ar X iv :1 91 2. 10 27 7v 1 [ m at h. L O ] 2 1 D ec 2 01 9 First-order swap structures semantics for some Logics of Formal Inconsistency Marcelo E. Coniglio1, Aldo Figallo-Orellano2 and Ana C. Golzio3 1Institute of Philosophy and the Humanities (IFCH) and Centre for Logic, Epistemology and The History of Science (CLE), University of Campinas (UNICAMP), Campinas, SP, Brazil. E-mail: coniglio@unicamp.br 2Departmento de Matemática, Universidad Nacional del Sur (UNS), Bahia Blanca, Argentina and Centre for Logic, Epistemology and The History of Science (CLE), University of Campinas (UNICAMP), Campinas, SP, Brazil. E-mail: aldofigallo@gmail.com 3São Paulo State University (Unesp), Marilia Campus, Brazil. E-mail: anaclaudiagolzio@yahoo.com.br Abstract The logics of formal inconsistency (LFIs, for short) are paraconsistent logics (that is, logics containing contradictory but non-trivial theories) having a consistency connective which allows to recover the ex falso quodlibet principle in a controlled way. The aim of this paper is considering a novel semantical approach to first-order LFIs based on Tarskian structures defined over swap structures, a special class of multialgebras. The proposed semantical framework generalizes previous aproaches to quantified LFIs presented in the literature. The case of QmbC, the simpler quantified LFI expanding classical logic, will be analyzed in detail. An axiomatic extension of QmbC called QLFI1◦ is also studied, which is equivalent to the quantified version of da Costa and D'Ottaviano 3-valued logic J3. The semantical structures for this logic turn out to be Tarkian structures based on twist structures. The expansion of QmbC and QLFI1◦ with a standard equality predicate is also considered. Keywords: First-order logics; Logics of formal inconsistency; Paraconsistent logics; Swap structures; Non-deterministic matrices; Twist structures 1 Introduction A logic is said to be paraconsistent if it contains in its language a negation and it has a contradictory theory (with respect to such negation) which is non-trivial. Such negation is called a paraconsistent or non-explosive negation. This is why paraconsistent logics are said to be tolerant to contradictions. The class of paraconsistent logics known as 1 logics of formal inconsistency (LFIs, for short) was introduced by W. Carnielli and J. Marcos in [11]. In its simplest form, they have a non-explosive negation ¬, as well as a (primitive or derived) consistency connective ◦ which allows to recover the explosion law in a controlled way. Defining interesting and ellucidative semantics for paraconsistent logics is a challenging task for paraconsistentists, taking into account that, in general, these logics are not algebraizable by means of the standard techniques. Several kinds of semantics of nondeterministic character were proposed for these logics, in particular for LFIs. Among them, the non-deterministic matrices (or Nmatrices), introduced by A. Avron and I. Lev in [4] (see also [5]) constitutes an interesting and useful semantical framework for dealing with such logics. The situation is even more delicate in the case of first-order paraconsistent logics. The aim of this paper is considering a novel semantical approach for first-order LFIs based on Tarskian structures defined over a special class of multialgebras called swap structures, which were introduced by Carnielli and Coniglio in [7]. The proposed semantical framework for quantified LFIs generalizes previous aproaches in the literature. In order to understand what lies behind the non-deterministic semantical framework for first-order LFIs proposed here, let us recall briefly the standard lattice-theoretic algebraic approach to first-order classical logic (CFOL, in short). To simplify the exposition, let us consider only sentences instead of formulas with free variables. Let us begin by recalling that the standard semantics for CFOL is given by Tarskian first-order structures. Given such a structure A, in order to evaluate sentences of the given language, it is used the standard two-valued logical matrix for propositional classical logic CPL based on the two-element Boolean algebra A2 with domain A2 = {0, 1}, expanded by quantification operators Q : (P(A2) − {∅}) → A2 for Q ∈ {∀, ∃} given, respectively, by ∀(X) = ∧ X and ∃(X) = ∨ X. It is known that this class of semantical structures can be enlarged by considering structures A = 〈U, IA〉 defined over a complete Boolean algebra A instead of the two-element Boolean algebra A2 (see, for instance, [28, Supplement–Section 8]. This means that IA(P ) is now a mapping IA(P ) : U n → A for any n-ary predicate symbol P , where A is the domain of A. The interpretation of sentences in CFOL is then given in the context (A,MA), where MA = 〈A, {1}〉 constitutes a logical matrix for CPL in which 1 is the only designated value, and where the quantifiers are interpreted mutatis mutandis as in the case of A2. This kind of algebraic approach to quantified logics was firstly proposed by Mostowski in [27], and later on generalized by Henkin in [24] and Rasiova and Sikorski (see [29, 28]). This approach was recently extended by Cintula and Noguera in [16] to first-order logics based on algebraizable logics in the sense of Blok and Pigozzi. What is proposed here for QmbC is a generalization of the algebraic approach to CFOL mentioned above, in which the logical matrix MA is replaced by a non-deterministic matrix M(B) for mbC, where B is a swap structure (a multialgebra of a special kind) for mbC which is induced by a complete Boolean algebra A. That is, sentences of QmbC will be interpreted in contexts (A,M(B)) for any swap structure B for mbC. The analogy with the semantics of CFOL is more accurate than it first appears: while CFOL is fully characterized by the class of contexts (A,M2) such that M2 = 〈A2, {1}〉 (which is equivalent, up to presentation, to the class of standard Tarskian structures with the usual semantics), QmbC can be characterized by contexts (A,M5) defined over the 5-element Nmatrix M5, which is the one induced by the greatest swap structure defined over A2, 2 as we shall see in Section 9. This is why semantical contexts for QmbC of the form (A,M5) are considered to be 'classical', in analogy to CFOL. It is worth noting that the semantics for QmbC given by the 'classical' contexts coincide, up to notational aspects, with the non-deterministic semantics proposed by Avron and Zamansky in [6]. An interesting feature of the mutialgebraic semantics sudied here is that, by adding axioms to a given LFI, conditions on the multioperations, and even on the domain of the swap structures, naturally arise. In particular, consider the 3-valued LFI1◦, which is equivalent (up to language) to several well-known 3-valued logics such as da CostaD'Ottaviano logic J3. As shown in [14], the swap structures for LFI1◦, which is an axiomatic extension of mbC, are deterministic, and as such they become twist structures, that is, ordinary agebras of a certain kind. From this, the first-order swap structures for QLFI1◦, the quantified extension of LFI1◦, become first-order twist structures. As we shall see, when the 'classical' structures (that is, the twist structures induced by the Boolean algebra A2) are considered, the corresponding first-order structures are defined over the characteristic 3-valued logical matrix for LFI1◦, hence this semantics is equivalent (up to presentation) with the early 3-valued model theory for quantified J3 introduced in [17], [18], [19] and [20]. This paper is organized as follows: in the first sections, the swap structures semantics for mbC will be recalled. In Section 6 a semantics based on swap stuctures for QmbC will be introduced, proving in the following sections the corresponding soundness and completeness theorems. As we shall see, the swap structures semantics generalizes the interpretation semantics for QmbC given in [7], as well as the Nmatrix semantics proposed in [6]. The extension of QmbC by adding a standard equality predicate will be analyzed in Sections 10 and 11. The analysis of QLFI1◦, whose semantics is based on twist structures, will be done in Sections 12 to 14. Some conclusions are given in the last section. 2 The logic mbC In this section, the notion of logics of formal inconsistency will be recalled, and the basic LFI called mbC will be briefly described. Let Σ′ be a propositional signature, and assume a denumerable set V = {p1, p2, . . .} of propositional variables. The propositional language generated by Σ′ from V will be denoted by LΣ′ Definition 2.1. Let L = 〈Σ′,⊢〉 be a Tarskian, finitary and structural logic defined over a propositional signature Σ′, which contains a negation ¬, and let ◦ be a (primitive or defined) unary connective. Then, L is said to be a logic of formal inconsistency with respect to ¬ and ◦ if the following holds:1 (i) φ,¬φ 0 ψ for some φ and ψ; (ii) there are two formulas α and β such that (ii.a) ◦α, α 0 β; (ii.b) ◦α,¬α 0 β; (iii) ◦φ, φ,¬φ ⊢ ψ for every φ and ψ. Condition (iii) states that ex falso quodlibet is controllably recovered in LFIs by assuming that the contradictory formula φ is consistent, i.e., ◦φ. 1The original definition of LFIs allows to consider a set ©(p) of formulas depending on a single propositional variable p, instead of a single formula ◦p. See [11, 9, 7]. 3 Definition 2.2. ([7, Definition 2.1.12]) Let Σ = {∧,∨,→,¬, ◦} be a propositional signature. The calculus mbC over the propositional language LΣ is defined by means of the following Hilbert calculus: Axiom schemas: (A1) α→ (β → α) (A2) (α→ (β → γ)) → ((α→ β) → (α→ γ)) (A3) α→ (β → (α ∧ β)) (A4) (α ∧ β) → α (A5) (α ∧ β) → β (A6) α→ (α ∨ β) (A7) β → (α ∨ β) (A8) (α→ γ) → ((β → γ) → ((α ∨ β) → γ)) (A9) α ∨ (α → β) (A10) α ∨ ¬α (A11) ◦α→ (α→ (¬α → β)) Inference rule: (MP) α α → β β Observe that mbC is obtained from positive classical logic by adding axioms (A10) and (A11) governing the new connectives ¬ (paraconsistent negation) and ◦ (consistency operator). It is easy to see that mbC is an LFI. Indeed, it is the least LFI which contains propositional classical logic CPL (see Remark 8.2). 3 Swap structures for mbC It is well-known that mbC, as well as several axiomatic extensions of it, are neither agebraizable (see [11, Section 3.12]), nor characterizable by a single finite logical matrix (see for instance [9, Theorems 121 and 125]). In this section a non-deterministic semantics for mbC based on multialgebras called swap structures, introduced in [7, Chapter 6], will be briefly recalled. Definition 3.1. Let Ω be a propositional signature. A multialgebra (or hyperalgebra) over Ω is a pair A = 〈A, σ〉 such that A is a nonempty set (the universe or support of A) and σ is a mapping assigning to each n-ary connective c, a function (called multioperation or hyperoperation) cA : An → (P(A) − {∅}). In particular, ∅ 6= cA ⊆ A if c is a constant symbol. Definition 3.2. Let Ω be a propositional signature. A non-deterministic matrix (or Nmatrix) is a pair M = 〈A, D〉 such that A = 〈A, σ〉 is a multialgebra over Ω with support A, and D is a subset of A. The elements in D are called designated elements. 