On the Way to a Wider Model Theory:Completeness Theorems for First-Order Logics

of Formal Inconsistency

Walter Carnielli, Marcelo E. Coniglio,Rodrigo Podiacki and Tarcísio Rodrígues

Centre for Logic, Epistemology and the History of Science – CLE

and

Department of Philosophy – UNICAMP

P.O. Box 6133, 13083-970

Campinas, SP, Brazil

walter.carnielli@cle.unicamp.br, coniglio@cle.unicamp.br

podiacki@gmail.com, tarcisio.genaro@gmail.com

AbstractThis paper investigates the question of characterizing first-order LFIs (logics

of formal inconsistency) by means of two-valued semantics. LFIs are powerfulparaconsistent logics that encode classical logic and permit a finer distinction be-tween contradictions and inconsistencies, with a deep involvement in philosophicaland foundational questions. Although focused on just one particular case, namely,the quantified logic QmbC, the method proposed here is completely general forthis kind of logics, and can be easily extended to a large family of quantifiedparaconsistent logics, supplying a sound and complete semantical interpretationfor such logics. However, certain subtleties involving term substitution and re-placement, that are hidden in classical structures, have to be taken into accountwhen one ventures into the realm of non-classical reasoning. This paper showshow such difficulties can be overcome, and offers detailed proofs showing that asmooth treatement of semantical characterization can be given to all such logics.Although the paper is well-endowed in technical details and results, it has a sig-nificant philosophical aside: it shows how slight extensions of classical methodscan be used to construct the basic model theory of logics that are weaker than tra-ditional logic due to the absence of certain rules present in classical logic. Severalsuch logics, however, as in the case of the LFIs treated here, are notorious fortheir wealth of models precisely because they do not make indiscriminate use ofcertain rules; these models thus require new methods. In the case of this paper, byjust appealing to a refined version of the Principle of Explosion, or Pseudo-Scotus,some new constructions and crafty solutions to certain non-obvious subtleties areproposed. The result is that a richer extension of model theory can be inaugurated,with interest not only for paraconsistency, but hopefully to other enlargements oftraditional logic.

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IntroductionLogics of Formal Inconsistency from now on LFIs are logics able to internal-ize, in a precise sense, the notions of “consistency” and “inconsistency” at the object-language level, be it by introducing primitive unary connectives, or by means of appro-priate definitions using the familiar propositional connectives. Such logics are para-consistent in the following sense: given a contradiction of the form (ϕ ∧ ¬ϕ), it is notpossible in general to deduce an arbitrary formula ψ from the contradiction. That is,such logics do not fall into deductive triviality when exposed to a contradiction. Thismeans that the Principle of Explosion, or Pseudo-Scotus, is not valid for such logics ingeneral. However, LFIs may “explode” if, besides ϕ being contradictory, there is an ad-ditional stipulation, namely, that ϕ is consistent, or that ϕ behaves classically. LFIs aretherefore submitted to a more restricted principle of explosion, called in [7] the Gen-tle Principle of Explosion: an LFI explodes if ϕ, ¬ϕ and ◦ϕ occur simultaneusly, forsome arbitrary ϕ, such that ‘◦ϕ’ expresses the fact that ϕ is consistent. This constitutesa wide generalization of the well-known C-systems introduced by da Costa throughthe hierarchy of systems Cn, for n ≥ 1 (see [12]). In C1, for instance, consistency (orwell-behavior, in da Costa’s words) is defined by the formula ◦ϕ = ¬(ϕ ∧ ¬ϕ).

In their beginnings, paraconsistent logics were mainly developed syntactically, thatis, presented through Hilbert calculi, without committing to a semantical interpretation.The first semantics for propositional paraconsistent logics (that is, for the calculi Cn ofda Costa) had to wait until the 70s, and were known as valuation semantics (see [14]).Nevertheless, the problem concerning a convenient interpretation for first-order para-consistent logic persisted. In 1984, Alves proposed a method which can be calledpre-structural semantics (see [1]).

The present paper proposes an axiomatization and first-order semantics for QmbC,the first-order extension of mbC, which is the simplest logic in the hierarchy of LFIsproposed in [8, 7]. The semantics, adapted from [25], differs slightly from the oneadopted in [23]. This improved formulation is preferred, as it can be better and morenaturally extended to other cases.

The structure of the paper is as follows: in Section 1 the basic system QmbC isintroduced, and some useful theorems are proved. Section 2 establishes some metathe-orems of QmbC. The (Tarskian) 2-valued semantics for QmbC is introduced in Sec-tion 3. The soundness and completeness theorems of QmbC with respect to the pro-posed semantics are obtained in sections 4 and 5, respectively. In Section 6 somefundamental theorems of Model Theory for QmbC are given: Compactness and theLowenhëim-Skolem Theorems. In Section 7, the system QmbC is expanded with thepredicate for standard equality, as is usually done in Model Theory. Section 8 presentsthe axiomatic extension of QmbC (possibly expanded with equality) to other LFI’sstudied in [8, 7]. In Section 8 a brief survey of previous approaches to first-order LFI’sproposed in the literature is presented. The final section discusses what was done froma conceptual point of view.

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1 The logic QmbCThis section introduces the logic QmbC, the first-order LFI to be investigated in detailin this paper.

Definition 1.1. Assume the set of connectives {¬, ◦,∧,∨,→ } for negation, consistency,conjunction, disjunction and implication, as well as the symbols ∀ (universal quanti-fier) and ∃ (existential quantifier), and punctuation marks (commas and parentheses).Let Var = {v1, v2, . . .} be a denumerable set of individual variables. A first-order sig-nature Σ for LFIs is composed of the following elements:

• a set C of individual constants;

• for each n ≥ 1, a set of function symbols of arity n;

• for each n ≥ 1, a set of predicate symbols of arity n.

As usual, given a signature Σ, it is assumed that it has at least one predicate symbol.The set of terms and of formulas of Σ (which are defined recursively, as usual) aredenoted by TΣ and LΣ, respectively. Also, the notions of subformula, scope of anoccurrence of a quantifier in a formula, free and bound occurrence of a variable in aformula, and of a term free for a variable in a formula, are the usual ones (the reader isreferred to [26] and [21]).

The set of atomic formulas and of sentences (i.e., formulas without free variables)of Σ are denoted by AtΣ and S LΣ

, respectively.The notation ϕ[x/t] will stand for the formula obtained from ϕ by substituting every

free occurrence of variable x by the term t.

Definition 1.2. Let ϕ and ψ be formulas. If ϕ can be obtained from ψ by means ofaddition or deletion of void quantifiers, or by renaming bound variables (keeping thesame free variables in the same places), then ϕ and ψ are said to be variants of eachother.

The logic mbC was introduced in [7] as a basic LFI, meaning that its axioms em-body a minimum proof power sufficient to preserve the positive theorems of classicalpropositional logic, while at the same time being capable of avoiding trivialization inthe presence of contradictions. The extension of mbC to first-order logic is calledQmbC, and is defined as follows:

Definition 1.3. Let Σ be a first-order signature. The logic QmbC (over Σ) is definedby the following Hilbert calculus:

Axiom Schemas

(Ax1) α→ (β→ α)

(Ax2) (α→ β)→ ((α→ (β→ γ))→ (α→ γ))

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(Ax3) α→ (β→ (α ∧ β))

(Ax4) (α ∧ β)→ α

(Ax5) (α ∧ β)→ β

(Ax6) α→ (α ∨ β)

(Ax7) β→ (α ∨ β)

(Ax8) (α→ γ)→ ((β→ γ)→ ((α ∨ β)→ γ))

(Ax9) α ∨ (α→ β)

(Ax10) α ∨ ¬α

(Ax11) ◦α→ (α→ (¬α→ β))

(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) ∀x(α→ β)→ (α→ ∀xβ), if x is not free in α

(Ax15) α→ β, whenever α is a variant of β

Inference Rules

MP: α, α→ β / β

∀-In: α→ β / α→ ∀xβ, if x is not free in α

∃-In: α→ β / ∃xα→ β, if x is not free in β

The consequence relation of QmbC will be denoted by `. Thus, if Γ ∪ {ϕ} ⊆ LΣ

then Γ ` ϕ will denote that there exists a derivation in QmbC of ϕ from Γ.It is worth noting that (Ax1)-(Ax11) plus MP (considered in a propositional lan-

guage) is a Hilbert calculus for the propositional logic mbC, while (Ax1)-(Ax9) plusMP is a Hilbert calculus for positive propositional classical logic (see [7]).

As it was proved in [7], the logic mbC can be characterized in terms of valuationsover {0, 1}, or bivaluations:

Definition 1.4. Let L be the algebra of formulas of mbC. A function v : L → {0, 1} isa valuation for mbC if it satisfies the following clauses:

(vAnd) v(α ∧ β) = 1 ⇐⇒ v(α) = 1 and v(β) = 1

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(vOr) v(α ∨ β) = 1 ⇐⇒ v(α) = 1 or v(β) = 1

(vImp) v(α→ β) = 1 ⇐⇒ v(α) = 0 or v(β) = 1

(vNeg) v(α) = 0 =⇒ v(¬α) = 1

(vCon) v(◦α) = 1 =⇒ v(α) = 0 or v(¬α) = 0.

The semantical consequence relation w.r.t. valuations for mbC is defined as ex-pected: Γ �mbC ϕ iff, for every valuation v for mbC, if v(γ) = 1 for every γ ∈ Γ thenv(ϕ) = 1.

Theorem 1.5. ([7]) For every set of formulas Γ ∪ {ϕ} ⊆ L:

Γ `mbC ϕ if and only if Γ �mbC ϕ .

2 Some useful (meta)theorems of QmbCIn this section, some useful theorems and meta-theorems of QmbC will be established.Throughout this section, a fixed first-order signature Σ will be assumed.

Theorem 2.1.

1. For every formula α: ` α→ α.

2. If Γ ` α→ γ and Γ ` β→ γ then Γ, α ∨ β ` γ.

3. If Γ ` ¬α→ γ and Γ ` α→ γ then Γ ` γ.

4. If Γ ` φ then Γ ` ∀x φ.

5. If x is not free in β then ∀x(α→ β) ` ∃xα→ β.

6. If x is not free in α then ∀x(α→ β) ` α→ ∀xβ.

7. For every formulas α, β, γ: ` (α→ β)→ ((β→ γ)→ (α→ γ)).

8. For every formulas α, β, γ: ` (α→ (β→ γ))→ (β→ (α→ γ)).

9. If Γ ` α→ β then Γ, α ` β.

10. α→ β, β→ γ ` α→ γ.

11. If ` α→ β then ` (γ → α)→ (γ → β), for every formula γ.

Proof. Item 1: It is well known that any Hilbert calculus containing axiom schemas (Ax1),(Ax2) and (MP) derives the schema α → α. The following derivation was takenfrom [21]:

1. (α→ ((α→ α)→ α))→ ((α→ (α→ α))→ (α→ α)) (Ax2)

2. α→ ((α→ α)→ α) (Ax1)

3. (α→ (α→ α))→ (α→ α) (MP 1,2)

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4. α→ (α→ α) (Ax1)

5. α→ α (MP 3,4)

Item 2: By hypothesis, Γ, α ∨ β ` α → γ and Γ, α ∨ β ` β → γ hold. But ` (α →γ) → ((β → γ) → ((α ∨ β) → γ)), by (Ax8), and so Γ, α ∨ β ` (α ∨ β) → γ, by usingMP twice. By applying MP once again it follows that Γ, α ∨ β ` γ.

Item 3: It is a consequence of Item 2, by observing that ` α ∨ ¬α, by (Ax10).

