\ {\varphi (\overrightarrow{d})}\right\ ^{{\varvec{A}}}_{\mathbf M_i}\) for every quantifierfree formula.
Therefore, unions of elementary chains preserve the values of all formulas. It is also interesting to consider formulas that are preserved by all unions of chains.
Definition 11
We say that a formula \(\varphi (x_1,\ldots ,x_n)\) is preserved under unions of chains if whenever we have a chain of models \(\{\langle {{\varvec{A}}, \mathbf{M}_i}\rangle \mid i < \gamma \}\) such that for every i, \(\left\ {\varphi (\overrightarrow{d})}\right\ ^{\varvec{A}}_{\mathbf M_i} = \overline{1}^{\varvec{A}} ( i < \gamma )\) for some sequence \(\overrightarrow{d}\) of elements of \(M_0\), then \(\left\ {\varphi (\overrightarrow{d})}\right\ ^{\varvec{A}}_{\mathbf M} = \overline{1}^{\varvec{A}}\), where \(\langle {{\varvec{ A}}, \mathbf{M}}\rangle \) is the union of the chain.
Let a be the element of \({\varvec{A}}\) immediately above \(\overline{0}^{\varvec{A}}\).
Lemma 9
\(\forall _2\)formulas are preserved under unions of chains.
Proof
Let \( (\forall \overrightarrow{x})(\exists \overrightarrow{y}) \phi \) be a \(\forall _2\)formula, \(\langle {{{\varvec{A}}}},{\mathbf {M}}\rangle \) be the union of a chain \(\{\langle {{{\varvec{A}}}},{\mathbf {M}_i}\rangle \mid i < \gamma \}\), and \(\overrightarrow{c}\) some sequence of elements of \(M_0\). Assume that for every \(i < \gamma \), \( \left\ {(\forall \overrightarrow{x})(\exists \overrightarrow{y}) \phi (\overrightarrow{c})}\right\ ^{\varvec{A}}_{\mathbf {M}_i} = \overline{1}^{\varvec{A}}\). Let \(\overrightarrow{d} \in M\), we show that
Next, we provide the amalgamation result that will allow us to prove a version of Chang–Łoś–Suszko theorem for graded model theory.
Proposition 10
Proof
Now, we are ready to prove the promised analogue of Robinson (1959, Theorem 1.2).
Theorem 11
 (i)
\(\varPhi (\overrightarrow{x})\) is preserved under unions of chains of models of T.
 (ii)
\(\varPhi (\overrightarrow{x})\) is 1equivalent modulo T to a set of \(\forall _2\)formulas.
Proof
So, we start with \(\langle {{\varvec{A}}, \mathbf M_0}\rangle \) being an arbitrary model of \( T \cup (T \cup \varPhi (\overrightarrow{x}))_{\forall _2}\). Now, assuming that we have \(\langle {{\varvec{A}}, \mathbf M_i}\rangle \) which is an elementary extension of \(\langle {{\varvec{A}}, \mathbf M_0}\rangle \). We first need to find a model \(\langle {{\varvec{A}}, \mathbf M_i^{\prime }}\rangle \) of the theory \(T \cup \varPhi (\overrightarrow{x})\) such that \(\langle {{\varvec{A}}, \mathbf M_i}\rangle \Rrightarrow _{\exists _2} \langle {{\varvec{A}}, \mathbf M_i^{\prime }}\rangle \) and then quote the \(\exists _2\)amalgamation theorem to obtain a model \(\langle {{\varvec{A}}, \mathbf N_i}\rangle \) of \(T \cup \varPhi (\overrightarrow{x})\) into which \(\langle {{\varvec{A}}, \mathbf M_i}\rangle \) can be strongly embedded in such a way that all \(\forall _1\)formulas are preserved by such strong embedding.
As a consequence, we can again obtain a result for single formulas, using the compactness of the consequence relation.
Corollary 12
 (i)
\(\varphi \) is preserved under unions of chains of models of T.
 (ii)
\(\varphi \) is 1equivalent modulo T to a set of \(\forall _2\)formulas.
5 Conclusions
In this paper, we have provided some necessary steps in the systematic study of syntactic characterizations of classes of graded structures and their corresponding preservations theorems. Work in progress in the same line includes the study of the universal Horn fragment of predicate fuzzy logics and the classes axiomatized by sets of Horn clauses. Moreover, in the general endeavour of graded model theory, we believe that, among others, future works should focus on the study of types, with the construction of saturated models and typeomission theorems, the study of particular kinds of graded structures that are relevant for computer science applications and, also, the development of Lindströmstyle characterization theorems for predicate fuzzy logics that may lead to the creation of a nonclassical abstract model theory.
Notes
Acknowledgements
Open access funding provided by Austrian Science Fund (FWF). The authors are indebted to two anonymous referees and to the editor for their critical and interesting remarks that have helped improving the presentation the paper. Costa, Dellunde, and Noguera received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Curie Grant Agreement No. 689176 (SYSMICS project). Badia is supported by the Project I 1923N25 (New perspectives on residuated posets) of the Austrian Science Fund (FWF). Costa is also supported by the Grant for the recruitment of earlystage research staff (FI2017) from the Generalitat de Catalunya. Dellunde is also partially supported by the Project RASO TIN201571799C21P, CIMBVAL TIN201789758R, and the Grant 2017SGR172 from the Generalitat de Catalunya. The research leading to these results has received funding from AppPhilRecerCaixa. Finally, Noguera is also supported by the Project GA1704630S of the Czech Science Foundation (GAČR).
Compliance with ethical standards
Conflict of interest
The authors declare they have no conflict of interest.
Human and animal rights
This article does not contain any studies with human participants or animals performed by any of the authors.
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