4 Notation 3.3. Let A be a Boolean algebra with domain A. If x ∈ A×A×A then (x)i (or simply xi) will denote the ith-projection of x, that is, πi(x), where πi is the ith-canonical projection for i = 1, 2, 3. Definition 3.4. ([7, Definition 6.4.1]) Let A = (A,∧,∨,→, 0, 1) be a Boolean algebra, and BA = {x ∈ A× A× A : x1 ∨ x2 = 1 and x1 ∧ x2 ∧ x3 = 0}. A swap structure for mbC over A is any multialgebra B = (B, ∧, ∨, →, ¬, ◦) over Σ such that B ⊆ BA and where the multioperations satisfy the following, for every x and y in B: (i) ∅ 6= x#y ⊆ {z ∈ B : z1 = x1#y1}, for each # ∈ {∧,∨,→}; (ii) ∅ 6= ¬x ⊆ {z ∈ B : z1 = x2}; (iii) ∅ 6= ◦x ⊆ {z ∈ B : z1 = x3}. Definition 3.5. The full swap structure for mbC over A, denoted by BA, is the unique swap structure for mbC over A with domain BA, in which '⊆' is replaced by '=' in items (i)-(iii) of Definition 3.4. Observe that BA is the greatest swap structure for mbC over A (see [14]). The elements of a given swap structure are called snapshots. This terminology is inspired by its use in computer science to refer to states. Accordingly, a triple (a, b, c) of a swap structure B keeps track simultaneously of the value a of a given formula φ, a possible value b for ¬φ, and a possible value c for ◦φ. Given that any swap structure is a multialgebra, the consequence relation over swap structures will be defined by means of non-deterministic matrices, in analogy with the corresponding notion for twist structures. Definition 3.6. ([7, Definition 6.4.3]) Given a Boolean algebra A and a swap structure B for mbC over A with domain B, let DB = {x ∈ B : x1 = 1} be the set of designated elements. The non-deterministic matrix associated to B is M(B) = (B, DB) (or simply M(B) = (B, D)). The Nmatrix associated to BA will be denoted by MA. The class of all the Nmatrices defined by swap structures for mbC will be denoted by MmbC, that is: MmbC = {M(B) : B is a swap structure for mbC over A, for some A}. Definition 3.7. Let M(B) ∈ MmbC. A valuation over M(B) is a function v : LΣ → |B| such that, for every φ, ψ ∈ LΣ: (i) v(φ#ψ) ∈ v(φ)#v(ψ), for every # ∈ {∧,∨,→}; (ii) v(¬φ) ∈ ¬v(φ); (iii) v(◦φ) ∈ ◦v(φ). Definition 3.8. Let M(B) ∈ MmbC, and let Γ∪{φ} ⊆ LΣ. We say that φ is a consequence of Γ in M(B), denoted by Γ |=M(B) φ, if the following holds: for every valuation v over M(B), if v[Γ] ⊆ D then v(φ) ∈ D. In particular, φ is valid in M(B), denoted by |=M(B) φ, if v(φ) ∈ D for every valuation v over M(B). The swap consequence relation |=MmbC for mbC is given by: Γ |=MmbC φ whenever Γ |=M(B) φ for every M(B) ∈ MmbC. Theorem 3.9. (Adequacy of mbC w.r.t. swap structures, [7, Theorem 6.4.8] and [14, Theorem 7.1]) For every Γ ∪ {φ} ⊆ LΣ: Γ ⊢mbC φ iff Γ |=MmbC φ. 5 4 The 5-valued characteristic Nmatrix M5 for mbC It is illustrative to compare Theorem 3.9 with the adequacy of classical propositional logic CPL w.r.t. Boolean algebras semantics. As it is well known, it is enough to consider just one Boolean algebra to semantically characterize CPL, namely the twoelement Boolean algebra A2 with domain A2 = {0, 1} and the associated logical matrix with 1 as designated value. In the case of swap structures semantics, it is enough to consider the Nmatrix M5 = M ( BA2 ) induced by the full swap structure BA2 defined over A2. The Nmatrix M5 was originally introduced by A. Avron in [1] to semantically characterize mbC. Observe that BA2 = { T, t, t0, F, f0 } where T = (1, 0, 1), t = (1, 1, 0), t0 = (1, 0, 0), F = (0, 1, 1), and f0 = (0, 1, 0). The set D of designated elements of M5 is D = {T, t, t0}, while ND = { F, f0 } is the set of non-designated truth-values. The multioperations proposed by Avron over the set BA2 coincide with the corresponding ones for BA2 , and so his 5-valued Nmatrix coincides with M ( BA2 ) . Observe that the swap structure of M5 is defined as follows: ∧M5 T t t0 F f0 T D D D ND ND t D D D ND ND t0 D D D ND ND F ND ND ND ND ND f0 ND ND ND ND ND ∨M5 T t t0 F f0 T D D D D D t D D D D D t0 D D D D D F D D D ND ND f0 D D D ND ND →M5 T t t0 F f0 T D D D ND ND t D D D ND ND t0 D D D ND ND F D D D D D f0 D D D D D ¬M5 T ND t D t0 ND F D f0 D ◦M5 T D t ND t0 ND F D f0 ND Theorem 4.1. (Adequacy of mbC w.r.t. M5, [1, Theorem 3.6]) For every set of formulas Γ ∪ {φ} ⊆ LΣ: Γ ⊢mbC φ iff Γ |=M5 φ. A new proof of this result was obtained in [7, Corollary 6.4.10], by using bivaluations for mbC in connection with the Nmatrix M ( BA2 ) . Definition 4.2. ([9, Definition 54]) A function ρ : LΣ → { 0, 1 } is a bivaluation for mbC if it satisfies the following clauses: (vAnd) ρ(α ∧ β) = 1 iff ρ(α) = ρ(β) = 1 (vOr) ρ(α ∨ β) = 1 iff ρ(α) = 1 or ρ(β) = 1 (vImp) ρ(α→ β) = 1 iff ρ(α) = 0 or ρ(β) = 1 (vNeg) ρ(¬α) = 0 implies ρ(α) = 1 (vCon) ρ(◦α) = 1 implies ρ(α) = 0 or ρ(¬α) = 0. The consequence relation of mbC w.r.t. bivaluations, denoted by |=2mbC, is given by: Γ |=2mbC φ iff ρ(φ) = 1 for every bivaluation for mbC such that ρ[Γ] ⊆ {1}. Theorem 4.3. (Adequacy of mbC w.r.t. bivaluations, [9, Theorems 56 and 61]) For every set of formulas Γ ∪ {φ} ⊆ LΣ: Γ ⊢mbC φ iff Γ |= 2 mbC φ. 6 Theorem 4.4. ([7, Theorem 6.4.9]) Any bivaluation ρ for mbC induces a valuation vρ over the Nmatrix M5 given by v ρ(α) def = (ρ(α), ρ(¬α), ρ(◦α)) such that, for every formula α: ρ(α) = 1 iff vρ(α) ∈ D. From the previous result, Theorem 4.1 follows easily (see [7, Corollary 6.4.10]). The characteristic Nmatrix M5 of mbC can be considered as the 'classical' model of it, since it is based on the 'classical' Boolean algebra A2. In Section 9 it will be shown that QmbC, the first-order version of mbC, can be characterized by first-order structures defined over M5. These structures can be considered as 'classical' in this sense. 5 The logic QmbC In this section the first-order logic QmbC, introduced in [10] (see also [7]) as an extension of mbC to first-order languages, will be briefly recalled. In Section 6 a new semantics of first-order swap structures for QmbC will be defined. Definition 5.1. Assume the propositional signature Σ = {∧,∨,→,¬, ◦} for mbC, as well as the symbols ∀ (universal quantifier) and ∃ (existential quantifier), with the punctuation marks (commas and parentesis). Let V ar = {v1, v2, . . .} be a denumerable set of individual variables. A first-order signature Θ for QmbC is composed by the following elements: a set C of individual constants; for each n ≥ 1, a set Fn of function symbols of arity n, for each n ≥ 1, a set Pn of predicate symbols of arity n. 2 Notation 5.2. Let Θ be a first-order signature for QmbC. The sets of terms and formulas generated by Θ from V ar we will denoted by Ter(Θ) and For(Θ), respectively. The set of sentences (formulas without free variables) and the set of closed terms (terms without variables) over Θ are denoted by Sen(Θ) and CTer(Θ), respectively. Given a formula φ, the formula obtained from φ by substituting every free occurrence of a variable x by a term t will be denoted by φ[x/t]. The notions of subformula, scope of an occurrence of a quantifier in a formula, free and bound occurrences of a variable in a formula, and of term free for a variable in a formula, are the usual ones (see, for instance, [26]). Definition 5.3. ([7, Definition 7.1.4]) Let φ and ψ be formulas. If φ can be obtained from ψ by means of addition or deletion of void quantifiers, or by renaming bound variables (keeping the same free variables in the same places), we say that φ and ψ are variant of each other. Definition 5.4. ([7, Definition 7.1.5]) Let Θ be a first-order signature. The logic QmbC is obtained from the Hilbert calculus mbC extended by the following axioms and rules: Axiom schemas: 2It will be assumed, as usual, that Θ has at least one predicate symbol, in order to have a nonempty set of formulas. For instance, it could be assumed from the beginning an equality predicate (see Section 10). 7 (Ax12) φ[x/t] → ∃xφ, if t is a term free for x in φ (Ax13) ∀xφ → φ[x/t], if t is a term free for x in φ (Ax14) α → β, whenever α is a variant of β Inference rules: (∃-In) φ→ ψ ∃xφ→ ψ , where x does not occur free in ψ (∀-In) φ→ ψ φ→ ∀xψ , where x does not occur free in φ Definition 5.5. If Γ ∪ {φ} ⊆ For(Θ), then Γ ⊢QmbC φ will denote that there exists a derivation in QmbC of φ from Γ. In [7] it was proved that the logic QmbC enjoys the Deduction meta-theorem (DMT), as usually presented in first-order logics: Theorem 5.6 (Deduction Meta-Theorem (DMT) for QmbC). Suppose that there exists in QmbC a derivation of ψ from Γ ∪ {φ}, such that no application of the rules (∃-In) and (∀-In) have, as their quantified variables, free variables of φ (in particular, this holds when φ is a sentence). Then Γ ⊢QmbC φ→ ψ. 6 First-Order Swap Structures The traditional approach to first-order structures based on algebraic structures (see for instance [27, 24, 29]) will be adapted to swap structures semantics. Thus, from now on the Boolean algebras to be considered are assumed to be complete.3 Definition 6.1. Let M(B) = (B, D) be a non-deterministic matrix defined by a swap structure B for mbC over a complete Boolean algebra A, and let Θ be a first-order signature (see Definition 5.1). A (first-order) structure over M(B) and Θ is pair A = 〈U, IA〉 such that U is a nonempty set (the domain of the structure) and IA is an interpretation function which assigns: to each individual constant c ∈ C, an element IA(c) of U ; to each function symbol f of arity n, a function IA(f) : U n → U ; to each predicate symbol P of arity n, a function IA(P ) : U n → |B|. From now on, the expressions cA, fA and PA will be used instead of IA(c), IA(f) and IA(P ), for an individual constant symbol c, a function symbol f and a predicate symbol P , respectively. Definition 6.2. Let A be a structure over M(B) and Θ. A function μ : V ar → U is called an assignment over A. 3Instead of this we could consider arbitrary Boolean algebras, thus obtaining partial models in which some quantified formulas has no denotation because of the lack of infima and/or suprema of some subsets. Since every Boolean algebra can be completed (see Section 8), we decide to restrict the semantic to complete Boolean algebras. An interesting discussion concerning this topic can be found in [16, footnote 3]. 