Item 4: Consider the (meta)derivation below.

Γ ` φ HypothesisΓ ` φ→

(¬∀x φ→ φ

)Ax1

Γ ` ¬∀x φ→ φ MPΓ ` ¬∀x φ→ ∀x φ ∀-InΓ ` ∀x φ→ ∀x φ Item 1Γ ` ∀x φ Item 2

Item 5: Consider the derivation in QmbC below.

1. ∀x(α→ β) (premise)

2. ∀x(α→ β)→ (α→ β) (Ax13)

3. α→ β (MP 1,2)

4. ∃xα→ β (∃-In 3)

Item 6: It follows by (Ax14) and MP.

Item 7: By considering the semantics of bivaluations for mbC given above, it is easyto see that (α → β) → ((β → γ) → (α → γ)) is a valid formula. By completenessof mbC w.r.t. bivaluations, that formula is derivable in mbC, for every α, β, γ. SinceQmbC extends mbC, it follows that the schema (α → β) → ((β → γ) → (α → γ)) isderivable in QmbC.

Item 8: The proof is identical to that of Item 7.

Item 9: Consider the (meta)derivation below.

Γ ` α→ β HypothesisΓ, α ` α→ β MonotonicityΓ, α ` β MP

Item 10: It follows from Item 7 and Item 9 (used two times).

Item 11: From items 7 and 8 it follows that ` (α→ β)→ ((γ → α)→ (γ → β)). Then,by hypothesis and MP, ` (γ → α)→ (γ → β). �

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Recall from [7] that, in mbC, a strong (classical) negation can be defined as ∼βα =

α → ⊥β, where ⊥β = (β ∧ (¬β ∧ ◦β)) is a bottom formula,1 for any formula β. Inthe case of first-order languages, the formula β must be a sentence. For simplicity, aprivileged one will be chosen, and the subscript β will be omitted in ⊥β and ∼β fromnow on.

Proposition 2.2 (Strong Negation). The strong negation ∼ satisfies the following prop-erties in mbC (and, therefore, also in QmbC):

(i) ` ∼α→ (α→ ψ) for every α and ψ;

(ii) ` α ∨ ∼α

(iii) ` α→ ∼∼α and ` ∼∼α→ α

(iv) If (Γ ` α→ γ) and (∆, ` ∼α→ γ) then (Γ,∆ ` γ)

(v) ` (α→ β)→ (∼β→ ∼α) and so α→ β ` ∼β→ ∼α

(vi) ` (∼α→ ∼β)→ (β→ α) and so ∼α→ ∼β ` β→ α

(vii) ∼α→ β ` ∼β→ α

(viii) ` ∼(α→ β)→ (α ∧ ∼β)

(ix) ` ⊥ → α

(x) ` ∀x∼α→ ∼∃xα

(xi) If ` α→ β then ` ∀xα→ ∀xβ

(xii) ` ∀x(α→ β)→ (∃xα→ β), if x does not occur free in β.

Proof. Items (i)-(iii) and (v)-(ix): It follows from the completeness of mbC with re-spect to bivaluations (see Theorem 1.5), by observing that v(⊥) = 0, and v(∼α) = 1 iffv(α) = 0, for every formula α and every valuation v for mbC. An argument similar tothe proof of Theorem 2.1 Item 7 can then be used.

Item (iv): It follows from Item (ii) and from Item 2 of Theorem 2.1.

Item (x): From (Ax13) it holds that ` ∀x∼α → ∼α and so ` ∼∼α → ∼∀x∼α, byItem (v). Using Item (iii) and Theorem 2.1 Item 10, ` α → ∼∀x∼α. By rule (∃-In),` ∃xα → ∼∀x∼α whence ` ∼∼∀x∼α → ∼∃xα, using again Item (v). Finally, byItem (iii) and Theorem 2.1 Item 10, ` ∀x∼α→ ∼∃xα.

Item (xi): From (Ax13), ` ∀xα→ α and then, by hypothesis and Theorem 2.1 Item 10,` ∀xα→ β. By rule (∀-In ), ` ∀xα→ ∀xβ.

Item (xii): By items (v) and (xi) it follows that ` ∀x(α → β) → ∀x(∼β → ∼α).But ` ∀x(∼β → ∼α) → (∼β → ∀x∼α), by (Ax14), and then ` ∀x(α → β) →(∼β → ∀x∼α), by Theorem 2.1 Item 10. By Item (x) and Theorem 2.1 Item 11,` (∼β → ∀x∼α) → (∼β → ∼∃xα). From this, ` ∀x(α → β) → (∼β → ∼∃xα),

1That is: ⊥β ` ψ for every ψ.

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by Theorem 2.1 Item 10. By Item (vi), ` (∼β → ∼∃xα) → (∃xα → β). Finally,` ∀x(α→ β)→ (∃xα→ β), using again Theorem 2.1 Item 10. �

The next step is to establish a particularly important meta-theorem of QmbC: theDeduction Meta-Theorem (DMT). As in the case of classical first-order logic, it doesnot hold in the general case, but does hold under certain assumptions concerning freevariables of the formula being discharged from the assumptions. The proof will bedone by adapting the case for classical logic presented in [21].

Definition 2.3. Let d = ϕ1, . . . , ϕn be a derivation in QmbC from a set of hypothesisΓ, and let ϕ ∈ Γ. Then ϕi is said to depend upon ϕ in d if:

• ϕi = ϕ; or

• ϕi is obtained from ϕ j and ϕk (with j, k < i) by (MP), where ϕ j or ϕk dependupon ϕ in d; or

• ϕi is obtained from ϕ j (with j < i) by (∃-In), where ϕ j depends upon ϕ in d; or

• ϕi is obtained from ϕ j (with j < i) by (∀-In), where ϕ j depends upon ϕ in d.

Of course the notion above can be adjusted to any Hilbert calculus. The next resultholds in any Hilbert calculus (see [21]).

Lemma 2.4. If ψ does not depend upon ϕ in the derivation of ψ from Γ ∪ {ϕ}, thenΓ ` ψ.

Theorem 2.5 (Deduction Meta-Theorem (DMT) for QmbC). Suppose that there existsin QmbC a derivation of ψ from Γ∪{ϕ}, such that no application of the rules (∃-In) and(∀-In) to formulas that depend upon ϕ have as their quantified variables free variablesof ϕ. Then Γ ` ϕ→ ψ.

Proof. Let d = ϕ1, . . . , ϕn be a derivation in QmbC of ψ from Γ ∪ {ϕ}, satisfying theconditions of the hypothesis of the theorem; then ϕn = ψ. It will be proven by inductionon n that Γ ` ϕ → ϕi for every 1 ≤ i ≤ n. From this it follows that Γ ` ϕ → ψ, asrequired.

The proof is identical to that presented in [21] for first-order classical logic, with theexception of the rules for quantification (which are different from the rules of QmbC),and so this case is the only one to be treated here (observe that the part of the proofin [21] omitted here uses that α → α is a theorem, as proved here in Item 1 of Theo-rem 2.1).

Thus, suppose that Γ ` ϕ → ϕ j for every 1 ≤ j < i, with i ≥ 2. By the considera-tions above, just two cases need to be analyzed:

1) There exists j < i such that ϕ j = α → β and ϕi = ∃xα → β (with x not free inβ) is obtained from ϕ j by (∃-In). By induction hypothesis, Γ ` ϕ → ϕ j and, by thehypothesis on d, either ϕ j does not depend upon ϕ or x does not occur free in ϕ. Thereare two subcases to be analyzed:1.1) ϕ j does not depend upon ϕ. By Lemma 2.4, Γ ` ϕ j, that is, Γ ` α → β. By apply-ing rule (∃-In) it follows that Γ ` ∃xα→ β, that is, Γ ` ϕi. From this, Γ ` ϕ→ ϕi.

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1.2) x does not occur free in ϕ. As Γ ` ϕ → ϕ j, that is, Γ ` ϕ → (α → β), thenΓ ` ϕ → ∀x(α → β), by applying rule (∀-In). By Proposition 2.2 Item (xii) it followsthat ` ∀x(α → β) → (∃xα → β) and then ` (ϕ → ∀x(α → β)) → (ϕ → (∃xα → β)),by Theorem 2.1 Item 11. From this, Γ ` ϕ→ (∃xα→ β), by MP. That is, Γ ` ϕ→ ϕi.

2) There exists j < i such that ϕ j = α → β and ϕi = α → ∀xβ (with x not free in α)is obtained from ϕ j by (∀-In). The proof of this case is quite similar to that of Case 1and so it will be omitted (it is worth noting that, in the second subcase, axiom (Ax14)is used instead of Proposition 2.2 Item (xii)). �

The version of (DMT) presented above is very general, but for this reason it can bea bit complicated to determine the conditions under which it can be applied.A partic-ular case is given below, which is simpler than the general case but enough for mostapplications.

Corollary 2.6 ((DMT), simplified version). Suppose that there exists in QmbC aderivation 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 ϕis a sentence). Then Γ ` ϕ→ ψ.

The Deduction Meta-Theorem simplifies considerably the derivations in QmbC.The proof of the following result, which will be used in the sequel, takes profit ofDMT:

Theorem 2.7. If α ` β then γ → α ` γ → β and β→ γ ` α→ γ, for every γ.

Proof. Consider the following (meta) derivation in QmbC:

γ → α, γ ` α MPα ` β Hypothesis

γ → α, γ ` β Transitivity

Therefore, by (DMT), γ → α ` γ → β.Finally, the following (meta)derivation above can be considered:

α ` β Hypothesisβ→ γ, α ` β Monotonicityβ→ γ, α ` β→ γ Reflexivityβ→ γ, α ` γ MP

Therefore, by (DMT), β→ γ ` α→ γ. �

Some variations of the inference rules ∃-In and ∀-In will now be discussed. Theseresults are essential in order to prove the Completeness theorem for QmbC, more pre-cisely when proving that non-trivial theories can be conservatively extended to non-trivial Henkin theories (Theorem 5.3). First, however, some technical results must beobtained, recalling that ∼ denotes the strong negation.

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Lemma 2.8. In QmbC the following hold:

(i) ` (α→ β) implies ` (∃xα→ ∃xβ)(ii) ` ∃x∼∼α→ ∃xα

(iii) ` ∼∀x α→ ∃x ∼α

(iv) ` ∼∀x∼α→ ∃xα

(v) ` (∀xα→ β)→ ∃x(α→ β) if x does not occur free in β

(vi) ` (α→ ∃x β)→ ∃x(α→ β) if x does not occur free in α.