8 Definition 6.3. Let A be a structure over M(B) and Θ, and let μ : V ar → U be an assignment. For each term t, we define [[t]]Aμ in U such that: - [[c]]Aμ = c A if c is an individual constant; - [[x]]Aμ = μ(x) if x is a variable; - [[f(t1, . . . , tn)]] A μ = f A([[t1]] A μ , . . . , [[tn]] A μ) if f is a function symbol of arity n and t1, . . . , tn are terms. Definition 6.4. Consider a structure A over M(B) and Θ. The diagram language of A is the set of formulas For(ΘU), where ΘU is the signature obtained from Θ by adding a new individual constant ā for each element a of the domain U of A. Definition 6.5. The structure Â = 〈U, I Â 〉 over ΘU is the structure A over Θ extended by I Â (ā) = a for every a ∈ U . Observe that sÂ = sA if s is a symbol (individual constant, function symbol or predicate symbol) of Θ. Notation 6.6. For any formula φ, FV (φ) will denote the set of free variables of φ. The set of (closed) sentences (formulas without free variables) of the diagram language of A will be denoted by Sen(ΘU), and the set of terms and of closed terms over ΘU will be denoted by Ter(ΘU) and CTer(ΘU), respectively. Remark 6.7. Clearly, if t is a closed term then the value of [[t]]Aμ does not depend on the assignment μ, that is: [[t]]Aμ = [[t]] A μ′ , for every assignments μ and μ ′. Thus, if t is a closed term we can write [[t]]A instead of [[t]]Aμ , for any assignment μ. Definition 6.8. Let Â = 〈U, I Â 〉 be as above. Any assignment μ over Â induces a function μ : (Ter(ΘU)∪For(ΘU)) → (Ter(ΘU)∪For(ΘU)) given by μ(s) = s[x1/μ(x1), . . . , xn/μ(xn)], if either s ∈ Ter(ΘU) such that V ar(s) ⊆ {x1, . . . , xn} or s ∈ For(ΘU) such that FV (s) ⊆ {x1, . . . , xn}. It is worth observing that μ(φ) ∈ Sen(ΘU) if φ ∈ For(ΘU), and μ(t) ∈ CTer(ΘU) if t ∈ Ter(ΘU). The next step is to define the notion of interpretation (or denotation) of a formula φ ∈ For(ΘU) in a given (extended) structure Â and assignment μ, which could be denoted by [[φ]]Aμ (being coherent with the previous notation). Is exactly at this point when non-determinism enters. Observe that, in the traditional (truth-functional or algebraic) first-order semantical approach, any structure and assignment induce together a (unique) denotation for any formula. In the present framework, this is also true for atomic formulas, since predicates are interpreted by means of functions, and taking into account that the denotation of any term is uniquely determined given a structure and an assignment. However, the denotation of complex formulas is possibly non-deterministic (i.e., ambiguous), given that it involves logical symbols (connectives and quantifiers) to be evaluated over a non-deterministic matrix. As happens with the propositional case, we are not interested in assigning sets of truth-values to single formulas: instead of this, valuations (legal valuations, in Avron and Lev's terminology) are used in order to choose, in a coherent way, a single truth-value for any formula.4 The definition of (legal) valuations over 4As we shall see in Section 12, in the case of the first-order logic LFI1◦, which is an axiomatic extension of QmbC based on a truth-functional 3-valued logic, any structure and assignment will determine a unique truth-value for any single formula. In this case, it will not necessary to consider valuations. 9 first-order swap structures involves an additional technical complication with respect to the propositional case: the validity of the Substitution Lemma – a crucial result which allows to substitute a universally quantified variable by any term free for such variable in a given formula – is far from being true in our non-deterministic environment. Indeed, this technical result is trivially true for first-order logics in which the semantics is obtained by algebraic manipulations over the interpretation of the subformulas of the formula being interpreted. Since in QmbC it is necessary to introduce the valuations as intermediaries between the formulas and the multioperators of the swap structures, such valuations must satisfy additional requirements in order to guarantee the validity of the Substitution Lemma (namely, clause (vi) in Definition 6.9 below). To summarize, in order to interpret formulas in the present non-deterministic framework, it is necesary a structure, an assignment, and a (first-order) valuation over the underlying swap structure, which will be called a QmbC-valuation. Given an assignment μ over a structure A, a variable x and a ∈ U , the assignment μxa over A is given by μxa(y) = a, if y = x, and μ x a(y) = μ(y) otherwise. Thus, the previous considerations lead us to the following notion: Definition 6.9. (QmbC-valuations) Let M(B) = (B, D) be a non-deterministic matrix defined by a swap structure B for mbC, and let A be a structure over Θ and M(B). A function v : Sen(ΘU)→ |B| is a valuation for QmbC (or a QmbC-valuation) over A and M(B), if it satisfies the following clauses: (i) v(P (t1, . . . , tn)) = P A([[t1]] Â, . . . , [[tn]] Â), if P (t1, . . . , tn) is atomic; 5 (ii) v(#φ) ∈ #v(φ), for every # ∈ {¬, ◦}; (iii) v(φ#ψ) ∈ v(φ)#v(ψ), for every # ∈ {∧,∨,→}; (iv) v(∀xφ) ∈ {z ∈ |B| : z1 = ∧ {π1(v(φ[x/ā])) : a ∈ U}}; (v) v(∃xφ) ∈ {z ∈ |B| : z1 = ∨ {π1(v(φ[x/ā])) : a ∈ U}}; (vi)] Let t be free for z in φ and ψ, μ an assignment and b = [[t]]Âμ . Then: (vi.1) If v(μ(φ[z/t])) = v(μ(φ[z/b])), then v(μ(#φ[z/t])) = v(μ(#φ[z/b])), for every # ∈ {¬, ◦}; (vi.2) If v(μ(φ[z/t])) = v(μ(φ[z/b])) and v(μ(ψ[z/t])) = v(μ(ψ[z/b])), then v(μ(φ#ψ[z/t])) = v(μ(φ#ψ[z/b])), for every # ∈ {∧,∨,→}; (vi.3) Let x be such that x 6= z and x does not occur in t. If v(μxa(φ[z/t])) = v(μ x a(φ[z/b])), for every a ∈ U , then v(μ((Qxφ)[z/t])) = v(μ((Qxφ)[z/b])), for every Q ∈ {∀, ∃}; (vii) If φ and φ′ are variant, then v(φ) = v(φ′). Observe that clause (i) in the previous definition is the only one that uses the information of the structure A, and it allows to interpret the atomic formulas. In order to obtain a single denotation for a complex formula, the valuation is used to choose (coherently) a denotation for the formula from the denotation of its components. Clause (vi) guarantees the validity of the Substitution Lemma, a crucial step for obtaining the soundness of the proposed semantics. Definition 6.10. Let v be a QmbC-valuation over A and M(B). Given an assignment μ over A, we define vμ : For(ΘU)→ |B| as vμ(φ) = v(μ(φ)). Definition 6.11. Let A be a structure over Θ and M(B). If Γ ∪ {φ} ⊆ For(ΘU), φ is said to be a semantical consequence of Γ over (A,M(B)), denoted by Γ |=(A,M(B)) φ, if the following holds: for every QmbC-valuation v over (A,M(B)), if vμ(γ) ∈ D, for every formula γ ∈ Γ and every assignment μ, then vμ(φ) ∈ D, for every assignment μ. 5For the notation used here, recall Remark 6.7. 10 Definition 6.12. (Semantical consequence relation in QmbC w.r.t. swap structures) If Γ ∪ {φ} ⊆ For(Θ), φ is said to be a semantical consequence of Γ in QmbC w.r.t. first-order swap structures, denoted by Γ |=QmbC φ, if Γ |=(A,M(B)) φ for every (A,M(B)). As mentioned in the Introduction, the semantical contexts (A,M(B)) for QmbC generalize the semantical contexts (A,MA) for first-order classical logic CFOL, where MA = 〈A, {1}〉. The latter, by its turn, generalize the class of standard Tarskian structures for CFOL with the usual semantics, by taking the two-element Boolean algebra A2. 7 Soundness of QmbC w.r.t. swap structures In this section the soundness of QmbC w.r.t. first-order swap structures semantics for QmbC will be proved. As mentioned in the previous section, a key result for proving soundness is the Substitution Lemma, which can be proved easily by induction on the complexity of φ. Theorem 7.1. (Substitution Lemma) Let v be a QmbC-valuation over A and M(B) and let μ be an assignment. If t is a term free for z in φ and b = [[t]]Âμ , then vμ(φ[z/t]) = vμ(φ[z/b]). A useful property of the semantics of the universal quantifier can be obtained now. The easy proof is ommited. Proposition 7.2. Let v be a QmbC-valuation over A and M(B), and let φ be a formula such that FV (φ) ⊆ {x1, . . . , xn}. Then, v(∀x1 . . .∀xnφ) ∈ D iff vμ(φ) ∈ D, for every μ. If α and β are formulas in For(Θ) then α↔ β will denote the formula (α→ β)∧(β → α) in For(Θ). Proposition 7.3. (i) α, α→ β |=QmbC β; (ii) α→ β |=QmbC ∃xα → β, if x /∈ FV (β); (iii) α→ β |=QmbC α → ∀xβ, if x /∈ FV (α); (iv) |=QmbC ∀xα → α[x/t], if t is a term free for x in α; (v) |=QmbC α[x/t] → ∃xα, if t is a term free for x in α; (vi) |=QmbC α ↔ α ′, if α and α′ are variant. Proof. (i): It is obvious from the definitions. (ii): Let v be a QmbC-valuation over (A,M(B)) such that vμ(α → β) ∈ D, for every assignment μ. Hence (vμ(α))1 ≤ (vμ(β))1, for every μ. From this, for every a ∈ U : (vμxa(α))1 ≤ (vμxa(β))1 = (vμ(β))1, since x /∈ FV (β). But then: (vμ(∃xα))1 =∨ a∈U(vμ(α[x/a]))1 = ∨ a∈U(vμxa(α))1 ≤ (vμ(β))1. Hence, vμ(∃xα → β) ∈ D, for every μ. This shows that α→ β |=QmbC ∃xα → β. Item (iii) is proved analogously. (iv): Assume that t is a term free for x in α. Let v be a QmbC-valuation over (A,M(B)) and let μ be an assignment. If b = [[t]]Aμ , then, by Theorem 7.1, vμ(α[x/t]) = vμ(α[x/b]). Then, (vμ(∀xα))1 = ∧ a∈U (vμ(α[x/ā))1 ≤ (vμ(α[x/b]))1 = (vμ(α[x/t]))1. Hence, vμ(∀xα → α[x/t]) ∈ D. Item (v) is proved analogously. (vi): Let v be a QmbC-valuation and let μ be an assignment. If α and α′ are variant, so are μ(α) and μ(α′). By Definition 6.9(vii), vμ(α↔ α ′) ∈ D. 11 Corollary 7.4. Let v be a QmbC-valuation over A and M(B). Then: (1) If α is an instance of a QmbC axiom schema then vμ(α) ∈ D, for every assignment μ. (2) If α and β are formulas such that vμ(α) ∈ D and vμ(α → β) ∈ D for every assignment μ, then vμ(β) ∈ D for every assignment μ. (3) If α and β are formulas such that vμ(α → β) ∈ D for every assignment μ, and if x does not occur free in β, then vμ(∃xα → β) ∈ D for every μ. (4) If α and β are formulas such that vμ(α → β) ∈ D, for every μ, and if x does not occur free in α, then vμ(α → ∀xβ) ∈ D for every μ. Proof. Item (1) follows by Theorem 2.2.2 in [7] and by Proposition 7.3(iv)-(vi). The rest of the proof follows by Proposition 7.3(i)-(iii). From this corollary it follows easily: Theorem 7.5. (Soundness of QmbC w.r.t. first-order swap structures) For every set Γ ∪ {φ} ⊆ For(Θ): if Γ ⊢QmbC φ, then Γ |=QmbC φ. 8 Completeness of QmbC w.r.t. swap structures In this section the completeness of QmbC w.r.t. first-order swap structures semantics for QmbC will be obtained. In order to do this, some definitions and results given in Sections 7.5.1 and 7.5.2 of [7] for proving the completeness theorem for QmbC w.r.t. interpretations will be adapted. In addition, the technique for proving the completeness of mbC w.r.t. swap structures presented in [14, Theorem 7.1] will be also used. The first step is considering a notion of C-Henkin theory a bit stronger than the one proposed in [7, Definition 7.5.1]. Definition 8.1. Consider a theory ∆ ⊆ Sen(Θ) and a nonempty set C of constants of the signature Θ. Then, ∆ is called a C-Henkin theory in QmbC if it satisfies the following: for every formula φ with (at most) a free variable x, there exists a constant c in C such that ∆ ⊢QmbC ∃xφ → φ[x/c]. Remark 8.2. Recall by [7, Section 2.4] that ⊥β def = β ∧ (¬β ∧ ◦β) is a bottom in mbC, hence ∼βα def = α → ⊥β is a classical negation in mbC. This construction does not depend on β (up to logical equivalence), hence we will write ∼α instead of ∼βα. This can be also done in QmbC. By [7, Proposition 7.2.2] it follows that ∃x∼φ→ ∼φ[x/c] ⊢QmbC φ[x/c] → ∀xφ. Thus, if ∆ is a C-Henkin theory in QmbC and φ is a formula with (at most) a free variable x then there is a constant c in C such that ∆ ⊢QmbC φ[x/c] → ∀xφ. Definition 8.3. Let ΘC be the signature obtained from Θ by adding a set C of new individual constants. The consequence relation ⊢CQmbC is the consequence relation of QmbC over the signature ΘC . Recall that, given a Tarskian and finitary logic L = 〈For,⊢〉 (where For is the set of formulas of L), and given a set Γ ∪ {φ} ⊆ For, the set Γ is said to be maximally nontrivial with respect to φ in L if the following holds: (i) Γ 0 φ, and (ii) Γ, ψ ⊢ φ for every ψ /∈ Γ. Proposition 8.4. ([7, Corollary 7.5.4]) Let Γ ∪ {φ} ⊆ Sen(Θ) such that Γ 0QmbC φ. Then, there exists a set of sentences ∆ ⊆ Sen(Θ) which is maximally non-trivial with respect to φ in QmbC (by restricting ⊢QmbC to sentences) and such that Γ ⊆ ∆. 12 Definition 8.5. Let ∆ ⊆ Sen(Θ) be non-trivial in QmbC, that is: there is some sentence φ in Sen(Θ) such that ∆ 0QmbC φ. Let ≡∆ ⊆ Sen(Θ) 2 be the relation in Sen(Θ) defined as follows: α ≡∆ β iff ∆ ⊢QmbC α↔ β. By adapting the proof of [14, Theorem 7.1] it follows that ≡∆ is an equivalence relation. Moreover, in the quotient set A∆ def = Sen(Θ)/≡∆ it is possible to define binary operators ∧, ∨, → as follows: [α]∆#[β]∆ def = [α#β]∆ for any # ∈ {∧,∨,→}, where [α]∆ denotes the equivalence class of formula α w.r.t. ≡∆. Using the axioms of QmbC coming from mbC it follows: Proposition 8.6. The structure A∆ def = 〈A∆, ∧, ∨, →, 0∆, 1∆〉 is a Boolean algebra with 0∆ def = [φ ∧ (¬φ ∧ ◦φ)]∆ and 1∆ def = [φ ∨ ¬φ]∆, for any sentence φ. In order to construct the canonical model for QmbC w.r.t. ∆, the Boolean algebra A∆ needs to be completed. Recall (see, for instance, [23, Chapter 25]) that a Boolean algebra B is a completion of a Boolean algebra A if: (1) B is complete, and (2) B includes A as a dense subalgebra (that is: every element in B is the supremum, in B, of some subset of A). As a consequence of the definition, it follows that B preserves all the existing infima and suprema in A. In formal terms: there exists a monomorphism of Boolean algebras (therefore an injective mapping) ∗ : A → B such that ∗( ∨ A X) = ∨ B ∗[X ] for every X ⊆ A such that the supremum ∨ A X exists, where ∗[X ] = {∗(a) : a ∈ X}. Analogously, ∗( ∧ A X) = ∧ B ∗[X ] for every X ⊆ A such that the infimum ∧ A X exists. By the (independent) results of MacNeille and Tarski, it is known that every Boolean algebra has a completion; moreover, the completion is unique up to isomorphisms. Thus, let CA∆ be the completion of A∆ and let ∗ : A∆ → CA∆ be the associated monomorphism. Definition 8.7. Let CA∆ be the complete Boolean algebra defined as above. The full swap structure for mbC over CA∆ (recall Definition 3.5) will be denoted by B∆. The associated Nmatrix (recall Definition 3.6) will be denoted by M(B∆) def = (B∆, D∆). Notice that (∗([α]∆), ∗([β]∆), ∗([γ]∆)) ∈ D∆ iff ∆ ⊢QmbC α. Definition 8.8. (Canonical Structure) Let Θ be a signature with some individual constant. Let ∆ ⊆ Sen(Θ) be non-trivial in QmbC, let M(B∆) be as in Definition 8.7, and let U = CTer(Θ). The canonical structure induced by ∆ is the structure A∆ = 〈U, IA∆〉 over M(B∆) and Θ such that: cA∆ = c, for each individual constant c; fA∆ : Un → U is such that fA∆(t1, . . . , tn) = f(t1, . . . , tn), for each function symbol f of arity n; PA∆(t1, . . . , tn) = (∗([φ]∆), ∗([¬φ]∆), ∗([◦φ]∆)) with φ = P (t1, . . . , tn), for each predicate symbol P of arity n. Notice that [φ]∆∨[¬φ]∆ = [φ ∨ ¬φ]∆ = 1 and [φ]∆∧[¬φ]∆∧[◦φ]∆ = [φ ∧ ¬φ ∧ ◦φ]∆ = 0. Thus ∗([φ]∆) ∨ ∗([¬φ]∆) = ∗([φ]∆∨[¬φ]∆) = 1, and ∗([φ]∆) ∧ ∗([¬φ]∆) ∧ ∗([◦φ]∆) = ∗([φ]∆∧[¬φ]∆∧[◦φ]∆) = 0. Hence P A∆(t1, . . . , tn) ∈ |B∆| and so A∆ is indeed a structure over M(B∆) and Θ. Definition 8.9. Let (*)⊲ : (Ter(ΘU) ∪ For(ΘU)) → (Ter(Θ) ∪ For(Θ)) be the mapping such that ( s )⊲ is the expression obtained from s by substituting every occurrence of a constant t by the term t itself, for t ∈ CTer(Θ). 13 For instance, ( P (f(c, x)) ∧Q ( f(c, x), z ) )⊲ = P (f(c, x)) ∧Q(f(c, x), z). Definition 8.10. (Canonical valuation) Let ∆ ⊆ Sen(Θ) be a set of sentences over a signature Θ such that ∆ is a C-Henkin theory in QmbC for a nonempty set C of individual constants of Θ, and ∆ is maximally non-trivial with respect to φ in QmbC, for some sentence φ. The canonical QmbC-valuation induced by ∆ over A∆ and M(B∆) is the function v∆ : Sen(ΘU) → |B∆| such that v∆(ψ) = (∗([(ψ) ⊲]∆), ∗([¬(ψ) ⊲]∆), ∗([◦(ψ) ⊲]∆)), for every sentence ψ over ΘU . Notice that v∆(ψ) ∈ D∆ iff ∆ ⊢QmbC (ψ) ⊲. Lemma 8.11. Let ∆ ⊆ Sen(Θ) be as in Definition 8.10. Then, for every formula ψ(x) in which x is the unique variable (possibly) occurring free, it holds: (1) [∀xψ]∆ = ∧ A∆ {[ψ[x/t]]∆ : t ∈ CTer(Θ)}, where ∧ A∆ denotes an existing infimum in the Boolean algebra A∆; (2) [∃xψ]∆ = ∨ A∆ {[ψ[x/t]]∆ : t ∈ CTer(Θ)}, where ∨ A∆ denotes an existing supremum in the Boolean algebra A∆. Proof. (2) Observe that, in A∆, [α]∆ ≤ [β]∆ iff ∆ ⊢QmbC α → β. Let α be a formula in which x is the unique variable (possibly) occurring free. Then [α[x/t]]∆ ≤ [∃xα]∆ for every t ∈ CTer(Θ), by (Ax12). Let β be a sentence such that [α[x/t]]∆ ≤ [β]∆ for every t ∈ CTer(Θ). That is, ∆ ⊢QmbC α[x/t] → β for every t ∈ CTer(Θ). Since ∆ is a C-Henkin theory, there is a constant c of Θ such that ∆ ⊢QmbC ∃xα → α[x/c]. By hypothesis, ∆ ⊢QmbC α[x/c] → β and so ∆ ⊢QmbC ∃xα → β. That is, [∃xα]∆ ≤ [β]∆. This shows that [∃xα]∆ = ∨ A∆ {[α[x/t]]∆ : t ∈ CTer(Θ)}. Item (1) is proved analogously, but now by using Remark 8.2. Theorem 8.12. The canonical QmbC-valuation v∆ is a QmbC-valuation over A∆ and M(B∆). Proof. Let us see that v∆ satisfies all the requirements of Definition 6.9. (i) If φ is an atomic formula P (t1, . . . , tn) then: v∆(φ) = (∗([(φ) ⊲]∆), ∗([¬(φ) ⊲]∆), ∗([◦(φ) ⊲]∆)) = P A∆((t1) ⊲, . . . , (tn) ⊲) = PA∆([[t1]] Â∆ , . . . , [[tn]] Â∆). (ii) v∆(¬ψ) = (∗([¬(ψ) ⊲]∆), ∗([¬¬(ψ) ⊲]∆), ∗([◦¬(ψ) ⊲]∆)) ∈ ¬v∆(ψ). On the other hand, v∆(◦ψ) = (∗([◦(ψ) ⊲]∆), ∗([¬◦(ψ) ⊲]∆), ∗([◦◦(ψ) ⊲]∆)) ∈ ◦v∆(ψ). (iii) Since ∗([δ#ψ]∆) = ∗([δ]∆) # ∗([ψ]∆), then v∆(δ#ψ) ∈ v(δ)#v∆(ψ), for every # ∈ {∧,∨,→}. (iv) By Lemma 8.11 (and recalling that U = CTer(Θ)), [∀xψ]∆ = ∧ A∆ {[ψ[x/t]]∆ : t ∈ U} and so ∗([∀xψ]∆) = ∧ CA∆ {∗([ψ[x/t]]∆) : t ∈ U}. Then, (v∆(∀xψ))1 = ∗([(∀xψ)⊲]∆) = ∧ t∈U ∗([(ψ[x/t]) ⊲]∆) = ∧ t∈U(v∆(ψ[x/t]))1. (v) The case ∃xψ is treated analogously. (vi) Let t be free for z in φ, μ an assignment and b = [[t]]Â∆μ . By induction on the complexity of φ it can be proved that (μ(φ[z/t]))⊲ = (μ(φ[z/b]))⊲. Hence, v∆(μ(φ[z/t])) = v∆(μ(φ[z/b])), by definition of v∆. From this, it is obvious that v∆ satisfies conditions (vi.1)-(vi.3). (vii) If φ and φ′ are variant, so are (φ)⊲ and (φ′)⊲; (¬φ)⊲ and (¬φ′)⊲; and (◦φ)⊲ and (◦φ′)⊲. From this, v∆(φ) = v∆(φ ′), by axiom (Ax14). 14 Theorem 8.13. (Completeness of QmbC restricted to sentences w.r.t. first-order swap structures) Let Γ ∪ {φ} ⊆ Sen(Θ). If Γ |=QmbC φ then Γ ⊢QmbC φ. Proof. Let Γ ∪ {φ} ⊆ Sen(Θ) such that Γ 0QmbC φ. Then, by recalling Definition 8.1 and by Theorem 7.5.3 in [7],6 there exists a C-Henkin theory ∆H over ΘC in QmbC for a nonempty set C of new individual constants such that Γ ⊆ ∆H and, for every α ∈ Sen(Θ): Γ ⊢QmbC α iff ∆ H ⊢CQmbC α. Hence, ∆ H 0 C QmbC φ. Now, by Proposition 8.4, there exists a set of sentences ∆H in ΘC extending ∆ H which is maximally non-trivial with respect to φ in QmbC (defined over Sen(ΘC)), such that ∆H is a C-Henkin theory over ΘC in QmbC. Let M(B∆H), A∆H and v∆H be as in Definitions 8.7, 8.8 and 8.10, respectively. Then, v∆H (α) ∈ D∆H iff ∆ H ⊢CQmbC α, for every α in Sen(ΘC). From this, v ∆H [Γ] ⊆ D ∆H and v ∆H (φ) /∈ D ∆H . Finally, let A and v be the respective restrictions of A ∆H and v ∆H to Θ. Then, A is a structure over M(B ∆H ), and v is a valuation for QmbC over A and M(B∆H ) such that v[Γ] ⊆ D∆H but v(φ) /∈ D∆H . This shows that Γ 6|=QmbC φ. For any formula ψ in For(Θ) let (∀)ψ be the universal closure of ψ, defined as follows: if ψ is a sentence then (∀)ψ def = ψ; and if ψ has exactly the variables x1, . . . , xn occurring free then (∀)ψ def = (∀x1) * * * (∀xn)ψ. Note that (∀)ψ ∈ Sen(Θ). If Γ is a set of formulas in For(Θ) then (∀)Γ def = {(∀)ψ : ψ ∈ Γ}. It is easy to show that, for every Γ ∪ {φ} ⊆ For(Θ): (i) Γ ⊢QmbC φ iff (∀)Γ ⊢QmbC (∀)φ; and (ii) Γ |=QmbC φ iff (∀)Γ |=QmbC (∀)φ. From this, a general completeness result can be obtained: Corollary 8.14. (Completeness of QmbC w.r.t. first-order swap structures) Let Γ ∪ {φ} ⊆ For(Θ). If Γ |=QmbC φ then Γ ⊢QmbC φ. 9 Completeness of QmbC w.r.t. structures over M5 Recall from Section 4 the 5-valued Nmatrix M5 introduced by Avron in [1]. From the adequacy of QmbC w.r.t. first-order swap structures, and given that mbC can be characterized just with M5 = M ( BA2 ) , it is a natural question to determine if it is possible to extend the proof of [7, Theorem 6.4.9 and Corollary 6.4.10] (see Theorem 4.4 above) to QmbC. Namely, taking into account that QmbC can be characterized by standard Tarskian structures expanded with bivaluations which naturally extend the ones for mbC (see Theorem 9.3 below), it seems plausible to extend the technique of Theorem 4.4 to QmbC. In Theorem 9.6 below it will be shown that this is really the case, hence QmbC can be characterized by first-order structures over M5. Such structures, which were introduced by Avron and Zamansky in [6] (see Remark 9.4 below), can be considered as being 'classical', as discussed at the end of Section 4. Consider a standard Tarskian first-order structure A = 〈U, IA〉 over a first-order signature Θ (see, for instance, [26]). Observe that A is defined as in Definition 6.1, but now any predicate symbol P of arity n is interpreted as a subset IA(P ) of U n. The notions of diagram language For(ΘU), extended structure Â and Sen(ΘU) are defined as in Definitions 6.4 and 6.5, and Notation 6.6 above. In [7] the notion of bivaluations for mbC was extended to bivaluations for QmbC as follows: 6The proof of Theorem 7.5.3 in [7] also holds for the notion of C-Henkin theory adopted here in Definition 8.1 (which is stronger than the one proposed in [7, Definition 7.5.1]), as it can be easily verified. 15 Definition 9.1. (Bivaluations for QmbC, [7, Definition 7.3.5]) Let A = 〈U, IA〉 be a standard Tarskian first-order structure over Θ, and let Â = 〈U, I Â 〉 be the expansion of A to ΘU by setting IÂ(ā) = a for every a ∈ U . A bivaluation 7 for QmbC over A is a function ρ : Sen(ΘU) → {0, 1} satisfying the clauses of Definition 4.2 above plus the following: (vPred) ρ(P (t1, . . . , tn)) = 1 iff 〈[[t1]] Â, . . . , [[tn]] Â〉 ∈ IA(P ), for P (t1, . . . , tn) atomic (vVar) ρ(φ) = ρ(ψ) whenever φ is a variant of ψ (vUni) ρ(∀xφ) = 1 iff ρ(φ[x/ā]) = 1 for every a ∈ U (vEx ) ρ(∃xφ) = 1 iff ρ(φ[x/ā]) = 1 for some a ∈ U (vSubs) if μ is an assignment, t is a term free for z in φ and b = [[t]]Âμ , then: ρ(μ(φ[z/t])) = ρ(μ(φ[z/b]) implies ρ(μ(#φ[z/t])) = ρ(μ(#φ[z/b])), for # ∈ {¬, ◦}. Definition 9.2. ([7, Definitions 7.3.6 and 7.3.12]) An interpretation for QmbC over a signature Θ is a pair 〈A, ρ〉 such that A is a Tarskian first-order structure over Θ and ρ is a bivaluation for QmbC over A. The consequence relation |=2QmbC of QmbC w.r.t. interpretations is given by: Γ |=2QmbC φ if, for every interpretation 〈A, ρ〉: ρ(μ(γ)) = 1 for every γ ∈ Γ and every assignment μ implies that ρ(μ(φ)) = 1 for every assignment μ. Theorem 9.3. (Adequacy of QmbC w.r.t. interpretations, [7, Theorems 7.4.1. and 7.5.6]) If Γ ∪ {φ} ⊆ For(Θ) then: Γ ⊢QmbC φ iff Γ |= 2 QmbC φ. 8 Now, Theorem 4.4 will be extended to QmbC (see Theorem 9.5 below). Previous to this, it is worth observing the following: Remark 9.4. Consider once again the characteristic 5-valued Nmatrix M5 = M ( BA2 ) for mbC, and let A be a structure over Θ and M5. If v is a valuation for QmbC over A and M5, it is easy to see that clauses (iv) and (v) of Definition 6.9 are equivalent to the following: for every Q ∈ {∀, ∃}, v(Qxφ) ∈ Q ( {v(φ[x/ā]) : a ∈ U} ) where Q : (P(BA2) − {∅}) → (P(BA2) − {∅}) is ∀(X) = { D, if X ⊆ D ND, otherwise and ∃(X) = { D, if X ∩ D 6= ∅ ND, otherwise and BA2 = |M5| = { T, t, t0, F, f0 } , D = {T, t, t0} and ND = {F, f0} (recall Section 4). It is not hard to prove that the notions of structures over Θ and M5, and valuations over them, coincide with the corresponding notions introduced in [6]. Thus, the present framework generalizes, from A2 to arbitrary complete Boolean algebras, the semantical framework proposed in [6]. Theorem 9.5. Let I = 〈A, ρ〉 be an interpretation for QmbC over a signature Θ. Then, it induces a first-order structure AI over M5 and Θ, and a QmbC-valuation v ρ over AI and M5 given by v ρ(α) def = (ρ(α), ρ(¬α), ρ(◦α)) such that: ρ(α) = 1 iff vρ(α) ∈ D, for every sentence α ∈ Sen(ΘU). 