Proof. (i) Suppose that ` (α→ β). Then ` (β→ ∃xβ), by (∃-Ax), and so ` (α→ ∃xβ),by transitivity of→. The result follows by applying rule (∃-In).(ii) As ` ∼∼α→ α, by Proposition 2.2(iii), the result follows from item (i).(iii) By (Ax12), ` ∼α → ∃x∼α. By Proposition 2.2(vii), ` ∼∃x∼α → α. By ∀-In,` ∼∃x∼α→ ∀xα. Finally, by Proposition 2.2(vii), ` ∼∀xα→ ∃x∼α.(iv) By item (iii), ` ∼∀x∼α→ ∃x∼∼α. The result follows by item (ii) and by transitiv-ity of→.(v) By MP, (∀xα → β),∀xα ` β. But ` β → (α → β) and ` (α → β) → ∃x(α → β),by (Ax1) and (Ax12), respectively. Then, (∀xα → β),∀xα ` ∃x(α → β). On the otherhand, ` ∼α → (α → β), by Proposition 2.2(i), and so ` ∃x∼α → ∃x(α → β), byitem (i). But ` ∼∀xα → ∃x∼α, by item (iii) , therefore ` ∼∀xα → ∃x(α → β), bytransitivity of →. From this, (∀xα → β),∼∀xα ` ∃x(α → β). Thus, by (DMT) andProposition 2.2(iv), (∀xα → β) ` ∃x(α → β). The result follows by (DMT), as x doesnot occur free in β.(vi) From (α → ∃x β),∀x∼(α → β) it follows that (α → ∃x β),∼(α → β), by (Ax13),and from this (α → ∃x β), α,∼β, by Proposition 2.2(viii), (Ax4) and (Ax5). Fromthis ∃x β,∼β is obtained by MP. But ∼β = β → ⊥ and β → ⊥ ` ∃xβ → ⊥,by (∃-In). That is, ∼β ` ∼∃x β. Combining this with the inference above, from(α → ∃x β),∀x∼(α → β), it follows that ∃x β,∼∃x β and from this one obtains ⊥.Therefore, (α → ∃x β) ` ∼∀x∼(α → β), by (DMT) and the definition of ∼. Byitem (iv), (α → ∃x β) ` ∃x(α → β). As x does not occur free in α, the result followsagain by (DMT). �

Lemma 2.9. If x does not occur free in ϕ and ψ, the following holds in QmbC:

1. If Γ ` (φ→ ϕ)→ ψ then Γ ` (∀x φ→ ϕ)→ ψ

2. If Γ ` (φ→ ϕ) then Γ ` (∀x φ→ ϕ)

3. If Γ ` (ϕ→ φ)→ ψ then Γ ` (ϕ→ ∃x φ)→ ψ

4. If Γ ` (ϕ→ φ) then Γ ` (ϕ→ ∃x φ).

Proof. 1. By Lemma 2.8(v), ` (∀xφ → ϕ) → ∃x(φ → ϕ). Then, by Theorem 2.7,` (∃x(φ→ ϕ)→ ψ)→ ((∀xφ→ ϕ)→ ψ).

Thus, suppose that Γ ` (φ → ϕ) → ψ. Then Γ ` ∃x(φ → ϕ) → ψ, by (∃-In). Bythe observation above, Γ ` (∀xφ→ ϕ)→ ψ.

2. Consider the following derivation in QmbC:

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1. φ→ ϕ (premise)

2. ∀xφ (premise)

3. ∀xφ→ φ (Ax13)

4. φ (MP 2,3)

5. ϕ (MP 1,4)

Thus, φ→ ϕ,∀xφ ` ϕ, and so, by (DMT), it follows that φ→ ϕ ` ∀xφ→ ϕ. The resultfollows by transitivity of derivations.

3. By Lemma 2.8(vi), ` (ϕ → ∃x φ) → ∃x(ϕ → φ). By Theorem 2.7, it follows that` (∃x(ϕ→ φ)→ ψ)→ ((ϕ→ ∃x φ)→ ψ).

Now, suppose that Γ ` (ϕ→ φ)→ ψ. By (∃-In) it follows that Γ ` ∃x(ϕ→ φ)→ ψ.Then Γ ` (ϕ→ ∃x φ)→ ψ, by the observation above.

4. Consider the following derivation in QmbC:

1. ϕ→ φ (premise)

2. ϕ (premise)

3. φ (MP 1,2)

4. φ→ ∃xφ (Ax12)

5. ∃xφ (MP 3,4)

Then, ϕ → φ, ϕ ` ∃xφ. By (DMT), ϕ → φ ` ϕ → ∃xφ. Therefore, if Γ ` ϕ → φ thenΓ ` ϕ→ ∃xφ, by transitivity of derivations. �

This quick treatment of the syntax is already sufficient for the enterprise of seman-tics, and for clarifying certain subtleties therein.

3 First-order paraconsistent structuresThis section introduces the semantic interpretation for QmbC, which consists of the(usual) Tarskian structures endowed with paraconsistent valuations. In the followingsections the soundness and completeness of QmbC with respect to such interpretationswill be proved.

Definition 3.1 (Structures). Let Σ be a first-order signature (see Definition 1.1). Astructure over Σ is pair A = 〈A, IA〉 such that A is a non-empty set (the domain of thestructure) and IA is an interpretation mapping assigning, to each individual constantc ∈ C, an element IA(c) of A; to each function symbol f of arity n, a function IA( f ) :An → A; and to each predicate symbol P of arity n, a relation IA(P) ⊆ An.

A structure A over Σ defines an interpretation mapping (·)A : CTΣ → A from theset CTΣ of closed terms (that is, without variables) of Σ to the set A. This mapping isdefined recursively as follows:

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• cA = IA(c) if c is an individual constant;

• f (t1, . . . , tn)A = IA( f )(tA1 , . . . , tAn ) if f is a function symbol of arity n and t1, . . . , tn ∈

CTΣ.

From now on, the notaton f A and PA will be used, instead of IA( f ) and IA(P), for afunction symbol f and a predicate symbol P, respectively.

In order to interpret the quantifiers in a given structure A, it is useful to give a formalname to each element of the domain A of A by means of new individual constants.

Definition 3.2 (Diagram languages and extended structures). Consider a structure Aover Σ. The diagram language of A, denoted by LΣ(A), or simply L(A), is definedover the signature ΣA obtained from Σ by adding a new individual constant a for eachelement a of the domain A of A. The notation TΣ(A), or simply T (A), will be used todenote the set of terms of the diagram language of A.

The structure A can be naturally extended to a structure A = 〈A, IA〉 over ΣA by

defining IA

(a) = a for every a ∈ A.

In order to define QmbC-valuations able to interpret formulas in a given structure,it will be necessary for technical reasons to deal with some notions introduced in thefollowing definition:

Definition 3.3 (Multiple substitution). Let A be a structure for a signature Σ, and~x = x1, . . . , xn a sequence of different variables. The set of formulas of L(A) whose freevariables occur in the sequence ~x is denoted by L(A)~x, and ~x is said to be a context forthe formulas in L(A)~x. The set LΣ~x of all the formulas of LΣ with context ~x is definedanalogously. Given a sequence ~a = a1, . . . , an of elements in A and ϕ ∈ L(A)~x, thenotation ϕ[~x/~a] denotes the sentence of S L(A), obtained from ϕ by substituting simulta-neously every free occurrence of variable xi by the constant ai, for 1 ≤ i ≤ n. In thesame way, if t is a term over the signature ΣA whose variables occur in the sequence~x, then t[~x/~a] is the closed term obtained from t by substituting simultaneously everyoccurrence of variable xi by the constant ai, for 1 ≤ i ≤ n. The set of all the terms ofthe signature ΣA of L(A) with context ~x will be denotd by T (A)~x.

Remark 3.4. Observe that when n = 1, the notation introduced in Definition 3.3 isdifferent to that introduced in Section 1 concerning substitutions of terms for variables.In fact, according to the latter, ϕ[x/a] denotes the substitution of constant a for variablex. But, according to Definition 3.3, the same formula can be denoted by ϕ[x/a] (whenconsidering x as a context). As it will be convenient to identify (informally) an elementb of A with the constant b of ΣA, this duality is not problematic. (Notice that this dualityalready appears in the following definition.)

Definition 3.5 (QmbC-valuations). Let A be a structure over Σ with domain A. Amapping v : S L(A) → {0, 1} is a QmbC-valuation over A if it satisfies the followingclauses:

(vPred) v(P(t1, . . . , tn)) = 1 ⇐⇒ 〈 tA1 , . . . , tAn 〉 ∈ IA(P), for P(t1, . . . , tn) ∈ AtΣA

(vOr) v(α ∨ β) = 1 ⇐⇒ v(α) = 1 or v(β) = 1

12

(vAnd) v(α ∧ β) = 1 ⇐⇒ v(α) = 1 and v(β) = 1

(vImp) v(α→ β) = 1 ⇐⇒ v(α) = 0 or v(β) = 1

(vNeg) v(α) = 0 =⇒ v(¬α) = 1

(vCon) v(◦α) = 1 =⇒ v(α) = 0 or v(¬α) = 0

(vVar) v(φ) = v(ψ) whenever φ is a variant of ψ

(vEx) v(∃xφ) = 1 ⇐⇒ v(φ[x/a]) = 1 for some a ∈ A

(vUni) v(∀xφ) = 1 ⇐⇒ v(φ[x/a]) = 1 for every a ∈ A

(sNeg) For every context (~x; z) and (~x;~y), for every sequence (~a;~b) in A interpreting(~x;~y), for every ϕ ∈ L(A)~x;z and every t ∈ T (A)~x;~y such that t is free for z in ϕ, ifϕ[z/t] ∈ L(A)~x;~y and b = (t[~x;~y/~a;~b])A then:

v((ϕ[z/t])[~x;~y/~a;~b]) = v(ϕ[~x; z/~a; b]) =⇒ v((¬ϕ[z/t])[~x;~y/~a;~b]) = v(¬ϕ[~x; z/~a; b])

(sCon) For every context (~x; z) and (~x;~y), for every sequence (~a;~b) in A interpreting(~x;~y), for every ϕ ∈ L(A)~x;z and every t ∈ T (A)~x;~y such that t is free for z in ϕ, ifϕ[z/t] ∈ L(A)~x;~y and b = (t[~x;~y/~a;~b])A then:

v((ϕ[z/t])[~x;~y/~a;~b]) = v(ϕ[~x; z/~a; b]) =⇒ v((◦ϕ[z/t])[~x;~y/~a;~b]) = v(◦ϕ[~x; z/~a; b]).

Notice that, in particular,

v(P(a1, . . . , an)) = 1 ⇐⇒ 〈 a1, . . . , an 〉 ∈ PA

for every n-ary predicate symbol P and every a1, . . . , an ∈ A. On the other hand,

v(∼α) = 1 ⇐⇒ v(α) = 0

for every formula α.

Definition 3.6 (Interpretations). An interpretation for QmbC (over signature Σ) is apair 〈A, v〉, such that A is a structure over Σ and v : S L(A) → {0, 1} is a QmbC-valuation.

As would be expected from the previous definition, the notions of satisfiability (andthus the semantical consequence relation) are defined for sentences of the extendedlanguage.

Definition 3.7 (Semantical consequence relation). An interpretation 〈A, v〉 for QmbCover Σ is said to satisfy a sentence ϕ ∈ S L(A), denoted by A, v � ϕ, if v(ϕ) = 1.If Γ ⊆ S L(A), then A, v � Γ means that A, v � γ for every γ ∈ Γ. Finally, givenΓ ∪ {ϕ} ⊆ S L(A), we say that ϕ is a semantical consequence of Γ, denoted by Γ � ϕ, ifA, v � Γ implies that A, v � ϕ, for every interpretation 〈A, v〉.

Remark 3.8. Clauses (sNeg) and (sCon) are of a purely technical character, and theyestablish that if two related formulas in the diagram language involving substitutions

13

get the same truth value, then this equality must be preserved through the non-truth-functional connectives ¬ and ◦. For instance, let P be a symbol for a unary predicateand f a symbol for a unary function. Let ~x be the empty context (and so ~a is alsoempty); ~y = x (a single variable), ~b = a (a single element of A), ϕ = P(z) (where z is avariable) and t = f (x). Let b = (t[x/a])A = f (a)A = f A(a). Then,

v((P(z)[z/t])[x/a]) = v(P(t)[x/a]) = v(P( f (x))[x/a]) = v(P( f (a)))

whilev(P(z)[z/b]) = v(P(b)) = v

(P

(f A(a)

)).

By (vPred), v(P( f (a))) and v(P

(f A(a)

))coincide. However, only clauses (sNeg) and

(sCon) can guarantee that v(#P( f (a))) = v(#P

(f A(a)

))for # ∈ {¬, ◦}, as expected.

This feature will be fundamental in order to prove the Substitution Lemma (Theo-rem 3.13) which, in turn, is crucial in the proof of the soundness of QmbC with respectto interpretations.

It is important to note that in the absence of (sNeg) and (sCon), it is possible to findinterpretations falsifying axioms (Ax12) and (Ax13) (see Remark 4.2).

The semantical notions introduced above can be extended to general formulas, thatis, to formulas having free variables, by using some concepts from Definition 3.3.

Definition 3.9 (Extended valuation). Let A be a structure over Σ, ~x a context and ~a asequence of elements in A interpreting ~x. If v : S L(A) → {0, 1} is a QmbC-valuationover A, its extension v~a

~x : L(A)~x → {0, 1} is defined as follows: v~a~x(ϕ) = v(ϕ[~x/~a]), for

every ϕ ∈ L(A)~x.

Remarks 3.10.

(1) Clearly, if ϕ ∈ L(A)~x and ~y = (~x;~z) with~z = z1, . . . , zm then v~a;~b~y (ϕ) = v~a

~x(ϕ) for every

sequence ~b = b1, . . . , bm in A. In particular, v(ϕ) = v~a~x(ϕ) for every ~x and ~a, whenever

ϕ ∈ S L(A).

(2) The clauses for QmbC-valuations (see Definition 3.5) can be reintroduced in termsof extended valuations. The clauses for connectives and quantifiers are essentially thesame: it is enough to carry on the context ~x and the sequence ~a interpreting it. Forinstance, clause (vUni) changes to

(vUni)’ v~a~x(∀xφ) = 1 ⇐⇒ v~a

~x(φ[x/a]) = 1 for every a ∈ A,

observing that whether x occurs or not in ~x is irrelevant. In order to see this, notethat v~a

~x(∀xφ) = v((∀xφ)[~x/~a]) = v((∀y(φ[x/y]))[~x/~a]), where y is a variable that doesnot occur either in ~x or in φ, by clause (vVar) and the definition of substitution. But(∀y(φ[x/y]))[~x/~a] = ∀y(φ[x/y][~x/~a]), and so v~a

~x(∀xφ) = v(∀y(φ[x/y][~x/~a])) = 1 iffv(φ[x/y][~x/~a][y/a]) = v(φ[x/a][~x/~a]) = 1, for every a ∈ A. But the latter is equivalentto saying that v~a

~x(φ[x/a]) = 1, for every a ∈ A.On the other hand, the two clauses concerning substitution can be presented in

a simplified way. Thus, under the same notation and assumptions as for (sNeg) and

14

(sCon), the corresponding clauses for extended valuations are the following:

(sNeg)’ v~a;~b~x;~y(ϕ[z/t]) = v~a;b

~x;z (ϕ) =⇒ v~a;~b~x;~y(¬ϕ[z/t]) = v~a;b

~x;z (¬ϕ)

(sCon)’ v~a;~b~x;~y(ϕ[z/t]) = v~a;b

~x;z (ϕ) =⇒ v~a;~b~x;~y(◦ϕ[z/t]) = v~a;b

~x;z (◦ϕ).

Definition 3.11 (Extended semantical consequence relation). An interpretation 〈A, v〉is said to satisfy a formula ϕ ∈ L(A)~x, denoted by A, v �~x ϕ, if v~a

~x(ϕ) = 1 for everysequence ~a in A. If Γ ⊆ L(A)~x, then A, v �~x Γ means that A, v �~x γ, for every γ ∈ Γ.Finally, given Γ∪ {ϕ} ⊆ L(A)~x, ϕ is said to be a semantical consequence of Γ in context~x, denoted by Γ �~x ϕ, if A, v �~x Γ implies that A, v �~x ϕ, for every interpretation 〈A, v〉.

Remark 3.12. Observe that when Γ ∪ {ϕ} ⊆ S L(A), the notions �~x and � coincide.Moreover,

Γ �~x ϕ ⇐⇒ (∀)Γ � (∀)ϕ

where (∀)ϕ = ∀x1 · · · ∀xnϕ and (∀)Γ = {(∀)γ : γ ∈ Γ}.

This section concludes with the proof of a technical result which is fundamental inorder to state the soundness of QmbC with respect to the proposed semantics. In orderto lighten notation, and without loss of generality, extended valuations will be used.

Theorem 3.13 (Substitution Lemma). Let t be a term free for the variable z in theformula ϕ. Suppose that (~x; z) and (~x;~y) are contexts for ϕ and ϕ[z/t], respectively. Let〈A, v〉 be an interpretation for QmbC. If b = (t[~x;~y/~a;~b])A then:

v~a;~b~x;~y(ϕ[z/t]) = v~a;b

~x;z (ϕ) .

Proof. The proof is identical with that for classical logic, by induction on the complex-ity of ϕ ∈ L(A)~x;z.(a) ϕ = P(t1, . . . , tk), with P a symbol for predicate and t1, . . . , tk terms in T (A)~x;z. Then,ϕ[z/t] = P(t1[z/t], . . . , tk[z/t]). By the definition of extended valuation it follows that

v~a;~b~x;~y(ϕ[z/t]) = 1 iff 〈 ((t1[z/t])[~x;~y/~a;~b])A, . . . , ((tk[z/t])[~x;~y/~a;~b])A 〉 ∈ IA(P).

By induction on the complexity of the term u ∈ T (A)~x;z, it is easy to prove that

((u[z/t])[~x;~y/~a;~b])A = (u[~x; z/~a; b])A

for b = (t[~x;~y/~a;~b])A. From this,

〈 ((t1[z/t])[~x;~y/~a;~b])A, . . . , ((tk[z/t])[~x;~y/~a;~b])A 〉 ∈ IA(P)

if and only if〈 (t1[~x; z/~a; b])A, . . . , (tk[~x; z/~a; b])A 〉 ∈ IA(P) .

As〈 (t1[~x; z/~a; b])A, . . . , (tk[~x; z/~a; b])A 〉 ∈ IA(P) iff v~a;b

~x;z (ϕ) = 1

15

it follows thatv~a;~b~x;~y(ϕ[z/t]) = 1 iff v~a;b

~x;z (ϕ) = 1 .

That is, v~a;~b~x;~y(ϕ[z/t]) = v~a;b

~x;z (ϕ).

(b) ϕ = (α#β), with # ∈ {∨,∧,→}. Assuming that α and β satisfy the property (by in-duction hypothesis), then ϕ also satisfies the property, as v is truth-functional for theseconnectives.(c) ϕ = ∀xψ. If z does not occur free in ϕ, the result is obviously true. If z oc-curs free in ϕ then, as t is free for z in ϕ, it follows that x does not occur in t.Thus, ϕ[z/t] = (∀xψ)[z/t] = ∀x(ψ[z/t]). By definition of extended valuation and Re-mark 3.10(2), if y is a variable that does not occur in either ~x;~y; z or ψ, v~a;~b

~x;~y(ϕ[z/t]) =

v~a;~b~x;~y(∀x(ψ[z/t])) = v((∀x(ψ[z/t]))[~x;~y/~a;~b]) = v(∀y(ψ[z/t][x/y][~x;~y/~a;~b])). Then, by

(vUni) and the equations above,

v~a;~b~x;~y(ϕ[z/t]) = v(∀y(ψ[z/t][x/y][~x;~y/~a;~b])) = 1

if and only ifv(ψ[z/t][x/y][~x;~y/~a;~b][y/a]) = 1 for every a ∈ A.

But ψ[z/t][x/y][~x;~y/~a;~b][y/a] = (ψ[x/y])[z/t][~x;~y; y/~a;~b; a], as individual constantsare being substituted for variables, and so the simultaneous substitution coincides withthe iterative substitution. Thus, v~a;~b

~x;~y(ϕ[z/t]) = 1 iff v((ψ[x/y])[z/t][~x;~y; y/~a;~b; a]) =

v~a;~b;a~x;~y;y ((ψ[x/y])[z/t]) = 1, for every a ∈ A.

By the induction hypothesis applied to ψ[x/y], and given that b = (t[~x;~y/~a;~b])A =

(t[~x;~y; y/~a;~b; a])A (as y is new),

v~a;~b;a~x;~y;y ((ψ[x/y])[z/t]) = 1 iff v~a;b;a

~x;z;y (ψ[x/y]) = 1 .

On the other hand,

v~a;b;a~x;z;y (ψ[x/y]) = 1 for every a iff v~a;b

~x;z (∀yψ[x/y]) = v~a;b~x;z (∀xψ) = v~a;b

~x;z (ϕ) = 1

and sov~a;~b~x;~y(ϕ[z/t]) = 1 iff v~a;b

~x;z (ϕ) = 1 .

That is, v~a;~b~x;~y(ϕ[z/t]) = v~a;b

~x;z (ϕ).

(d) ϕ = ∃xψ. This is a consequence of the fact that v(∃xδ) = v(∼∀x∼δ).(e) ϕ = #ψ, with # ∈ {¬, ◦}. By induction hypothesis,

v~a;~b~x;~y(ψ[z/t]) = v~a;b

~x;z (ψ)

and then, by clauses (vNeg)’ and (vCon)’,

v~a;~b~x;~y(ϕ[z/t]) = v~a;~b

~x;~y(#ψ[z/t]) = v~a;b~x;z (#ψ) = v~a;b

~x;z (ϕ) .

�

The importance of clauses (vNeg) and (vCon) is clear, therefore, from the proof ofthe above theorem.

16

4 Soundness of QmbCThe next step is to prove that the semantics of paraconsistent interpretations is adequatefor the logic QmbC, presented as a Hilbert calculus. For simplicity, the result will beproved just for sentences (i.e., for formulas without free variables). That is, if ∆ ∪ {ϕ}is a set of sentences, then

∆ ` ϕ ⇐⇒ ∆ � ϕ .

It should be observed that, despite the fact that the premises (the set ∆) and the conclu-sion (the formula ϕ) are sentences, a given derivation of ϕ from ∆ can involve formulaswith free variables, and so the use of extended valuations will be required. On the otherhand, there is no loss of generality by proving soundness and completeness just for sen-tences, by virtue of Remark 3.12 (which obviously also holds in the Hilbert calculusQmbC).

In this section, the soundness of QmbC is stated:

Theorem 4.1 (Soundness of QmbC with respect to interpretations). For every set ofsentences ∆ ∪ {ϕ}: if ∆ ` ϕ then ∆ � ϕ.

Proof. By induction on the length n of a derivation ϕ1, . . . , ϕn of ϕ from ∆ in QmbC,it will be proved that given a structure A, each QmbC valuation v over A such thatA, v � ∆, satisfies the following condition: v~a

~x(ϕi) = 1 for every sequence ~a in A andevery i ≤ i ≤ n, where ~x is a context for every ϕi (1 ≤ i ≤ n). In particular, it will beproved that v(ϕ) = 1, as desired.

It is clear that, in order to get the desired result, it is enough to prove the following:

(i) v~a~x(ψ) = 1 for every ~a and every instance ψ of an axiom schema of QmbC

(ii) if v~a~x(ψ1) = 1 and v~a

~x(ψ1 → ψ2) = 1 for every ~a then v~a~x(ψ2) = 1 for every ~a

(iii) if v~a;b~x;y(ψ1 → ψ2) = 1 for every (~a; b), and if the variable y does not occur free in

ψ1, then v~a~x(ψ1 → ∀yψ2) = 1 for every ~a

(iv) if v~a;b~x;y(ψ1 → ψ2) = 1 for every (~a; b), and if the variable y does not occur free in

ψ1, then v~a~x(∃yψ1 → ψ2) = 1 for every ~a.