7It was called QmbC-valuation in [7, Definition 7.3.5]. 8Rigourously speaking, in [7, Theorem 7.5.6] it was obtained completeness of QmbC w.r.t. interpretations, but only for sentences. However, completeness of QmbC (in the full language) w.r.t. interpretations follows easily, as we have done here in Corollary 8.14. 16 Proof. Given I = 〈A, ρ〉 consider the first-order structure AI over M5 and Θ obtained from A by taking the same domain U ; IAI coincides with IA for every individual constant and function symbol; and IAI (P ) : U n → |M5| is given by IAI(P )(a1, . . . , an) = vρ(P (ā1, . . . , ān)) for every predicate symbol P of arity n, where v ρ : Sen(ΘU ) → |M5| is defined by vρ(α) = (ρ(α), ρ(¬α), ρ(◦α)), for every α ∈ Sen(ΘU). Clearly ρ(α) = 1 iff vρ(α) ∈ D, for every α ∈ Sen(ΘU). Thus, it remains to prove that v ρ is indeed a QmbC-valuation over AI and M5. It is clear that clauses (i)-(iii) of Definition 6.9 are satisfied, since [[t]]ÂI = [[t]]Â for every closed term t, and by Theorem 4.4. With respect to clause (iv), suppose that {vρ(φ[x/ā]) : a ∈ U} ⊆ D. Then ρ(φ[x/ā]) = 1 for every a ∈ U and so ρ(∀xφ) = 1, by (vUni). Hence vρ(∀xφ) ∈ D. This means that vρ(∀xφ) ∈ ∀ ( {vρ(φ[x/ā]) : a ∈ U} ) . If {vρ(φ[x/ā]) : a ∈ U} 6⊆ D then, by a similar reasoning, it is shown that, once again, vρ(∀xφ) ∈ ∀ ( {vρ(φ[x/ā]) : a ∈ U} ) . Analogously it can be proven that vρ satisfies clause (v). Clause (vi) is satisfied by vρ, since [[t]]ÂIμ = [[t]] Â μ for every term t, and by the fact that ρ satisfies the Substitution Lemma: ρ(μ(φ[z/t])) = ρ(μ(φ[z/b])) for b = [[t]]Âμ . Clause (vii) is also satisfied, since ρ satisfies (vVar). This concludes the proof. Theorem 9.6. (Adequacy of QmbC w.r.t. first-order structures over M5) For every set Γ ∪ {φ} ⊆ For(Θ): Γ ⊢QmbC φ iff Γ |=(A,M5) φ for every structure A over Θ and M5. Proof. 'Only if' part (Soundness): It is a consequence of Theorem 7.5. 'If' part (Completeness): Suppose that Γ |=(A,M5) φ for every structure A over Θ and M5. Let I = 〈A, ρ〉 be an interpretation for QmbC over Θ such that ρ(μ(γ)) = 1 for every γ ∈ Γ and every assignment μ. Let AI and v ρ be as in Theorem 9.5. Then vρμ(γ) ∈ D, for every formula γ ∈ Γ and every assignment μ. By hypothesis, v ρ μ(φ) ∈ D, for every assignment μ. This implies that ρ(μ(φ)) = 1, for every assignment μ. That is, Γ |=2QmbC φ. Therefore Γ ⊢QmbC φ, by Theorem 9.3. Taking into account Remark 9.4, the last result is a restatement of the adequacy for QmbC w.r.t. first-order structures over M5 obtained in [6, Theorem 24]. 10 Adding standard equality to QmbC In this section a binary predicate ≈ for dealing with equality will be considered. As expected, this predicate will be always interpreted as the standard identity. This means that the predicate ≈ will be seen, from a semantical point of view, as a logical symbol. The resulting logic will be called QmbC≈. The definition of QmbC≈ will follows closely [7, Section 7.7]. Definition 10.1. Let Θ be a first-order signature. The induced signature with equality Θ≈ is obtained from Θ by adding a new binary predicate symbol ≈. The expression (t1 ≈ t2) will stands for the atomic formula ≈ (t1, t2). If φ is a formula and y is a variable free for the variable x in φ, φ[x ≀ y] denotes any formula obtained from φ by replacing some, but not necessarily all (maybe none), free occurrences of x by y. Definition 10.2. ([7, Definition 7.7.1]) Let Θ≈ be a first-order signature with equality. The logic QmbC≈ (over Θ≈) is the extension of QmbC over For(Θ≈) obtained by 17 adding to QmbC, besides all the new instances of axioms and inference rules involving the equality predicate ≈, the following axiom schemas: (AxEq1) ∀x(x ≈ x) (AxEq2) (x ≈ y) → (φ→ φ[x ≀ y]), if y is a variable free for x in φ Notice that the axioms for equality are the same considered for classical logic (see, for instance, [26]). Given that QmbC≈ is an axiomatic extension of QmbC, it satisfies the deduction meta-theorem DMT (recall Theorem 5.6). Let ⊢QmbC≈ be the consequence relation of the Hilbert calculus QmbC≈. The semantics of first-order swap structures for QmbC can be easily extended to the equality predicate. Definition 10.3. Let M(B) = (B, D) be a non-deterministic matrix defined by a swap structure B for mbC, and let Θ≈ be a first-order signature with equality. A (first-order) structure with standard equality over M(B) and Θ≈ is a structure over M(B) and Θ≈ such that IA(≈)(a, b) ∈ D iff a = b. In what follows, (a ≈A b) will stands for IA(≈)(a, b), for every structure A and any a, b ∈ U . Given a structure A, the signature obtained from Θ by adding a new individual constant for each element of U (recall Definition 6.4) will be denoted by Θ≈U . The set of formulas and sentences over Θ≈U will be denoted by For(Θ ≈ U) and Sen(Θ ≈ U), respectively. If A is a structure with standard equality over M(B) and v is a QmbC-valuation over A and M(B) then, by Definition 6.9 (i) it follows that, for every closed terms t1 and t2, v(t1 ≈ t2) ∈ D iff [[t1]] A = [[t2]] A. This guarantees the validity of axiom (AxEq1). However, in order to validate axiom (AxEq2), the valuations must be additionally restricted: Definition 10.4. (QmbC≈-valuations) Let A be a structure with standard equality over Θ≈ and M(B). A valuation for QmbC ≈ (or a QmbC≈-valuation) over A and M(B) is a QmbC-valuation v : Sen(Θ≈U) → |B| which satisfies, in addition, the following clause, for every μ: (viii) vμ((x ≈ y) → (φ→ φ[x ≀ y])) ∈ D, if y is a variable free for x in φ. The notion of φ being a ≈-semantical consequence of Γ over (A,M(B)), denoted by Γ |=≈(A,M(B)) φ, is as stated in Definition 6.11, but now restricted to structures with standard equality and QmbC≈-valuations over them. Thus, φ is a semantical consequence of Γ in QmbC≈ w.r.t. first-order swap structures, denoted by Γ |=QmbC≈ φ, if Γ |= ≈ (A,M(B)) φ for every of such pairs (A,M(B)). Observe that the Substitution Lemma still holds for QmbC≈, since it holds for any structure and any QmbC-valuation. From this, and from Definition 10.4, the following result can be easily derived by adapting the proof of Theorem 7.5: Theorem 10.5. (Soundness of QmbC≈ w.r.t. first-order swap structures with standard equality) For every set Γ ∪ {φ} ⊆ For(Θ≈): if Γ ⊢QmbC≈ φ, then Γ |=QmbC≈ φ. In order to prove completeness of QmbC≈ w.r.t. swap structures semantics, the proof given in Section 8 will be adapted, in accordance with the argument given in [7, Section 7.7]. We begin by observing that the notion of C-Henkin theory in QmbC≈ can be defined by adapting Definition 8.1 in an obvious way. The signature obtained from Θ≈ by adding a set C of new individual constants will be denoted by Θ≈C , and the consequence relation in QmbC≈ over that signature will be denoted by ⊢C QmbC≈ . Clearly, Proposition 8.4 also 18 holds for QmbC≈. This result, combined with [7, Theorem 7.5.3] (which can also be easily adapted to QmbC≈) produces the following: Proposition 10.6. Let Γ ∪ {φ} ⊆ Sen(Θ≈) such that Γ 0QmbC≈ φ. Then, there exists a set of sentences ∆ ⊆ Sen(Θ≈C), for some set C of new individual constants, such that Γ ⊆ ∆, it is a C-Henkin theory in QmbC≈, and it is maximally non-trivial with respect to φ in QmbC≈ (by restricting ⊢C QmbC≈ to sentences in Sen(Θ≈C)). Definition 10.7. Let ∆ ⊆ Sen(Θ≈) be non-trivial in QmbC ≈, that is: there is some sentence φ in Sen(Θ≈) such that ∆ 0QmbC≈ φ. Let ≡ ≈ ∆ ⊆ Sen(Θ≈) 2 be the relation in Sen(Θ≈) defined as follows: α ≡ ≈ ∆ β iff ∆ ⊢QmbC≈ α↔ β. Then ≡≈∆ is an equivalence relation which induces a Boolean algebra A ≈ ∆ whose domain is the quotient set A≈∆ def = Sen(Θ≈)/≡≈ ∆ such that the operations are defined as follows (here [α]≈∆ denotes the equivalence class of α w.r.t. ≡ ≈ ∆): [α] ≈ ∆ # [β] ≈ ∆ def = [α#β]≈∆ for any # ∈ {∧,∨,→}; 0≈∆ def = [φ ∧ ¬φ ∧ ◦φ]≈∆, and 1 ≈ ∆ def = [φ ∨ ¬φ]≈∆ (for any sentence φ). Definition 10.8. Let A≈∆ be a Boolean algebra defined as above, and let CA ≈ ∆ be its completion with monomorphism ∗ (recall Section 8). The full swap structure for mbC over CA≈∆ will be denoted by B ≈ ∆, and the associated Nmatrix will be denoted by M(B≈∆) def = (B≈∆, D ≈ ∆). Remark 10.9. Note that (∗([α]≈∆), ∗([β] ≈ ∆), ∗([γ] ≈ ∆)) ∈ D ≈ ∆ iff ∆ ⊢QmbC≈ α. Definition 10.10. (Canonical Structure in QmbC≈) Let ∆ ⊆ Sen(Θ≈C) be a non-trivial and C-Henkin theory in QmbC≈. Define in the set C of constants the following relation: c ≃ d iff ∆ ⊢C QmbC≈ (c ≈ d). By the axioms of equality it follows that ≃ is an equivalence relation. For any c ∈ C let c = {d ∈ C : c ≃ d} be the equivalence class of c w.r.t. ≃, and let U = {c : c ∈ C} be the corresponding quotient set. Let M(B≈∆) be as in Definition 10.8. The canonical structure induced by ∆ in QmbC≈ is the structure A≈∆ = 〈U, IA≈∆〉 over Θ ≈ C and M(B ≈ ∆) defined as follows: if c is an individual constant in Θ≈C then IA≈∆(c) = d, where d ∈ C is such that ∆ ⊢C QmbC≈ (c ≈ d); if f is a function symbol, IA≈ ∆ (f) : Un → U is given by IA≈ ∆ (f)(c1, . . . , cn) = c, where c ∈ C is such that ∆ ⊢C QmbC≈ (f(c1, . . . , cn) ≈ c); if P is a predicate symbol, then IA≈ ∆ (P ) is given by IA≈ ∆ (P )(c1, . . . , cn) = (∗([P (c1, . . . , cn)] ≈ ∆), ∗([¬P (c1, . . . , cn)] ≈ ∆), ∗([◦P (c1, . . . , cn)] ≈ ∆)). The proof that IA≈ ∆ is well-defined for individual constants and function symbols is similar to that for classical logic (see [12]). Let P be predicate symbol of arity n and let (c1, . . . , cn) ∈ U n. Let d1, . . . , dn ∈ C such that ci ≃ di for every 1 ≤ i ≤ n. Then ∆ ⊢C QmbC≈ (ci ≈ di) for 1 ≤ i ≤ n. It is immediate that ∆ ⊢ C QmbC≈ ( ∧n i=1(ci ≈ di)) → (P (c1, . . . , cn) ↔ P (d1, . . . , dn)), hence ∆ ⊢ C QmbC≈ (P (c1, . . . , cn) ↔ P (d1, . . . , dn)). Analogously it can be proven that ∆ ⊢C QmbC≈ (¬P (c1, . . . , cn) ↔ ¬P (d1, . . . , dn)) and ∆ ⊢ C QmbC≈ (◦P (c1, . . . , cn) ↔ ◦P (d1, . . . , dn)). This shows that IA≈ ∆ (P ) is well-defined. Moreover, by similar considerations to the ones given after Definition 8.8, it follows that IA≈ ∆ (P )(c1, . . . , cn) belongs to |B≈∆| for every (c1, . . . , cn) ∈ U n. 19 Proposition 10.11. Let ∆ ⊆ Sen(Θ≈C) be a non-trivial and C-Henkin theory in QmbC ≈ and let M(B≈∆) be as in Definition 10.8. Then the canonical structure A ≈ ∆ induced by ∆ in QmbC≈is a structure with standard equality over Θ≈C and M(B ≈ ∆). Proof. As it was shown above, the mapping IA≈ ∆ is well-defined, and IA≈ ∆ (P ) is a