In order to prove (i) it is enough to analyze the axioms involving quantifiers; the othersare true because of the soundness theorem of mbC for bivaluations (see Theorem 1.5).The same holds for item (ii) (concerning MP), which is obviously true. Thus considerthe following cases for item (i):(i.1) ψ = ∀zα → α[z/t] where t is free for z in α. Let ~x be a context formed by all thevariables occurring free in ∀zα and let (~x;~y) be a context formed by the variables occur-ring free in α[z/t]. Consider a sequence (~a;~b) in A interpreting (~x;~y). If v~a;~b

~x;~y(∀zα) = 0

then v~a;~b~x;~y(ψ) = 1. If, on the other hand, v~a;~b

~x;~y(∀zα) = v~a~x(∀zα) = 1 then v~a;b

~x;z (α) = 1 forevery b ∈ A (by the considerations above concerning simultaneous and iterated substi-tutions). In particular, v~a;b

~x;z (α) = 1 for b = (t[~x;~y/~a;~b])A. By the Substitution Lemma

17

(Theorem 3.13), v~a;~b~x;~y(α[z/t]) = v~a;b

~x;z (α), as t is free for z in α. From this v~a;~b~x;~y(α[z/t]) = 1,

as required.(i.2) ψ = α[z/t] → ∃zα where t is free for z in α. The proof is analogous to that ofitem (i.1).(i.3) ψ = ∀x(α → β) → (α → ∀xβ), where x does not occur free in α. Suppose thatv~a~x(∀x(α → β)) = v~a

~x(α) = 1. If a ∈ A then v~a~x((α → β)[x/a]) = 1 and v~a

~x(α[x/a]) = 1,since x does not occur free in α. From this, v~a

~x(β[x/a]) = 1 and so v~a~x(∀xβ) = 1. This

shows that v~a~x(ψ) = 1 for every v and ~a.

(i.4) ψ = α → β, where α is a variant of β. This is an obvious consequence of clause(vVar) and the fact that α[~x/~a] is a variant of β[~x/~a] whenever α is a variant of β.

Now, in order to prove (iii), suppose that v~a;b~x;y(ψ1 → ψ2) = 1 for every (~a; b), where

the variable y does not occur free in ψ1. Fix the sequence ~a. If v~a~x(ψ1) = 0 then

v~a~x(ψ1 → ∀yψ2) = 1. On the other hand, if v~a

~x(ψ1) = v~a;b~x;y(ψ1) = 1 then, by hypothesis,

v~a;b~x;y(ψ2) = v~a

~x(ψ2[y/b]) = 1, for every b ∈ A. From this, v~a~x(∀yψ2) = 1.

Item (iv) is proved in a similar way. �

Remark 4.2. As observed in Remark 3.8, clauses (sNeg) and (sCon) are crucial inorder to prove the soundness theorem above.

Consider, for instance, α = ¬P(z) and t = f (x, y), with P a symbol denoting aunary predicate. Suppose that va;b

x;y(∀zα) = v(∀z¬P(z)) = 1. Then,

v(¬P(e)) = 1 for every e ∈ A. (1)

In particular,v(¬P

(f A(a, b)

))= 1 . (2)

On the other hand,

va;bx;y(α[z/t]) = va;b

x;y(¬P( f (x, y))) = v(¬P( f (a, b))). (3)

In order to guarantee that v(¬P( f (a, b))) = 1, one has to ensure that

v(¬P

(f A(a, b)

))= v(¬P( f (a, b))). (∗)

But the latter is only obtained from the Substitution Lemma or, in this specific case, byclause (sNeg). In other words, without (sNeg) it would be possible to find a valuationv over a structure A such that va;b

x;y(∀z¬P(z)) = 1 but va;bx;y(¬P( f (x, y))) = 0. That is, it

would be possible to falsify the instance

∀z ¬P(z)→ ¬P( f (x, y))

of axiom schema (Ax13). By a similar argument, it would be possible to falsify theinstance

∀z ◦P(z)→ ◦P( f (x, y))

of axiom schema (Ax13) without the presence of clause (sCon).

18

5 Completeness of QmbCGiven a first-order signature Σ, any set of sentences in LΣ will be called a theory.

This section is dedicated to prove the completeness of QmbC with respect to in-terpretations. The proof will be analogous to that for classical logic: given a theory Γ

which does not deduce a given sentence ϕ (being, therefore, non-trivial), a canonicalinterpretation will be constructed which satisfies Γ but does not satisfy ϕ. Therefore, itwill be proved that:

Γ 0 ϕ =⇒ Γ 2 ϕ .

In order to do this, the original theory Γ will be conservatively extended to a Henkintheory ∆ in an extended signature, that is, to a theory containing a witness for eachexistential sentence. Since ∆ is a conservative extension of Γ, it does not derive ϕ.Thus, by using a classical and general result by Lindenbaum-Łos, ∆ will be extendedto a maximal theory ∆ which does not derive ϕ and is still a Henkin theory. Usinga canonical structure generated from ∆, the characteristic map of ∆ will constitute aQmbC-valuation which, as required, satisfies Γ but does not satisfy ϕ.

5.1 Henkin theoriesA Henkin theory is a theory designed to comply with the inference rules for quantifiers.

Definition 5.1 (Henkin theory). Given a theory ∆ ⊆ S L and a non-empty set C ofconstants of the signature Σ of L, ∆ is called a C-Henkin theory in QmbC if it satisfiesthe following: for every sentence of the form ∃xφ in S L, there exists a constant c in Csuch that if ∆ ` ∃xφ then ∆ ` φ[x/c].

The set C is called a set of witnesses of ∆. The next step is to prove that any theorycan be conservatively extended to a C-Henkin theory, for some C.

Theorem 5.2 (Theorem of Constants). Let ∆ ⊆ S L be a theory in QmbC over asignature Σ, and let `C be the consequence relation of QmbC over the signature ΣC ,which is obtained from Σ by adding a set C of new individual constants. Then, for everyϕ ∈ S L,

∆ ` ϕ iff ∆ `C ϕ.

That is, QmbC (over ΣC) is a conservative extension of QmbC (over Σ).

Proof. The proof is analogous to that for classical first-order logic: given a derivation πof ϕ from Γ in QmbC over ΣC , the constants of C occurring in π are replaced uniformlyby new variables, obtaining a finite sequence π′ of formulas over Σ. But the instancesover ΣC of axioms of QmbC occurring in π become instances over Σ of axioms ofQmbC, and the same holds for the instances of inference rules. Then, π′ is in fact aderivation of ϕ from Γ in QmbC over Σ. The converse is obvious. �

Theorem 5.3. Every theory ∆ ⊆ S L in QmbC over a signature Σ can be conservativelyextended to a C-Henkin theory ∆H in QmbC over a signature ΣC , as in Theorem 5.2.That is, ∆ ⊆ ∆H and if ϕ ∈ S L then ∆ ` ϕ iff ∆H `C ϕ. Additionally, any extension of∆H by sentences in the signature ΣC is also a C-Henkin theory.

19

Proof. Let us define an increasing denumerable sequence of signatures Σ0 ⊆ Σ1 ⊆

. . . such that each Σn+1 is obtained from Σn by adding new individual constants. Thelanguage LΣn generated by Σn will be denoted by Ln, and so Ln ⊆ Ln+1.

The definition of the signatures is as follows:

(i) Σ0 = Σ; then, L0 = LΣ0 = LΣ.

(ii) Σ1 is obtained from Σ0 by adding the set of new individual constants

C1 ={

c∃xα : ∃xα is a sentence of L0};

(iii) For n ≥ 1, Σn+1is obtained from Σn by adding the set of new individual constants

Cn+1 ={

c∃xα : ∃xα is a sentence of Ln \ Ln−1}

.

Let C =⋃

n≥1 Cn be the set of new individual constants, let ΣC =⋃

n≥0 Σn be thesignature obtained by adding the new constants, and let LC = LΣC .

Consider now the following sequence of sets of non-logical axioms over ΣC:

AX0 = ∅

AXn+1 = {∃xφ→ φ[x/c∃xφ] : ∃xφ ∈ S Ln } (for n ≥ 0).

Finally, let ∆H = ∆ ∪⋃

n≥1 AXn. Observe that ∆H ⊆ S LC , and that it extends ∆. Itwill be proved now that ∆H is a conservative extension of ∆. Thus, let φ ∈ S L such that∆H `C φ, and let π be a derivation of φ from ∆H in QmbC over ΣC . As π is finite, thereexists a finite set ∆H

0 ⊆ ∆H such that ∆,∆H0 `C φ. Let ∃xψ → ψ[x/c∃xψ] in ∆H

0 , andlet ∆H

1 = ∆H0 \ {∃xψ → ψ[x/c∃xψ]}. Given that ∆H

0 is a set of sentences, (DMT) can beapplied in order to obtain ∆,∆H

1 `C (∃xψ→ ψ[x/c∃xψ])→ φ. Observe that the constantc∃xψ only appears in the conclusion and so, by using the same technique employed inthe proof of Theorem 5.2, that constant can be substituted by a new variable, namely y.This means that ∆,∆H

1 `C (∃xψ→ ψ[x/y])→ φ. By Lemma 2.9(3), ∆,∆H1 `C (∃xψ→

∃y(ψ[x/y])) → φ. On the other hand, `C ∃x ψ → ∃y(ψ[x/y]), by axiom (Ax15). Fromthis, ∆,∆H

1 `C φ.By repeating this process, every element of ∆H

1 can be eliminated in a finite numberof steps, proving that ∆ `C φ. By Theorem 5.2, it is finally obtained that ∆ ` φ. Thisshows that ∆H is in fact a conservative extension of ∆.

To finish the proof, consider an extension ∆H ′ of ∆H (in particular, one can chosse∆H ′ = ∆H), formed by sentences of LC . Suppose that for some sentence ∃xϕ ∈ LC ,the following condition holds: ∆H ′ `C ∃xϕ. As ∆H ′ extends ∆H , then ∆H ′ `C ∃xϕ →ϕ[x/c∃xϕ] and so ∆H ′ `C ϕ[x/c∃xϕ]. This means that ∆H ′ (and, in particular, ∆H) is aC-Henkin theory in QmbC over ΣC . �

5.2 Maximal extensions: the Lindenbaum-Łos theoremThe next step towards the proof of completeness requires the notion of maximal theo-ries with respect to a sentence. In particular, a classical and very useful result due toLindenbaum and Łos will be crucial. Some well-known concepts, essential ingredientsfor the proofs, are here recalled.

20

Definition 5.4 (Tarskian Logic2). Let L be a logic defined over a language L and witha consequence relation `. Then L is said to be Tarskian if it satisfies the following, forevery Γ ∪ ∆ ∪ {α} ⊆ L:

(1) if α ∈ Γ then Γ ` α;

(2) if Γ ` α and Γ ⊆ ∆ then ∆ ` α;

(3) if ∆ ` α and Γ ` β for every β ∈ ∆ then Γ ` α.

A Tarskian logic is finitary if it satisfies, additionally, the following:

(4) if Γ ` α then there exists a finite subset Γ0 of Γ such that Γ0 ` α.

�

Definition 5.5. LetL be a Tarskian logic over the language L, and let Γ∪{ϕ} ⊆ L. Theset Γ is said to be maximally non-trivial with relation to ϕ in L if Γ 0 ϕ but Γ, ψ ` ϕ forany ψ ∈ L \ Γ.

�

A proof of the following classical result can be found in [27], Theorem 22.2.