function from Un to |B≈∆|. Let c1, c2 ∈ U . Then (c1 ≈ A≈ ∆ c2) ∈ D ≈ ∆ iff ∆ ⊢ C QmbC≈ (c1 ≈ c2), by Definition 10.10 and by Remark 10.9, iff c1 ≃ c2 iff c1 = c2. This shows that A ≈ ∆ is indeed a structure with standard equality. Definition 10.12. Let (*)⊳ : (Ter((Θ≈C)U) ∪ For((Θ ≈ C)U)) → (Ter(Θ ≈ C) ∪ For(Θ ≈ C)) be the mapping recursively defined as in Definition 8.9, but with the following difference:( c )⊳ = d for some d ∈ c previously chosen, for every c ∈ U . Observe that, if s ∈ Ter((Θ≈C)U) ∪ For((Θ ≈ C)U), then (s) ⊳ is the expression in Ter(Θ≈C) ∪ For(Θ≈C) obtained from s by substituting every occurrence of a constant c by a constant d ∈ c. Suppose that (*)⊳ ′ : (Ter((Θ≈C)U)∪For((Θ ≈ C)U)) → (Ter(Θ ≈ C)∪For(Θ ≈ C)) is defined as (*)⊳, but now ( c )⊳′ = d′ for another choice of d′ ∈ c (possibly different to d), for every c ∈ U . Then, it is easy to prove that ∆ ⊢C QmbC≈ (ψ)⊳ ↔ (ψ)⊳ ′ for every sentence ψ. This shows that the choice of each d ∈ c in order to define ( c )⊳ , for every c ∈ U , is irrelevant. Proposition 10.13. Let ∆ ⊆ Sen(Θ≈C) be a set of sentences over the signature Θ ≈ C such that ∆ is a C-Henkin theory in QmbC≈ which is also maximally non-trivial with respect to φ in QmbC≈, for some sentence φ. Then, the canonical QmbC-valuation induced by ∆ over A≈∆ and M(B ≈ ∆) (see Definition 8.10) is a QmbC ≈-valuation, which will be denoted by v≈∆, such that v ≈ ∆(ψ) ∈ D ≈ ∆ iff ∆ ⊢ C QmbC≈ (ψ)⊳. Proof. Observe that, by the very definitions, v≈∆(ψ) ∈ D ≈ ∆ iff ∆ ⊢ C QmbC≈ (ψ)⊳ (and, as observed above, 'v≈∆(ψ) ∈ D ≈ ∆' does not depend on the choices made by (*) ⊳). Hence, it suffices to prove that v≈∆ satisfies clause (viii) of Definition 10.4. Thus, let α = (x ≈ y) → (ψ → ψ[x≀y]) (where y is a variable free for x in φ) be an instance of axiom (AxEq2), and let μ be an assignment. Given that ∆ is a closed theory in QmbC≈ over Θ≈C , it follows that ∆ ⊢C QmbC≈ (μ(α))⊳. Then v≈∆(μ(α)) ∈ D ≈ ∆, showing that v ≈ ∆ satisfies clause (viii). Theorem 10.14. (Completeness of QmbC≈ restricted to sentences w.r.t. first-order swap structures with standard equality) Let Γ ∪ {φ} ⊆ Sen(Θ≈). If Γ |=QmbC≈ φ then Γ ⊢QmbC≈ φ. Proof. Let Γ ∪ {φ} ⊆ Sen(Θ≈) such that Γ 0QmbC≈ φ. By Proposition 10.6, there exists a set of sentences ∆ ⊆ Sen(Θ≈C), for some set C of new individual constants, such that Γ ⊆ ∆, ∆ is a C-Henkin theory in QmbC≈, and it is maximally non-trivial with respect to φ in QmbC≈ (by restricting ⊢C QmbC≈ to sentences in Sen(Θ≈C)). Now, let M(B≈∆), A ≈ ∆ and v ≈ ∆ be as in Definitions 10.8 and 10.10, and as in Proposition 10.13, respectively. Then, v≈∆(α) ∈ D ≈ ∆ iff ∆ ⊢ C QmbC≈ α, for every α in Sen(Θ≈C) (by observing that (α)⊳ = α if α ∈ Sen(Θ≈C)). From this, v ≈ ∆[Γ] ⊆ D ≈ ∆ and v ≈ ∆(φ) /∈ D ≈ ∆. Finally, let A and v be the restriction to Θ≈ of A ≈ ∆ and v ≈ ∆, respectively. Then, A is a structure with standard equality over M(B≈∆), and v is a valuation for QmbC ≈ over A and M(B≈∆) such that v[Γ] ⊆ D ≈ ∆ but v(φ) /∈ D ≈ ∆. This shows that Γ 6|=QmbC≈ φ. By using universal closure, as it was done in Corollary 8.14, a general completeness result can be obtained for QmbC≈: 20 Corollary 10.15. (Completeness of QmbC≈ w.r.t. first-order swap structures with standard equality) Let Γ ∪ {φ} ⊆ For(Θ≈). If Γ |=QmbC≈ φ then Γ ⊢QmbC≈ φ. 11 Completeness of QmbC≈ w.r.t. structures with standard equality over M5 In this section the adequacy of QmbC w.r.t. first-order swap structures stated in Theorem 9.6 will be extended to QmbC≈. In order to do this, some definitions taken from Section 9 will be adapted to QmbC≈, by following the approach in [7, Section 7.7] with small modifications. In particular, [7, Definition 7.7.3] will be slightly adapted as follows: Definition 11.1. An interpretation for QmbC≈ over a signature Θ≈ is a pair 〈A, ρ〉 such that A is a standard Tarskian first-order structure with standard equality over Θ≈ 9 and ρ is a bivaluation for QmbC over A. The consequence relation |=2 QmbC≈ of QmbC≈ w.r.t. interpretations is given by: Γ |=2 QmbC≈ φ if, for every interpretation 〈A, ρ〉 for QmbC≈: ρ(μ(γ)) = 1 for every γ ∈ Γ and every μ implies that ρ(μ(φ)) = 1 for every μ. Remark 11.2. (1) In [7, Definition 7.7.3] it was introduced the notion of QmbC≈-valuations, which are bivaluations for QmbC over standard Tarskian structures A over Θ≈ satisfying for ≈, instead of (vPred), the following clauses: (vEq1) ρ(t1 ≈ t2) = 1 iff [[t1]] Â = [[t2]] Â for every t1, t2 ∈ CTer(Θ ≈ U); (vEq2) ρ(ā ≈ b) = 1 implies ρ(α[x, y/ā, b]) = ρ(α[x ≀ y][x, y/ā, b]) for every a, b ∈ U , if y is a variable free for x in α. It is easy to see that (vEq2) is derivable from (vEq1). Indeed, suppose that ρ(ā ≈ b) = 1. Hence a = [[ā]]Â = [[b]]Â = b, by clause (vEq1). But then ā = b, which implies that α[x, y/ā, b] = α[x ≀ y][x, y/ā, b]. Thus, ρ(α[x, y/ā, b]) = ρ(α[x ≀ y][x, y/ā, b]), showing that ρ also satisfies (vEq2). (2) Let 〈A, ρ〉 be an interpretation for QmbC≈ as in Definition 11.1. Then, by (vPred) applied to ≈ (recall Definition 9.1), and by the fact that ≈A= {(a, a) : a ∈ U}, it follows that ρ satisfies clause (vEq1), hence it also satisfies (vEq2), by item (1) above. This means that 〈A, ρ〉 is an interpretation for QmbC≈ in the sense of [7, Definition 7.7.3]. Conversely, if 〈A, ρ〉 is an interpretation for QmbC≈ in the sense of [7, Definition 7.7.3] let A′ be the standard Tarskian structure over Θ≈ obtained from A by setting ≈ A′ def= {(a, a) : a ∈ U}. Hence 〈A′, ρ〉 is an interpretation for QmbC≈ as in Definition 11.1, since ρ satisfies (vEq1) and so it satisfies (vPred) applied to ≈. This shows that our presentation is equivalent to that of [7]. Theorem 11.3. (Adequacy of QmbC≈ w.r.t. interpretations, [7, Theorem 7.7.5]) If Γ ∪ {φ} ⊆ For(Θ≈) then: Γ ⊢QmbC≈ φ iff Γ |= 2 QmbC≈ φ.10 Theorem 9.5 can be easily extended to QmbC≈: 9That is, A is a standard Tarskian first-order structure over signature Θ≈, as considered in Section 9, in which the equality predicate ≈ is interpreted as the identity: ≈A def = {(a, a) : a ∈ U}. 10As in the case of QmbC, in [7, Theorem 7.7.5] it was obtained adequacy of QmbC≈ w.r.t. interpretations, but only for sentences. Once again, adequacy for general formulas follows from that result by using universal closure. 21 Theorem 11.4. Let I = 〈A, ρ〉 be an interpretation for QmbC≈ over a signature Θ≈. Then, it induces a first-order structure with standard equality AI over M5 and Θ≈, and a QmbC≈-valuation vρ≈ over AI and M5 given by v ρ ≈(α) def = (ρ(α), ρ(¬α), ρ(◦α)) such that: ρ(α) = 1 iff vρ≈(α) ∈ D, for every sentence α ∈ Sen(Θ ≈ U). Proof. Let I = 〈A, ρ〉 be an interpretation for QmbC≈ over signature Θ≈. Consider the first-order structure AI over M5 and Θ≈ obtained from A as in the proof of Theorem 9.5. In particular, IAI (≈)(a1, a2) = v ρ ≈(ā1 ≈ ān), where v ρ ≈ : Sen(Θ ≈ U) → |M5| is given by vρ≈(α) = (ρ(α), ρ(¬α), ρ(◦α)), for every α ∈ Sen(Θ ≈ U). It is immediate to see that ρ(α) = 1 iff vρ≈(α) ∈ D, for every α ∈ Sen(Θ ≈ U). From this, IAI(≈)(a1, a2) = v ρ ≈(ā1 ≈ ān) ∈ D iff ρ(ā1 ≈ ā2) = 1 iff a1 = a2, since ρ satisfies clause (vEq1), by Remark 11.2. This shows that AI is a first-order structure with standard equality over M5 and Θ≈. In order to see that vρ≈ is a QmbC ≈-valuation vρ≈ over AI and M5 observe that vρ≈ satisfies clause (i) of Definition 6.9 for every predicate symbol in Θ. Concerning the equality predicate ≈ it is easy to prove that that, by the axioms of equality, the properties of bivaluations for QmbC, and the fact that a = [[ā]]ÂI = [[ā]]Â for every a ∈ U (in particular, for a = [[t]]ÂI , for t ∈ CTer(Θ≈U)), v ρ ≈(t1 ≈ t2) = v ρ ≈ ( [[t1]]ÂI ≈ [[t2]]ÂI ) = IAI(≈) ( [[t1]] ÂI , [[t2]] ÂI ) . This shows that vρ≈ also satisfies clause (i) of Definition 6.9 for the equality predicate ≈. By the proof of Theorem 9.5, it follows that vρ≈ satisfies the other clauses of Definition 6.9 for QmbC-valuation over AI and M5. It remains to prove that vρ≈ is a QmbC ≈-valuation over AI and M5, that is, v ρ ≈(μ(α)) ∈ D for every instance α of axiom (AxEq2) and every assignment μ. But this is easy to prove, by the properties of ρ and by an argument similar to that presented in Remark 11.2(1). This concludes the proof. As an immediate consequence of Theorems 10.5, 11.3 and 11.4: Theorem 11.5. (Adequacy of QmbC≈ w.r.t. first-order structures with standard equality over M5) For every set Γ ∪ {φ} ⊆ For(Θ≈): Γ ⊢QmbC≈ φ iff Γ |= ≈ (A,M5) φ for every structure A with standard equality over Θ≈ and M5. 12 First order twist structures based on the logic LFI1◦ The generalization of swap structures semantics to other quantified LFIs, defined as axiomatic extensions of QmbC, can be easily obtained. Indeed, by analyzing the swap structures semantics for axiomatic extensions of mbC given in [7, Section 6.5] (see also [14]), as well as the first-order version of such extensions proposed in [7, Section 7.8], it is immediate how to obtain first-order swap structures for all these logics. Thus, it is immediate to define QmbCciw, QmbCci, QbC and QCi, the quantified version of mbCciw, mbCci, bC and Ci, respectively, as well as the corresponding extensions of them by adding the standard equality. All these logics are characterized by means of first-order structures defined over 3-valued swap structures.11 11As proved by Avron in [2], the extension mbCcl of mbC by adding da Costa's axiom (cl): ¬(φ ∧ ¬φ) → ◦φ cannot be characterized by a single finite Nmatrix. This negative result also holds for Cila, the presentation of a Costa's system C1 in the language of LFIs, hence it holds for C1 itself. Thus, these systems lies ouside the scope of the present framework. A discussion about this question can be found 22 Instead of analyzing in this section these axiomatic extensions of QmbC, together with the corresponding swap structures semantics (a straightforward exercise), the firstorder version of a quite interesting axiomatic extension of mbC, the logic LFI1◦, will be analyzed with full detail. This logic can be semantically characterized by a 3-valued logical matrix called LFI1', which is equivalent (up to language) to several 3-valued paraconsistent logics such as the well-known da Costa-D'Ottaviano logic J3 and CarnielliMarcos-de Amo logic LFI1. From this, it follows that LFI1◦ is algebraizable in the sense of Blok and Pigozzi (see [7, Chapter 4] for a discussion about this logic). The interesting point is that, as proved in [14], the swap structures for LFI1◦ turn out to be deterministic, thus becoming twist structures, which represent the algebraic semantics for LFI1◦. 