Theorem 5.6 (Lindenbaum-Łos). Let L be a Tarskian and finitary logic over the lan-guage L. Let Γ∪{ϕ} ⊆ L such that Γ 0 ϕ. Then there exists a set ∆, such that Γ ⊆ ∆ ⊆ Lwith ∆ maximally non-trivial with relation to ϕ in L.

Clearly, any logic defined by means of a Hilbert calculus where the inference rulesare finitary is Tarskian and finitary, and so the theorem above holds for it. In particular,it holds for QmbC restricted to sentences: it is easy to see that the consequence relationof QmbC, when restricted to sentences, is Tarskian and finitary. Then, if the set L ofDefinition 5.5 is also restricted to sentences (that is, to S L), the following holds:

Corollary 5.7. Let Γ∪ {ϕ} ⊆ S L be a set of sentences such that Γ 0 ϕ in QmbC. Then,there exists a set of sentences ∆ ⊆ S L extending Γ which is maximally non-trivial withrelation to ϕ in QmbC (by restricting ` to sentences).

Theorem 5.8 (Canonical interpretation). Let ∆ ⊆ S L be a set of sentences over asignature Σ containing at least one individual constant. Assume that ∆ is a C-Henkintheory in QmbC for a non-empty set C of individual constants of Σ, and that ∆ isalso maximally non-trivial with relation to ϕ in QmbC, for some sentence ϕ. Then, ∆

induces a canonical structure A and a canonical QmbC-valuation v : S L(A) → {0, 1}over A such that, for every sentence ψ ∈ S L:

A, v � ψ ⇐⇒ ∆ ` ψ .

Proof. Let A = CTΣ be the set of closed terms (that is, without variables) over thesignature Σ. Then a structure A = 〈A, IA〉 over Σ can be defined as follows: IA(c) = c,if c is an individual constant; if f is a function symbol, then IA( f ) : An → A is such

2See, for instance, [27].

21

that IA( f )(t1, . . . , tn) = f (t1, . . . , tn) (and so tA = t for every t ∈ CTΣ). Finally, definethe interpretation of the predicate symbols as follows:

〈t1, . . . , tn〉 ∈ IA(P) ⇐⇒ ∆ ` P(t1, . . . , tn) .

Recall from Definition 3.2 the diagram language L(A) of A, its set T (A) of terms andthe extended structure A such that I

A(t) = t. Here, t is a new constant introduced for

each closed term t ∈ CTΣ (which is as an element of the domain of D). Let CT (A) bethe set of closed terms of the language L(A), and consider a mapping ∗ : CT (A)→ CTΣ

defined recursively as follows: ( t )∗ = t if t ∈ CTΣ; c∗ = c if c is an individual constantof Σ; and ( f (t1, . . . , tn))∗ = f (t∗1, . . . , t

∗n) if f is a function symbol of Σ. It is clear that

t∗ = tA for every t ∈ CT (A). This mapping can be naturally extended to sentences:let ∗ : S L(A) → S L be defined recursively as follows: (P(t1, . . . , tn))∗ = P(t∗1, . . . , t

∗n) if

P(t1, . . . , tn) is atomic; (#ψ)∗ = #(ψ∗) if # ∈ {¬, ◦}; (ϕ#ψ)∗ = (ϕ∗#ψ∗) if # ∈ {∧,∨,→};and (Qxψ)∗ = Qx(ψ∗) if Q ∈ {∀,∃}. Clearly, ϕ∗ is the sentence of LΣ obtained from ϕby substituting every occurrence of a constant t by the term t itself.

Finally, the mapping v : S L(A) → {0, 1} can be defined as follows:

v(ϕ) = 1 ⇐⇒ ∆ ` ϕ∗ .

By construction of v, it is clear that for every sentence ϕ ∈ S L:

A, v � ϕ ⇐⇒ ∆ ` ϕ .

The proof will be completed by showing that v is in fact a QmbC-valuation (recallDefinition 3.5).In order to prove (vPred), if P(t1, . . . , tn) is an atomic sentence of L(A), then

v(P(t1, . . . , tn)) = 1 ⇐⇒ ∆ ` P(t∗1, . . . , t∗n) .

But this happens iff 〈t∗1, . . . , t∗n〉 ∈ IA(P), by definition of IA(P). Given that t∗ = tA for

every t ∈ CT (A), then

v(P(t1, . . . , tn)) = 1 ⇐⇒ 〈tA1 , . . . , tAn 〉 ∈ IA(P) .

In order to see that v satisfies clauses (vOr), (vAnd), (vImp), (vNeg) and (vCon), thereader can consult [7], where the corresponding proof is provided for mbC, and so itholds mutatis mutandis for QmbC (by using the definition of ∗).The satisfaction of clause (vVar) follows from axiom (Ax15) and the definition of ∗.In order to prove (vEx), firstly observe that, if ∃xφ ∈ S L then ∆ ` ∃xφ implies that∆ ` φ[x/c] for some constant c of C (which is an element of CTΣ), as ∆ is a C-Henkintheory in QmbC. On the other hand, if ∆ ` φ[x/t] for some closed term t in CTΣ then∆ ` ∃xφ, in virtue of (Ax12) and MP. Consider now a sentence in L(A) of the form∃xφ. Then v(∃xφ) = 1 iff ∆ ` (∃xφ)∗ iff ∆ `L ∃x(φ)∗, by definition of ∗. From this, andby the observation above, one infers that v(∃xφ) = 1 iff ∆ ` ((φ)∗)[x/t] for some closedterm t in CTΣ. On the other hand, it is easy to prove by induction on the complexity of

22

φ that ((φ)∗)[x/t] = (φ[x/t])∗, for every t ∈ CTΣ. Thus, v(∃xφ) = 1 iff ∆ ` (φ[x/t])∗ forsome t of CTΣ. From this it follows that v(∃xφ) = 1 iff v(φ[x/t]) = 1 for some elementt of CTΣ.Concerning (vUni), as v satisfies the clauses for the propositional connectives thenv(∼ϕ) = 1 iff v(ϕ) = 0. On the other hand, v(∀xϕ) = v(∼∃x∼ϕ), because of the the-orems ` ∀xϕ → ∼∃x∼ϕ and ` ∼∃x∼ϕ → ∀xϕ of QmbC (and by the SoundnessTheorem). From this, and using clause (vEx), it can be immediately seen that v satis-fies clause (vUni).Finally, it will be proved that the pair 〈A, v〉 satisfies the Substitution Lemma (see The-orem 3.13) and so the mapping v satisfies the clauses (sNeg) and (sCon).Facts: Let t be a term free for a variable z in a formula ϕ. Suppose that (~x; z) and (~x;~y)are contexts for ϕ and ϕ[z/t], respectively, and let b = (t[~x;~y/~a;~b])∗. Then:(i) ((u[z/t])[~x;~y/~a;~b])∗ = (u[~x; z/~a; b])∗, for every term u ∈ T (A)~x;z.(ii) ((ϕ[z/t])[~x;~y/~a;~b])∗ = (ϕ[~x; z/~a; b])∗.Item (i) can be easily proved by induction on the complexity of u. (Notice that this factwas already used in item (a) of the proof of Theorem 3.13, given that uA = u∗ for everyterm u).Item (ii) is proved by induction on the complexity of ϕ. If ϕ is atomic, the result fol-lows immediately by item (i). The propagation of the induction hypothesis throughthe connectives ∧, ∨, →, ¬ and ◦ is obvious. The propagation of the induction hy-pothesis through the quantifiers is a consequence of the fact that t is free for z inϕ. Therefore, x does not occur in t when ϕ = Qxψ, with Q ∈ {∀,∃}. From this,((Qxψ)[z/t])[~x;~y/~a;~b] = Qx((ψ[z/t])[~x;~y/~a;~b]), and the result follows by inductionhypothesis and the definition of ∗. This concludes the proof of the Facts.

Now, let ϕ be a formula of L(A), and t a term free for the variable z in ϕ such that (~x; z)and (~x;~y) are contexts for ϕ and ϕ[z/t], respectively. Then

v~a;~b~x;~y(ϕ[z/t]) = v((ϕ[z/t])[~x;~y/~a;~b])

and sov~a;~b~x;~y(ϕ[z/t]) = 1 ⇐⇒ ∆ ` ((ϕ[z/t])[~x;~y/~a;~b])∗ .

On the other hand, v~a;b~x;z (ϕ) = v(ϕ[~x; z/~a; b]) therefore

v~a;b~x;z (ϕ) = 1 ⇐⇒ ∆ ` (ϕ[~x; z/~a; b])∗ .

Finally, by taking b = (t[~x;~y/~a;~b])∗, it follows by Facts(ii) that

v~a;~b~x;~y(ϕ[z/t]) = v~a;b

~x;z (ϕ)

as desired. From this, the mapping v satisfies clauses (sNeg) and (sCon).This proves that the pair 〈A, v〉 is an interpretation with the required properties. �

Theorem 5.9 (Completeness of QmbC with respect to interpretations). For every setof sentences ∆ ∪ {ϕ} over a signature Σ, if ∆ � ϕ then ∆ ` ϕ.

23

Proof. Suppose that ∆ ∪ {ϕ} ⊆ S L such that ∆ 0 ϕ. By Theorem 5.3, there exists a C-Henkin theory ∆H defined over a signature ΣC which conservatively extends ∆, that is:for every sentence ψ ∈ S L, ∆ ` ψ iff ∆H `C ψ, recalling that `C denotes the consequencerelation of the Hilbert calculus QmbC over signature ΣC . By Corollary 5.7, there existsa set of sentences ∆H over the signature ΣC which extends ∆ and is maximally non-trivial with relation to ϕ in QmbC (as a calculus defined over ΣC). By Theorem 5.3,∆H is also a C-Henkin theory. By Theorem 5.8, a canonical interpretation 〈A, v〉 overΣC can be defined such that, for every sentence ψ over ΣC ,

A, v � ψ ⇐⇒ ∆H `C ψ .

In particular, A, v � ∆ (as ∆ ⊆ ∆H) and A, v 2 ϕ (as ∆H 0C ϕ). Finally, let A bethe reduct of A to the signature Σ.3 That is, I

Acoincides with IA over Σ (and so just

‘forgets’ the interpretation of the individual constants in C). Let v be the restrictionof v to the set of sentences S L(A ) of the diagram language of A. Clearly, A, v � ψ iff

A, v � ψ, for every sentence ψ ∈ S L(A ). Therefore, 〈A, v〉 is an interpretation for QmbCover Σ such that A, v � ∆ but A, v 2 ϕ. This shows that ∆ 2 ϕ, as required. �

6 Compactness and Lowenhëim-Skolem TheoremsThis section is devoted to establishing some fundamental theorems of Model Theoryfor QmbC, namely: Compactness and the Lowenhëim-Skolem Theorems. Here aresome basic definitions and results to make the arguments more clear and self contained:

Definition 6.1. Let Σ be a first-order signature for LFIs. Consider the sets

C = {c : c is an individual constant of Σ}

F = { f : f is a function symbol of arity n of Σ, for some n ≥ 1}

P = {P : P is a predicate symbol of arity n of Σ, for some n ≥ 1}.

The cardinal of Σ, denoted by ||Σ||, is the cardinal of the set

ω ∪ C ∪ F ∪ P

where ω denotes the set of natural numbers.

Definition 6.2. Let A = 〈A, IA〉 be a structure. The cardinal of A, denoted by ||A||, isthe cardinal of the set A. Given an interpretation 〈A, v〉, its cardinal is, by definition,the cardinal of the structure A.