12 Because of this, the first-order structures for the first-order extension of LFI1◦ presented here are based on twist structures. Definition 12.1. ([7, Definition 4.4.39]) Let MLFI1 = 〈ALFI1, {1, 1 2 }〉 be the logical matrix such that ALFI1 is the 3-valued algebra over Σ with domain M = {1, 1 2 , 0}, where the operators are defined as follows: ∧ 1 12 0 1 1 12 0 1 2 1 2 1 2 0 0 0 0 0 ∨ 1 12 0 1 1 1 1 1 2 1 1 2 1 2 0 1 12 0 → 1 12 0 1 1 12 0 1 2 1 1 2 0 0 1 1 1 ¬ 1 0 1 2 1 2 0 1 ◦ 1 1 1 2 0 0 1 The logic associated to the logical matrix MLFI1 is called LFI1'. Recall that, by definition, the consequence relation |=LFI1′ of LFI1' is given as follows: for every Γ ∪ {α} ⊆ LΣ, Γ |=LFI1′ α iff, for every homomorphism v : LΣ → ALFI1 of algebras over Σ, if v[Γ] ⊆ {1, 1 2 } then v(α) ∈ {1, 1 2 }. A sound and complete Hilbert calculus for LFI1', called LFI1◦, was introduced in [7]. This calculus is an axiomatic extension of mbC. Definition 12.2. ([7, Definition 4.4.41]) The Hilbert calculus LFI1◦ over Σ is obtained from mbC by adding the following axioms: (ci) ¬◦α→ (α ∧ ¬α) (dneg) ¬¬α ↔ α (neg∨) ¬(α ∨ β) ↔ (¬α ∧ ¬β) (neg∧) ¬(α ∧ β) ↔ (¬α ∨ ¬β) (neg →) ¬(α → β) ↔ (α ∧ ¬β) Theorem 12.3. ([7, Theorem 4.4.45]) The logic LFI1◦ is sound and complete w.r.t. the matrix semantics of LFI1': Γ ⊢LFI1◦ α iff Γ |=LFI1′ α Remark 12.4. By Propositions 3.1.10 and 3.2.3 in [7], and given that LFI1◦ contains axiom (ci), it follows that ⊢LFI1◦ ◦α↔ ∼(α∧¬α) (here, ∼ is the classical negation defined as in Remark 8.2). On the other hand, taking into account that ⊢mbC (α ∧ ¬α) → ¬◦α, it follows from (ci) that ⊢LFI1◦ ¬◦α ↔ (α ∧ ¬α). in [7, Section 6.5]. On the other hand, Cila is not algebraizable in the sense of Blok and Pigozzi (consult, for instance, [11, Section 3.12]). As a consequence of this, none of the logics mbC, mbCciw, mbCci, bC and Ci is algebraizable. 12Twist structures were independently introduced by Fidel [22] and Vakarelov [30]. However, as observed by Cignoli in [13], the basic algebraic ideas underlying twist structures were firstly introduced by Kalman in [25]. 23 Since LFI1◦ is an axiomatic extension of mbC, its first-order extension QLFI1◦ can be defined as an axiomatic extension of QmbC,13 hence the semantics of first-order swap structures for QmbC given in the previous sections can be adapted to QLFI1◦, obtaining so a semantics based on first-order twist structures. Indeed, as shown in [14], each multioperation in the corresponding swap structures for LFI1◦ is deterministic, and so these swap structures are twist structures (which are ordinary algebras presented in a particular form). Definition 12.5. Let A = 〈A,∧,∨,→, 0, 1〉 be a Boolean algebra. The twist domain generated by A is the set TA = {(z1, z2) ∈ A×A : z1 ∨ z2 = 1}. Definition 12.6. ([14, Definition 9.2]) Let A be a Boolean algebra. The twist structure for LFI1◦ over A is the algebra TA = 〈TA, ∧, ∨, →, ¬, ◦〉 over Σ such that the operations are defined as follows, for every (z1, z2), (w1, w2) ∈ TA: (i) (z1, z2) ∧ (w1, w2) = (z1 ∧ w1, z2 ∨ w2); (ii) (z1, z2) ∨ (w1, w2) = (z1 ∨ w1, z2 ∧ w2); (iii) (z1, z2) → (w1, w2) = (z1 → w1, z1 ∧ w2); (iv) ¬(z1, z2) = (z2, z1); (v) ◦(z1, z2) = (∼(z1 ∧ z2), z1 ∧ z2). 14 The intuitive meaning of a snapshot (z1, z2) in TA is that z1 represents a value, in a given Boolean algebra, for the evidence for φ, while z2 represents a value for the evidence against φ (or a value for the evidence for ¬φ). Definition 12.7. The logical matrix associated to the twist structure TA is MT A = 〈TA, DA〉 where DA = {(z1, z2) ∈ TA : z1 = 1} = {(1, a) : a ∈ A}. The consequence relation associated to MT A will be denoted by |=TA , namely: Γ |=TA α iff, for every homomorphism h : LΣ → TA of algebras over Σ, if h(γ) ∈ DA for every γ ∈ Γ then h(α) ∈ DA. Let MLFI1 be the class of logical matrices MT A, for any Boolean algebra A. The twist consequence relation for LFI1◦ is the consequence relation |=MLFI1 associated to MLFI1, namely: Γ |=MLFI1 α iff Γ |=TA α for every Boolean algebra A. Remark 12.8. In [14, Theorem 9.6] it was shown that LFI1◦ is sound and complete w.r.t. twist structures semantics, namely: Γ ⊢LFI1◦ α iff Γ |=MLFI1 α, for every set of formulas Γ ∪ {α}. On the other hand, if A2 is the two-element Boolean algebra with domain {0, 1} then TA2 consists of three elements: (1, 0), (1, 1) and (0, 1). By identifying these elements with 1, 1 2 and 0, respectively, then TA2 coincides with the 3-valued algebra ALFI1 underlying the matrix MLFI1 (recall Definition 12.1). Moreover, MT A2 coincides with MLFI1. Taking into consideration Theorem 12.3, this situation is analogous to the semantical characterization of mbC w.r.t. the 5-element swap structure over A2: it is enough to consider the structure induced by A2 in order to characterize the logic. A first-order version of LFI1◦, which will be called QLFI1◦, can be easily defined from QmbC. Definition 12.9. Let Θ be a first-order signature. The logic QLFI1◦ is obtained from QmbC by deleting axiom (Ax14) and by adding axioms (ci), (dneg), (neg∨), (neg∧) and (neg →) from LFI1◦, plus the following: 13As we shall see in Remark 12.10, axiom (Ax14) will be redundant. 14Here, ∼ denotes the Boolean complement ∼x = x → 0. 24 (Ax¬∃) ¬∃xφ ↔ ∀x¬φ (Ax¬∀) ¬∀xφ ↔ ∃x¬φ Remark 12.10. Observe that QLFI1◦ can be alternatively defined as the Hilbert calculus obtained from LFI1◦ by adding axioms (Ax12) and (Ax13) from Definition 5.4, (Ax¬∃) and (Ax¬∀) above, and the inference rules (∃-In) and (∀-In) from Definition 5.4. The fact that axiom (Ax14) is no longer required is justified by the fact that it can now be derived from the other axioms. This can be proved easily after obtaining the completeness of QLFI1◦ w.r.t. twist structures semantics, since axiom (Ax14) is valid w.r.t. that semantics. The consequence relation of QLFI1◦ will be denoted by ⊢QLFI1◦ . Since QLFI1◦ does not add inference rules to QmbC, it satisfies a deduction meta-theorem (DMT) analogous to QmbC (see Theorem 5.6). Now, the swap structures semantics for QmbC can be adapted to QLFI1◦, taking into account that the swap structures for LFI1◦ are exactly the twist structures introduced in Definition 12.6. This leads us to the following definition: Definition 12.11. let A be a complete Boolean algebra. Let MT A be the logical matrix associated to the twist structure TA for LFI1◦, and let Θ be a first-order signature. A (first-order) structure over MT A and Θ, or a QLFI1◦-structure over Θ, is a pair A = 〈U, IA〉 as in Definition 6.1, but now IA(P ) is a function from U n to TA, for each predicate symbol P of arity n. Notation 12.12. As it was done with QmbC, cA, fA and PA will denote the interpretation of an individual constant symbol c, a function symbol f and a predicate symbol P , respectively. The notion of assignment over a QLFI1◦-structure is as in Definition 6.2. The notion of interpretation [[t]]Aμ of a term t given a structure A and an assignment μ is identical to the one described in Definition 6.3. Given A, the structure Â = 〈U, I Â 〉 over ΘU is defined analogously to the case of QmbC (recall Definition 6.5). Notation 12.13. By adapting Notation 3.3, if z ∈ TA then (z)1 and (z)2, or simply z1 and z2, will denote the first and second coordinates of z, respectively. As it was discussed after Definition 6.8, in order to obtain a single denotation (truthvalue) for a formula in QmbC, a given interpretation and an assignment are not enough: valuations are necessary in order to choose a unique denotation, in case the formula is complex (that is, if it contains connectives or quantifiers). The case of QLFI1◦ is different, since twist structures are deterministic (that is, they are ordinary algebras). This being so, from a given denotation for the atomic formulas, the denotation for complex formulas is uniquely determined fom the denotation of its components, which is in line with the traditional approach to first-order algebraic logic originated by Mostowski. Because of this, valuations over structures are no longer necessary for QLFI1◦, and a structure A will assign a single denotation (truth-value), denoted by [[φ]]A, to each sentence φ. Thid lead us to the following definition: Definition 12.14. (QLFI1◦ interpretation maps) Let A be a complete Boolean algebra, and let A be a structure over MT A and Θ. The interpretation map for QLFI1◦ over A and MT A is a function [[*]] A : Sen(ΘU) → TA satisfying the following clauses (using 25 Notation 12.13 in clauses (iv) and (v)): (i) [[P (t1, . . . , tn)]] A = PA([[t1]] Â, . . . , [[tn]] Â), if P (t1, . . . , tn) is atomic; (ii) [[#φ]]A = #[[φ]]A, for every # ∈ {¬, ◦}; (iii) [[φ#ψ]]A = [[φ]]A # [[ψ]]A, for every # ∈ {∧,∨,→}; (iv) [[∀xφ]]A = (∧ a∈U([[φ[x/ā]]] A)1, ∨ a∈U([[φ[x/ā]]] A)2 ) ; (v) [[∃xφ]]A = (∨ a∈U([[φ[x/ā]]] A)1, ∧ a∈U([[φ[x/ā]]] A)2 ) . The definition of the interpretation of the quantifiers in QLFI1◦ is coherent with the fact that TA (ordered by: z ≤ w iff z1 ≤ w1 and z2 ≥ w2) is a complete lattice (since A is a complete Boolean algebra), in which ∧ i∈I zi = (∧ i∈I(zi)1, ∨ i∈I(zi)2 ) , and ∨ i∈I zi = (∨ i∈I(zi)1, ∧ i∈I(zi)2 ) for every family (zi)i∈I in TA. Note that 1TA =def (1, 0) and 0TA =def (0, 1) are the top and bottom elements of TA, respectively. Definition 12.15. Let A be a complete Boolean algebra, A a structure over MT A and Θ, and let μ be an assignment over A. The extended interpretation map [[*]]Aμ : For(ΘU) → TA is given by [[φ]]Aμ = [[μ(φ)]] A. Definition 12.16. Let A be a complete Boolean algebra, and let A be a structure over MT A and Θ. Given a set of formulas Γ ∪ {φ} ⊆ For(ΘU), φ is said to be a semantical consequence of Γ w.r.t. (A,MT A), denoted by Γ |=(A,MT A) φ, if the following holds: if [[γ]]Aμ ∈ D, for every formula γ ∈ Γ and every assignment μ, then [[φ]] A μ ∈ D, for every assignment μ. Definition 12.17. (Semantical consequence relation in QLFI1◦ w.r.t. twist structures) Let Γ∪{φ} ⊆ For(Θ) be a set of formulas. Then φ is said to be a semantical consequence of Γ in QLFI1◦ w.r.t. first-order twist structures, denoted by Γ |=QLFI1◦ φ, if Γ |=(A,MT A) φ for every (A,MT A). The soundness of QLFI1◦ w.r.t. first-order twist structures semantics can be easily obtained. The proof is analogous but much easier than the proof for QmbC given in Theorem 7.5, given that valuations are no longer necessary. Proposition 12.18. (i) α, α→ β |=QLFI1◦ β; (ii) α→ β |=QLFI1◦ ∃xα → β, if x is not free in β; (iii) α→ β |=QLFI1◦ α→ ∀xβ, if x is not free in α; (iv) |=QLFI1◦ ∀xα → α[x/t], if t is a term free for x in α; (v) |=QLFI1◦ α[x/t] → ∃xα, if t is a term free for x in α; (vi) |=QLFI1◦ ¬∃xφ ↔ ∀x¬φ; (vii) |=QLFI1◦ ¬∀xφ ↔ ∃x¬φ. Proof. The proofs for items (i)-(v) are easily obtained by adapting the corresponding proofs for QmbC given in Proposition 7.3. Items (vi) and (vii) follow easily from the definitions. 26 Corollary 12.19. Let A be a complete Boolean algebra, and let A be a structure over MT A and Θ. Then: (1) If α is an instance of an axiom schema of QLFI1◦ then [[α]] A μ ∈ DA, for every assignment μ. (2) If α and β are formulas such that [[α]]Aμ ∈ DA and [[α → β]] A μ ∈ DA for every assignment μ, then [[β]]Aμ ∈ DA for every μ. (3) If α and β are formulas such that [[α → β]]Aμ ∈ DA for every assignment μ, and if x does not occur free in β, then [[∃xα → β]]Aμ ∈ DA for every μ. (4) If α and β are formulas such that [[α → β]]Aμ ∈ DA, for every assignment μ, and if x does not occur free in α, then [[α → ∀xβ]]Aμ ∈ DA for every μ. Theorem 12.20. (Soundness of QLFI1◦ w.r.t. first-order twist structures) For every set Γ ∪ {φ} ⊆ For(Θ): if Γ ⊢QLFI1◦ φ, then Γ |=QLFI1◦ φ. Proof. By induction on the length n of a derivation of φ from Γ in QLFI1◦, taking into account Corollary 12.19. 13 Completeness of QLFI1◦ w.r.t. twist structures Now, the completeness of QLFI1◦ w.r.t. first-order twist structures semantics will be proved, by adapting the completeness proof for QmbC given in Section 8. The notion of C-Henkin theory in QLFI1◦ is analogous to the one for QmbC given in Definition 8.1. The consequence relation ⊢CQLFI1◦ is the consequence relation of QLFI1◦ over the signature ΘC . The following result can be proved by adapting the proof for QmbC (see Proposition 8.4): Proposition 13.1. Let Γ ∪ {φ} ⊆ Sent(Θ) such that Γ 0QLFI1◦ φ. Then, there exists a set of sentences ∆ ⊆ Sent(Θ) which is maximally non-trivial with respect to φ in QLFI1◦ (assuming that ⊢QLFI1◦ is restricted to sentences) and such that Γ ⊆ ∆. Definition 13.2. Let ∆ ⊆ Sen(Θ) be a non-trivial theory in QLFI1◦. Let ≡ 1 ∆ ⊆ Sen(Θ) 2 be the relation in Sen(Θ) given by: α ≡1∆ β iff ∆ ⊢QLFI1◦ α ↔ β. As in the case of QmbC, ≡1∆ is an equivalence relation. The equivalence class of a sentence α w.r.t. ≡1∆ will be denoted by |α|∆. By adapting the proof of Proposition 8.6 it is easy to prove the following: Proposition 13.3. The structure A∆ def = 〈A∆, ∧, ∨, →, 0∆, 1∆〉 is a Boolean algebra with the following operations: |α|∆#|β|∆ def = |α#β|∆ for any # ∈ {∧,∨,→}, 0∆ def = |φ ∧ (¬φ ∧ ◦φ)|∆ and 1∆ def = |φ ∨ ¬φ|∆, for any sentence φ. It is worth noting that ∼|α|∆ def = |∼α|∆ is the Boolean complement of |α|∆ in A∆. 15 As it was done with QmbC, the construction of the canonical model for QLFI1◦ w.r.t. ∆ requires the use of the MacNeille-Tarski completion of the Boolean algebra A∆. Then, consider the following: Definition 13.4. Let (CA∆, ∗) be the MacNeille-Tarski completion of A∆. The twist structure for LFI1◦ over CA∆ will be denoted by T (∆), and its domain will be denoted by T (∆). The associated logical matrix will be denoted by MT (∆) def = (T (∆), D∆). 15Observe that the occurrence of ∼ on the right-hand side of the definition denotes a classical negation definable in LFI1◦, as explained in Remark 8.2. 27 Remark 13.5. It is worth noting that (∗(|α|∆), ∗(|β|∆)) ∈ D∆ iff ∆ ⊢QLFI1◦ α. Definition 13.6. (Canonical Structure) Let Θ be a signature with some individual constant. Let ∆ ⊆ Sen(Θ) be non-trivial in QLFI1◦, let MT (∆) be the matrix as in Definition 13.4, and let U def = CTer(Θ). The canonical QLFI1◦-structure induced by ∆ is the structure A∆ = 〈U, IA∆〉 over MT (∆) and Θ such that: cA∆ = c, for every individual constant c ∈ C; fA∆ : Un → U is such that fA∆(t1, . . . , tn) = f(t1, . . . , tn), for every function symbol f of arity n; PA∆(t1, . . . , tn) = (∗(|φ|∆), ∗(|¬φ|∆)) with φ = P (t1, . . . , tn), for every predicate symbol P of arity n. Note that |φ|∆∨ |¬φ|∆ = |φ ∨ ¬φ|∆ = 1 and so ∗(|φ|∆) ∨ ∗(|¬φ|∆) = ∗(|φ ∨ ¬φ|∆) = 1. Hence, PA∆(t1, . . . , tn) ∈ T (∆) and so A∆ is indeed a structure over MT (∆) and Θ. Let (*)⊲ : (Ter(ΘU) ∪ For(ΘU)) → (Ter(Θ) ∪ For(Θ)) be the function introduced in Definition 8.9 such that ( s )⊲ is obtained from s by substituting every occurrence of a constant t by the term t itself. Clearly (t)⊲ = [[t]]Â∆ for every t ∈ CTer(ΘU). By adapting the proof of Lemma 8.11 it follows: Lemma 13.7. Let ∆ ⊆ Sen(Θ) be a set of sentences over a signature Θ such that ∆ is a C-Henkin theory in QLFI1◦ for a nonempty set C of individual constants of Θ, and ∆ is maximally non-trivial with respect to φ in QLFI1◦, for some sentence φ. Then, for every formula ψ(x) in which x is the unique variable (possibly) occurring free, it holds: (1) |∀xψ|∆ = ∧ A∆ {|ψ[x/t]|∆ : t ∈ CTer(Θ)}, where ∧ A∆ denotes an existing infimum in the Boolean algebra A∆; (2) |∃xψ|∆ = ∨ A∆ {|ψ[x/t]|∆ : t ∈ CTer(Θ)}, where ∨ A∆ denotes an existing supremum in the Boolean algebra A∆. Proposition 13.8. Let ∆ ⊆ Sen(Θ) be as in Lemma 13.7. Then, the interpretation map [[*]]A∆ : Sen(ΘU) → T (∆) is such that [[ψ]] A∆ = (∗(|(ψ)⊲|∆), ∗(|(¬ψ) ⊲|∆)) for every sentence ψ. Moreover, [[ψ]]A∆ ∈ D∆ iff ∆ ⊢QLFI1◦ (ψ) ⊲. In particular, [[ψ]]A∆ ∈ D∆ iff ∆ ⊢QLFI1◦ ψ for every ψ ∈ Sen(Θ). Proof. The proof is done by induction on the complexity of the sentence ψ in Sen(ΘU). If ψ = P (t1, . . . , tn) is atomic then, by using Definition 12.14, the fact that [[t]] Â∆ = (t)⊲ for every t ∈ CTer(ΘU), and Definition 13.6, we have: [[ψ]]A∆ = PA∆([[t1]] Â∆ , . . . , [[tn]] Â∆) = PA∆((t1) ⊲, . . . , (tn) ⊲) = (∗(|(ψ)⊲|∆), ∗(|(¬ψ) ⊲|∆)). If ψ = ¬β then, by Definition 12.14 and by induction hypothesis, [[ψ]]A∆ = ¬[[β]]A∆ = ¬(∗(|(β)⊲|∆), ∗(|(¬β) ⊲|∆)) = (∗(|(¬β) ⊲|∆), ∗(|(β) ⊲|∆)). But |(β)⊲|∆ = |(¬¬β) ⊲|∆, by (dneg). Hence, [[ψ]] A∆ = (∗(|(ψ)⊲|∆), ∗(|(¬ψ) ⊲|∆)). If ψ = ◦β then, by Definition 12.14, by induction hypothesis, the definition of the operations in A∆ and the fact that ∗ is a homomorphism of Boolean algebras, [[ψ]]A∆ = ◦[[β]]A∆ = ◦(∗(|(β)⊲|∆), ∗(|(¬β) ⊲|∆)) = (∗(|(∼(β ∧ ¬β))⊲|∆), ∗(|(β ∧ ¬β) ⊲|∆)). 28 But |(∼(β ∧ ¬β))⊲|∆ = |(◦β) ⊲|∆ and |(β ∧ ¬β) ⊲|∆ = |(¬◦β) ⊲|∆, by Remark 12.4. Hence, [[ψ]]A∆ = (∗(|(ψ)⊲|∆), ∗(|(¬ψ) ⊲|∆)). If ψ = α#β for # ∈ {∧,∨,→}, the proof is analogous, but now axioms (neg∨), (neg∧) and (neg →) are required. If ψ = ∀xβ then, by Lemma 13.7 and using that U = CTer(Θ), |∀xβ|∆ = ∧ A∆ {|β[x/t]|∆ : t ∈ U} and so ∗(|∀xβ|∆) = ∧ CA∆ {∗(|β[x/t]|∆) : t ∈ U}. Analogously, ∗(|∃xβ|∆) =∨ CA∆ {∗(|β[x/t]|∆) : t ∈ U}. Then, by Definition 12.14, by induction hypothesis and by axiom (Ax¬∀): [[∀xβ]]A∆ = (∧ t∈U ([[β[x/t]]] A∆)1, ∨ t∈U ([[β[x/t]]] A∆)2 ) = (∧ t∈U ∗(|(β[x/t]) ⊲|∆), ∨ t∈U ∗(|(¬β[x/t]) ⊲|∆) ) = ( ∗(|(∀xβ)⊲|∆), ∗(|(∃x¬β) ⊲|∆) ) = ( ∗(|(∀xβ)⊲|∆), ∗(|(¬∀xβ) ⊲|∆) ) . Hence, [[ψ]]A∆ = (∗(|(ψ)⊲|∆), ∗(|(¬ψ) ⊲|∆)). If ψ = ∃xβ, the proof is analogous to the previous case. This shows that [[ψ]]A∆ = (∗(|(ψ)⊲|∆), ∗(|(¬ψ) ⊲|∆)) for every sentence ψ. The rest of the proof follows by Remark 13.5. Theorem 13.9. (Completeness of QLFI1◦ restricted to sentences w.r.t. first-order twist structures) If Γ |=QLFI1◦ φ then Γ ⊢QLFI1◦ φ, for every Γ ∪ {φ} ⊆ Sen(Θ). Proof. Suppose that Γ ∪ {φ} ⊆ Sen(Θ) is such that Γ 0QLFI1◦ φ. By adapting Theorem 7.5.3 in [7] to QLFI1◦, 16 there exists a C-Henkin theory ∆H over ΘC in QLFI1◦ for some nonempty set C of new individual constants such that Γ ⊆ ∆H and, in addition: Γ ⊢QLFI1◦ α iff ∆ H ⊢CQLFI1◦ α, for every α ∈ Sen(Θ). In consequence, ∆ H 0 C QLFI1◦ φ. Because of Proposition 13.1, there is a set of sentences ∆H in ΘC containing ∆ H which is maximally non-trivial with respect to φ in QLFI1◦ (restricted to Sen(ΘC)), and such that ∆ H is also a C-Henkin theory over ΘC in QLFI1◦. Consider now MT (∆ H) and A ∆H as in Definitions 13.4 and 13.6, respectively. By Proposition 13.8, [[α]]A∆H ∈ D ∆H iff ∆H ⊢CQLFI1◦ α, for every α in Sen(ΘC). But then [[γ]] A ∆H ∈ D ∆H for every γ ∈ Γ and [[φ]]A∆H /∈ D ∆H . Now, let A the reduct of A ∆H to Θ. Hence, A is a structure over MT (∆H) and Θ such that [[γ]]A ∈ D∆H for every γ ∈ Γ but [[φ]] A /∈ D∆H . This means that Γ 6|=QLFI1◦ φ. Corollary 13.10. (Completeness of QmbC w.r.t. first-order twist structures) Let Γ ∪ {φ} ⊆ For(Θ). If Γ |=QLFI1◦ φ then Γ ⊢QLFI1◦ φ. 14 Completeness of QLFI1◦ w.r.t. structures over MLFI1 In Remark 12.8 it was observed that TA2 , the twist structure for LFI1◦ defined over the two-element Boolean algebra A2, coincides (up to names) with the 3-valued algebra ALFI1 underlying the matrix MLFI1 and, moreover, MT A2 coincides with the 3-valued characteristic matrix MLFI1 of LFI1'. Recall that 1, 1 2 and 0 are identified with (1, 0), (1, 1) and (0, 1), respectively. Let A be a QLFI1◦-structure over A2. If φ is a formula 16As observed in the proof of Theorem 8.13 above, Theorem 7.5.3 in [7] also holds for the definition of C-Henkin theory adopted in this paper. 29 in which x is the unique variable (possibly) occurring free, let X = {[[φ[x/ā]]]A : a ∈ U}. Then: [[∀xφ]]A =    1 if X = {1} 1 2 if X ⊆ {1, 1 2 } and 1 2 ∈ X 0 if 0 ∈ X [[∃xφ]]A =    1 if 1 ∈ X 1 2 if X ∩ {1, 1 2 } = {1 2 } 0 if X = {0} In Section 9 it was obtained a characterization of QmbC in terms of swap structures over the 5-element characteristic Nmatrix of mbC, which coincides with the one given in [6]. That result can be easily adapted to QLFI1◦, by proving that QLFI1◦ can be characterized by first-order structures defined over MLFI1. Indeed, it is possible to adapt Theorem 9.5 to QLFI1◦, taking into account that the bivaluations for QLFI1◦ satisfy aditional clauses, see [7, Definition 7.9.16]. This lead us to the following result, in view of Remark 12.8 (details of the proof will be omitted): Theorem 14.1. (Adequacy of QLFI1◦ w.r.t. first-order structures over MLFI1) For every set Γ ∪ {φ} ⊆ For(Θ): Γ ⊢QLFI1◦ φ iff Γ |=(A,MLFI1) φ for every structure A over Θ and MLFI1. The latter result is a variant (up to language) of the adequacy theorem of first-order J3 w.r.t. first-order structures given in [18] (see also [17, 19, 20]). Indeed, the semantics in terms of first-order structures over MLFI1 is equivalent to the 3-valued first-order structures proposed by D'Ottaviano in [18] for a quantified version of J3, given that LFI1◦ is equivalent, up to language, to J3. This shows that the twist-structures semantics for QLFI1◦ constitutes a generalization, to any complete Boolean algebra, of the above mentioned semantics for first-order J3. The extension of QLFI1◦ with standard equality is straigtforward, taking into account the construction for QmbC≈ presented in Section 10. It is worth noting that, when restricted to structures over MLFI1, there are differences with D'Ottaviano's approach to first-order J3 with equality. Indeed, she assumes that the equality must be classical, that is, every formula ◦(t1 ≈ t2) is valid in her system, contrary to what happens in QLFI1◦ with equality. 15 Final remarks In this paper, the semantical frameworks for QmbC already proposed in the literature were extended to a vast class of models based on the non-deterministic algebras known as swap structures. Indeed, the Nmatrix semantics proposed in [6] and the semantics given by interpretations (i.e., standard Tarskian structures plus bivaluations) considered in [10] and [7] coincide, and are particular cases of the swap structures semantics introduced here, as it was shown along this paper. The advantage of considering models based on a class of swap structures instead of 'classical' models based on a finite Nmatrix (as done in [6]) is that this enlarged class of models allows us to consider applications to another fields such, for instance, algebraic logic (as done in [14]) or paraconsistent set theory. Concerning the latter, the Boolean valued models for set theory could be generalized to this setting, obtaining so swap structures models for several paraconsistent set theories based on LFIs, along the lines of the twist-valued models introduced in [8]. 30 Two important model-theoretic results for QmbC (and some of its axiomatic extensions) were obtained by Ferguson in [21]: Loś' ultraproducts theorem, and a suitable version of the Keisler-Shelah isomorphism theorem, which states that two QmbC-models are strongly elementarily equivalent iff there exists an ultrafilter U such that the corresponding ultrapowers over U are strongly isomorphic. The notions of strong elementary equivalence and strong isomorphism were introduced in [21], as well as an adaptation of the method of atomization introduced by Skolem, which was used in order to prove the Keisler-Shelah theorem for quantified LFIs. 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