Definition 6.3. Let Γ be a set of sentences over a signature Σ. We say that Γ is satis-fiable if there exists an interpretation 〈A, v〉 such that A, v � Γ (recall Definition 3.7).Otherwise, Γ is said to be unsatisfiable. If A, v � Γ then 〈A, v〉 is called a model of Γ.

3As usual, if Σ is a subsignature of another signature Σ′, then any structure A over Σ′ can be consideredas a structure over Σ, by ‘forgetting’ the interpretation of the symbols in Σ′ that do not belong to Σ. Such astructure over Σ is called the reduct of A to Σ (see [10]).

24

By recalling that a strong negation ∼ can be defined in QmbC, it is immediate toprove the following:

Proposition 6.4. Let Γ ∪ {ϕ} be a set of sentences over a signature Σ. Then, Γ � ϕ ifand only if Γ ∪ {∼ϕ} is unsatisfiable.

Proof. The proof is entirely analogous to that for classical logic, taking into accountthe definition of the consequence relation � (see Definition 3.7) and the fact thatA, v � ϕif and only if A, v 2 ∼ϕ. �

Profiting from the previous result, the following can easily be proved:

Proposition 6.5. Let Σ be a signature. Then, the following statements are equivalentin QmbC:

(i) For every set of sentences Γ over Σ the following holds: every finite subset of Γ issatisfiable if and only if Γ is satisfiable;

(ii) For every set of sentences Γ ∪ {ϕ} over Σ the following holds: Γ � ϕ if and only ifΓ0 � ϕ for some finite subset Γ0 of Γ.

Proof. (i) ⇒ (ii): Assuming (i), suppose that Γ � ϕ. Then Γ ∪ {∼ϕ} is unsatisfiable,by Proposition 6.4. By (i), there exists a finite subset ∆0 of Γ ∪ {∼ϕ} such that ∆0 isunsatisfiable. Let Γ0 = ∆0 \ {∼ϕ}. Then, Γ0∪{∼ϕ} is also unsatisfiable (since it contains∆0). Using again Proposition 6.4, it follows that Γ0 � ϕ, where Γ0 is a finite subset ofΓ. The converse is obvious.

(ii) ⇒ (i): Assuming (i), suppose that Γ is a set of sentences over Σ which is unsatis-fiable. Since clearly Γ is non-empty, there exists some sentence, say ϕ, belonging to Γ.Given that A, v � ϕ if and only if A, v � ∼∼ϕ, for every interpretation 〈A, v〉, it followsthat Γ ∪ {∼∼ϕ} is unsatisfiable and so Γ � ∼ϕ, by Proposition 6.4. Using (ii) one infersthat Γ0 � ∼ϕ for some finite subset Γ0 of Γ. Being so, and using Proposition 6.4 again,it follows that Γ0 ∪ {∼∼ϕ} is unsatisfiable. But then Γ0 ∪ {ϕ} is unsatisfiable, by the ob-servation above, where Γ0 ∪ {ϕ} is a finite subset of Γ. The converse is immediate. �

Because of the last result, the compactness of QmbC can be stated as follows:

Definition 6.6. The logic QmbC is (semantically) compact if it satisfies statement (i)(or, equivalently, statement (ii)) of Proposition 6.5.

Theorem 6.7 (Compactness of QmbC). The logic QmbC is (semantically) compact.

Proof. It will be proved that QmbC satisfies statement (ii) of Proposition 6.5. Supposethat Γ ∪ {ϕ} is a set of sentences such that Γ � ϕ. By the Completeness Theorem 5.9it follows that Γ ` ϕ. Hence, since the syntactical consequence relation ` is finitary,there is some finite subset Γ0 of Γ such that Γ0 ` ϕ and so Γ0 � ϕ, by the Sound-ness Theorem 4.1. The converse is immediate and so QmbC satisfies statement (ii) ofProposition 6.5, being therefore compact. �

Definition 6.8. Given a theory Γ over Σ, that is, a set of sentences in LΣ, Γ is said tobe non-trivial if Γ 2 ϕ for some sentence ϕ over Σ.

25

Theorem 6.9 (Downward Lowenhëim-Skolem Theorem for QmbC). Let Σ be a sig-nature. Every non-trivial theory Γ over Σ has a model of cardinal equal to ||Σ||.

Proof. Suppose that Γ is a non-trivial theory over Σ. Then, there is some sentence ϕover Σ such that Γ 2 ϕ. By the proof of the Completeness Theorem 5.9, there exists aninterpretation 〈A, v〉 for QmbC over Σ which is a model for Γ, such that the domain Aof the structure A is the set of closed terms over the signature ΣC . It is routine to provethat the cardinal of A is ||Σ||. �

Lemma 6.10. Let Σ be a signature, and let Γ be a non-trivial theory over Σ. If Γ has amodel of cardinal κ, then it has a model of cardinal κ′, for every cardinal κ′ greater orequal than κ.

Proof. Let 〈A, v〉 be a model of Γ of cardinal κ such that A = 〈A, IA〉. Let κ′ be a car-dinal strictly greater than κ, and let A′ be a set of cardinality κ′ such that A ⊂ A′. Fixan element a of A, and define a structure A′ = 〈A′, IA′〉 over Σ as follows. If c is anindividual constant of Σ then IA′ (c) = IA(c). If P is a predicate symbol of Σ of arity nand 〈a′1, . . . , a

′n〉 ∈ (A′)n then 〈a′1, . . . , a

′n〉 ∈ IA′ (P) if and only if 〈a1, . . . , an〉 ∈ IA(P),

where ai is a′i , if a′i ∈ A, or ai is a otherwise, for every 1 ≤ i ≤ n. If f is a function sym-bol of Σ of arity n and 〈a′1, . . . , a

′n〉 ∈ (A′)n then IA′ ( f )(a′1, . . . , a

′n) = IA( f )(a1, . . . , an),

where each ai is defined as above. Finally, consider a QmbC-valuation v′ over A′

which extends v, such that v′(ϕ[~x/~a′]) = v(ϕ[~x/~a]) for every ϕ with context ~x, every~a′ = a′1, . . . , a

′n and every ~a = a1, . . . , an such that each ai is defined from a′i as above (it

is easy to see that it is always possible to define such a valuation from v). Then, 〈A′, v′〉is a model of Γ of cardinal κ′. �

Theorem 6.11 (Upward Lowenhëim-Skolem Theorem for QmbC). Let Σ be a signa-ture. Every non-trivial theory Γ over Σ has a model of cardinal κ, for every cardinal κgreater or equal than ||Σ||.

Proof. It is a direct consequence of Theorem 6.9 and Lemma 6.10. �

7 QmbC with equalityIn order to develop higher-level applications of the quantified version QmbC of mbC,such as paraconsistent model theory or paraconsistent set theory, it is necessary toconsider a binary predicate ≈ for the equality relation satisfying the usual axioms,which should be interpreted as the identity relation. As such, the predicate ≈ will beconsidered as a logical symbol (like the connectives and the quantifiers), not belongingto the signatures. By writing, as usual, (t ≈ t′) instead of ≈ (t, t′) (where t and t′ areterms of the language), the following definitions are in order:

Definition 7.1. Let Σ be a first-order signature for LFIs (recall Definition 1.1). Theset of formulas with equality ≈ over Σ, denoted by L≈

Σ, is defined as usual, but now ex-

pressions of the form (t ≈ t′) (where t and t′ are terms of the language) are also atomicformulas. In other words, a new symbol ≈ for a binary predicate denoting equalityis added to the set of given logical symbols (connectives, quantifiers and punctuation

26

marks). The set of sentences of L≈Σ

will be denoted by S ≈Σ

. On the other hand, thediagram language of A and the corresponding sets of sentences, when including theequality symbol ≈, will be denoted by L≈(A) and S L≈(A) respectively.

If α is a formula and y is a variable free for the variable x in α, then α[x o y] denotes anyformula obtained from α by replacing some, but not necessarily all, free occurrencesof x by y.

Definition 7.2. Let Σ be a first-order signature for LFIs. The logic QmbC≈ (over Σ)is the extension of QmbC over L≈

Σobtained by adding to QmbC, besides all the new

instances of axioms and inference rules involving the equality predicate ≈, the follow-ing axiom schemas:

(AxEq1) ∀x(x ≈ x)

(AxEq2) (x ≈ y)→ (α→ α[x o y]), if y is a variable free for x in α

It is worth noting that axiom (AxEq2) depends on each α and each specific α[x o y].

Definition 7.3. The semantics for QmbC≈ is given by interpretations 〈A, v〉 (recallDefinition 3.6) such that the QmbC-valuation v : S L≈(A) → {0, 1} satisfies, additionally,the following clauses:

(vEq1) v(t1 ≈ t2) = 1 ⇐⇒ tA1 = tA2 , for every t1, t2 ∈ CT (A) (the set of closedterms of the language L(A))

(vEq2) v(a ≈ b) = 1 =⇒ v(α[x, y/a, b]) = v((α[xoy])[x, y/a, b]) for every a, b ∈ A,if y is a variable free for x in α.

Since v(a ≈ a) = 1 for every a ∈ A, by (vEq1), then v(∀x(x ≈ x)) = 1, by (vUni).However, it is possible to have v(¬(t ≈ t)) = 1, that is, A, v � ¬(t ≈ t), for someinterpretation 〈A, v〉 and some term t. In other words, it is not required that v(◦(t ≈t)) = 1 is always the case.

From the clauses (vEq1) and (vEq2), it is clear that the Substitution Lemma (The-orem 3.13) can be extended to QmbC≈, as it clearly holds for atomic formulas of theform (t ≈ t′) (and the proof is done by induction on the complexity of formulas).

Remark 7.4. At first sight, it would seem that the clause for valuations correspondingto (AxEq1) should be simply v(∀x(x ≈ x)) = 1 or, equivalently, v(a ≈ a) = 1 forevery a ∈ A. However, in order to ensure the validity of the Substitution Lemma, thestronger condition (vEq1) must be required. In fact, recall from Remark 4.2 that thevalidity of the Substitution Lemma is necessary to guarantee the soundness of (Ax13).Consider again the terms t1 = f A(a, b) and t2 = f (a, b) of Remark 4.2. If one simplyrequires for the QmbC≈-valuations the condition v(a ≈ a) = 1 for every a ∈ A, thereis no guarantee that v(t1 ≈ t2) = 1 despite tA1 = tA2 . This situation would violate theSubstitution Lemma, and consequently also the Soundness theorem of QmbC≈ withrespect to interpretations, as observed above.

27

Now, it is easy to extend the previous results in order to prove the following sound-ness and completeness theorem for QmbC≈. Thus, by denoting by `≈ the relationconsequence of the Hilbert calculus QmbC≈ and by �≈ the semantical consequencerelation with respect to interpretations (see Definition 7.3), the following holds:

Theorem 7.5 (Soundness and Completeness of QmbC≈ with respect to interpreta-tions). For every set of sentences ∆ ∪ {ϕ} ⊆ S ≈

Σin a language with equality over a

signature Σ:∆ `≈ ϕ ⇐⇒ ∆ �≈ ϕ .

Proof. Soundness can be easily established from the considerations above. For com-pleteness, the proof of Theorem 5.9 is adapted as follows: by assuming that ∆ ∪ {ϕ} ⊆

S ≈L is a set of sentences with equality over a signature Σ such that ∆ 0≈ ϕ, let ∆H be aset of sentences with equality over the signature ΣC which extends ∆ and is maximalnon-trivial with relation to ϕ in QmbC (as a calculus defined over ΣC) and is also aC-Henkin theory. A canonical interpretation 〈A, v〉 over ΣC will be defined now suchthat, for every sentence ψ over ΣC ,

A, v � ψ ⇐⇒ ∆H `≈C ψ .

Define, thus, the following relation in the set C of constants : c ' d iff ∆H `≈C (c ≈ d).Then ' is an equivalence relation. Let c = {d ∈ C : c ' d} for c ∈ C, and letA = {c : c ∈ C}. The structure A over ΣC with domain A is defined as follows:

(i) if c is an individual constant in ΣC then IA(c) = d, where d ∈ C is such that∆H `≈C (c ≈ d);

(ii) if f is a function symbol, then IA( f ) : An → A is such that IA( f )(c1, . . . , cn) = cwhere c ∈ C is such that ∆H `≈C ( f (c1, . . . , cn) ≈ c);

(iii) if P is a predicate symbol, then

〈c1, . . . , cn〉 ∈ IA(P) ⇐⇒ ∆H `≈C P(c1, . . . , cn) .

The proof that IA is well-defined is similar to that for classical logic (see, for instance,[10]).

Now, let v : S L≈(A) → {0, 1} be the mapping defined as follows:

v(ψ) = 1 ⇐⇒ ∆H `≈C ψ∗

where ψ∗ is the sentence over ΣC obtained from ψ by substituting every occurrence ofa constant c by the constant c. Thus, for every sentence ψ over ΣC ,

A, v � ψ ⇐⇒ ∆H `≈C ψ .

Then, it is proved that v is a QmbC≈-valuation over A. Finally, the reduct 〈A, v〉 of〈A, v〉 to Σ is an interpretation for QmbC≈ over Σ such that A, v � ∆ but A, v 2 ϕ,showing that ∆ 2≈ ϕ. �

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Finally, it is easy to adapt the definitions and results of the previous section to thelogic QmbC≈. Thus, the logic QmbC≈ is compact (by using a notion of compact-ness similar to that of Definition 6.6), and the two versions of the Lowenhëim-SkolemTheorem hold for QmbC≈. From this, it is easy to prove the following:

Proposition 7.6. If a theory Γ of QmbC≈ has arbitrarily large finite models, then ithas an infinite model.

Proof. Given a theory Γ of QmbC≈ over a signature Σ with arbitrarily large finitemodels, consider a denumerable set C = {cn : n ≥ 0} of new individual constants. LetΣC be the signature obtained from Σ by adding the set C of individual constants and let∆ be the following theory over ΣC:

∆ = Γ ∪ {∼(cn ≈ cm) : n < m}.

Under the given hypothesis over Γ, it is easy to prove that every finite subset of ∆ issatisfiable. By the Compactness Theorem for QmbC≈, the theory ∆ has a model 〈A, v〉and so the domain A of A must be infinite. Let 〈A′, v′〉 such that A′ is the reduct of Ato Σ and v′ is the corresponding restriction of v to S L(A′). Since Γ ⊆ S L then 〈A′, v′〉 isa model of Γ which is infinite. �

8 First-order characterization of other quantified LFIsIn the previous sections QmbC, the first-order extension of mbC, which constitutesthe simplest propositional LFI analyzed in [7], has been carefully studied. There existseveral propositional extensions of mbC proposed and studied in [8] and [7], to whichthe concepts and techniques employed in the previous sections could be readily appliedin order to obtain the corresponding first-order versions. Some extensions of mbC willbe briefly mentioned below.

(i) The logic mCi is the extension of mbC obtained by adding the following axiomschemas:

(ci) ¬◦ϕ→ (ϕ ∧ ¬ϕ)

(ccn) ◦¬n◦ϕ (for n ≥ 0)

(ii) The logic Ci is obtained from mCi by adding the axiom

(cf) ¬¬ϕ→ ϕ

or, equivalently, by adding to mbC the axioms (ci) and (cf).

(iii) The system Cil is obtained from Ci by adding the axiom

(cl) ¬(ϕ ∧ ¬ϕ)→ ◦ϕ

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A basic feature of da Costa’s C-systems (see [12]) is that ‘well-behavior’ (that is, con-sistency) is propagated from simpler to more complex formulas. This motivates thefollowing:

(iv) The logic Cia is obtained by the addition of the following axiom schemas to Ci:

(ca1) (◦α ∧ ◦β)→ ◦(α ∧ β)

(ca2) (◦α ∧ ◦β)→ ◦(α ∨ β)

(ca3) (◦α ∧ ◦β)→ ◦(α→ β)

(v) The logic Cila is obtained by the addition of the axiom schema (cl) to Cia or,equivalently, of the axioms (ca1)–(ca3) to Cil.4

Observe that all the extensions of mbC presented above consist in the addition ofsome axiom schemas. On the other hand, the corresponding clauses for the associ-ated bivaluations can be straightforwardly associated to such axioms. From this, itis possible to extend the soundness and completeness theorem of mbC to all thesepropositional systems (see [7]). Being so, the first-order version of each of the LFIsintroduced above is straightforwardly defined by adding to QmbC the correspondingaxiom schemas from the list above. Thus, for instance, QCi is obtained from QmbCby adding axiom schemas (ci) and (cf). Of course, the valuations of the interpretationstructures must satisfy the additional clauses for bivaluations required in each case.Thus, a QCi-valuation over a structure A is a QmbC valuation (recall Definition 3.5)satisfying, additionally, the following clauses:

(vCon)’ v(¬◦α) = 1 =⇒ v(α) = 1 and v(¬α) = 1

(vNeg)’ v(¬¬α) = 1 =⇒ v(α) = 1.

From this perspective, the proof of soundness and completeness theorems of QmbCstated above can be easily extended to the new quantified LFIs. Clearly, all of them canalso be equipped with an equality predicate ≈, as it was done for QmbC in Section 7.The details of these constructions are left to the diligent reader .

9 Related workThere exist several proposals in the literature concerning the development of first-orderLFI’s. In his famous monograph [12] (see also [13]), da Costa introduced the first-order version C∗n of each calculus Cn (recall that, as observed in [7], the calculi Cn arespecial cases of LFI’s). The first-order axioms and rules are, as in our case, the classical

4It is worth noting that the only difference between Cila and C1 is that the consistency connective ◦ wasnot taken as primitive in C1, but as an abbreviation given by the formula ¬(α∧¬α). It can be proven that C1is equivalent to Cila up to translations (see [7]).

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ones, with just one difference: da Costa required the propagation of consistency for thequantifiers (in this way, generalizing the propagation of consistency for conjunctionand disjunction, by taking into account that the universal and the existencial quantifierscan be seen as arbitrary conjunctions and disjunctions, respectively). The extension ofthese systems to include the (standard) equality predicate was also considered by daCosta, obtaining the hierarchy of calculi called C=

n . Subsequently, E. Alves obtainedseveral basic results of model theory for such calculi (see [1]).

The semantics for the calculi C∗n and C=n is, as the one proposed in the present pa-

per, 2-valued. Correspondingly, the usual Tarskian first-order structures are equippedwith paraconsistent bivaluations. In her Phd thesis [15] and in a series of papers([16, 17, 18]), I. D’Ottaviano developed the basic model theory of the first-order ver-sion of the well-known 3-valued paraconsistent logic J3, introduced in [19]. It is worthnoting that J3 was reintroduced in [9] as an LFI (in a different signature contaninga consistency connective) called LFI1 (see also [7]). The semantics proposed byD’Ottaviano is given again by usual Tarskian first-order structures, now equiped witha 3-valued paraconsistent valuation. A generalization of the quasi-truth theory intro-duced in [22] has been more recently proposed (cf. [11]). The proposed system isa 3-valued first-order paraconsistent LFI whose semantics is given by Tarskian first-order structures in which the predicate symbols of arity n are interpreted as triples ofpairwise disjoint subsets of Dn (where D is the domain of the Tarskian structure) whoseunion is Dn. It is proved in [11] that the proposed logic coincides with D’Ottaviano’sfirst-order version of J3, and so it is a first-order version of LFI1, which also coincideswith the first-order version of LFI1 called LFI1* studied in [9]. As a consequence, thelogic LFI1* coincides with the first-order version of J3 proposed by D’Ottaviano. Byits turn, these systems are conservative extensions of G. Priest’s first-order version ofthe logic of paradox LP proposed in [24] (see [11]).

In an independent research line, A. Avron and I. Lev introduced in [2, 3] a gen-eralization of the concept of matrix semantics called non-deterministic matrices, orNmatrices. In a series of papers, Avron and his collaborators introduced Nmatrices forseveral logic systems, including all the LFI’s studied in [8, 7], as well as for new LFI’sproposed by them. In [4] a semantics based on Nmatrices for several first-order LFI’sis proposed. The first-order axioms and rules added to the propositional systems, guar-anteeing a careful treatment of the substitution lemma, coincide with the ones usedin the present paper (despite the fact that they do not include axiom (Ax14)). Se-mantically, they consider Tarskian first-order structures enriched with valuations oversuitable Nmatrices. Since the semantics of the considered Nmatrices coincides with thecorresponding bivaluation semantics, both proposals are equivalent. The convenienceof enriching the Tarskian structures with bivaluations or with valuations over Nma-trices is a matter of discussion. Moreover, the truth-values as well as the operationsof the Nmatrices are obtained from an analysis of the bivaluations, as it was shownin [6], and so both approaches are conceptually very close. It can be argued that theuse of bivaluations allows us to consider a model theory closer to the classical one:the structures considered here define a conservative extension of the logic associatedto the usual ones, by adding two (non-truth-functional) new connectives, namely theparaconsistent negation ¬ and the consistency operator ◦. Being so, our treatment offirst-order LFI’s extends the original proposal of da Costa and Alves to several LFI’s,

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on the one hand, and it is more ‘palatable’ to the classically-oriented logicians, on theother.

10 Some methodological considerationsThis paper introduces first-order extension of several LFI’s in such a way that thesemantical structures are as close as possible to the classical ones. Some essential re-sults on model theory, showing the validity of the compactness and of the Lowenhëim-Skolem theorems, were also obtained. It was also shown how the underlying first-orderlanguage of such systems can be extended with a standard equality predicate, whichaffords richer theories. For instance, in [5] several paraconsistent versions of ZF set-theory based on different LFI’s were investigated. The corresponding first-order logicwith equality for each system is based on the results obtained of the present paper.

There is still a good deal of work to be done by employing the methods developedhere, or extensions thereof, to investigate, for instance, the validity of other “grandtheorems" of model theory, as put in [20]. For example, the Lyndon interpolationtheorem, the omitting types theorem and the initial model theorem, not to mentionelimination of quantifiers. It may happen that theorems of this type would be of lessinterest for paraconsistent model theory, and that other, currently unheard of, propertieswill emerge.

We believe, however, that there is a modest, but solid lesson to be learned fromour approach, of interest to the philosophy and methodology of logic: regarded interms of their methods and mathematical properties, the borders between the so-calledclassical and non-classical logics are too vague to grant an absolute distinction betweenclassicality and non-classicality when referring to logic.

Indeed, as this paper has attested, with a bit of generalization some well-establishedconstructions in logic can be suited to meet the requirements of more expressive logicswhich, as in our case, are genuine enlargements of the logic space, and not any exoticconcoction. And if the distinction from the mathematical perspective is as faint as itappears, much more is needed on the philosophical side to maintain the demarcationbetween classical and non-classical logics, at least in some cases. From this perspec-tive, the logics we have treated here, and for whose model theory we have started aninvestigation, are perfectly classical.

Acknowledgments: We thank the anonymous referees for several suggestions thathelped to improve the final version of the paper. This research was financed by FAPESP(Thematic Project LogCons 2010/51038-0, Brazil) and by individual research grantsfrom The National Council for Scientific and Technological Development (CNPq),Brazil.

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