1 Introduction and Preliminaries

In 1923, Hilbert proposed studying the implicative fragment of classical propositional calculus. This fragment is well-known as positive implicative propositional calculus and its study was started by Hilbert and Bernays in 1934. The following axiom schemas define this calculus:

(E1) α→(β→α),

(E2) (α→(β→γ))→((α→β)→(α→γ)),

and the inference rule modus ponens is:

(MP) α,α→ββ.

In 1950, Henkin introduced the implicative models as algebraic models of the positive implicative calculus. Later, A. Monteiro renamed them as Hilbert algebras and his Ph. D. student Diego ([⁸]) made one of the most important contributions to these algebraic structures.

In particular, this author proved that the class of Hilbert algebras is an equational class, that is to say, it is possible to characterize the class via certain equations.

Moreover, Diego proved that the positive implicative propositional calculus is decidable by means of using algebraic technical tools.

On the other hand, Thomas in [²⁶] considered the n-valued positive implicative calculus, with signature {→,1}, as a calculus that has a characteristic matrix 〈A,{1}〉 where {1} is the set of designated elements and the algebra A=(ℂn,→,1) is defined as follows:

and

This author proved that for this calculus we must add the following axiom to the positive implicative calculus:

(E3) Tn(α0,⋯,an−1)=βn−2→(βn−3→(⋯→(β0→α0)⋯)), where

The algebraic counterpart of n-valued positive implicative calculus was studied in [¹⁵] where the axiom (E3) is translated by the equation Tn=1 to the ones of Hilbert algebras. In particular, in the n=3 case, the variety is generated by an algebra that has this set ℂ3={0,12,1} as support and an implication → defined by the following table 1.

It is clear that 3-valued Hilbert algebras are Hilbert algebras that verify the following identity:

(IT3) ((x→y)→z)→(((z→x)→z)→z)=1.

It is important to note that the implication defined in Table 1 characterizes the implication of 3-valued Gödel logic that we call G3.

Paraconsistent extensions of 3-valued Gödel logic were studied as a tool for knowledge representation and nonmonotonic reasoning, [²¹, ²⁰]. Particularly, Osorio and his collaborators showed that some of these logics can be used to express interesting nonmonotonic semantics. In addition, these paraconsistent systems were also studied under a mathematical logic point of view as we can see in the following papers: [²², ¹², ¹⁷, ¹⁹, ¹⁸]. To see other applications of three-valued logic to other fields the reader can consult [⁵].

In this paper, we will study implicative fragments of G3 enriched with certain modal operators that we call Moisil’s operators. In this setting, recall that Moisil introduced 3-valued Łukasiewicz algebras (or 3-valued Łukasiewicz-Moisil algebras) as algebraic models of 3-valued logic proposed by Łukasiewicz. It is well-known, and part of folklore, that the class of 3-valued Łukasiewicz algebras is term equivalent to the one of 3-valued MV-algebras (see, for instance, [²]). Recall that an algebra (A,∧,∨,~,∇,0,1) is a 3-valued Łukasiewicz algebra if the following conditions hold: (L0) , (L1) x∧(x∨y)=x, (L2) x∧(y∨z)=(z∧x)∨(y∧x), (L3) ~~x=x, (L4) ~(x∧y)= ~x ∨~y, (L5) ~x∨∇x=1, (L6) ~x∧x= ~x∧∇x, and (L7) ∇(x∧y)=∇x∧∇y. It is well known that each 3-valued Łukasiewicz algebra is a De Morgan algebra because equations (L0) to (L4) hold, [², Definition 2.6]. In general, to see more technical aspects of Łukasiewicz-Moisil algebras, the reader can consult [²].

On the other hand, the characteristic matrix of logic from trivalent Łukasiewicz algebras has the operators ∧, ∨, ~, ∇ ( possibility operator) and △ (necessity operator) over the chain ℂ3={0,12,1}, and they are defined by the next table:

In addition, the implication → defined in Table 1 can be obtained from the operators ∧, ∨, ~, ∇ and △ by the following formula:

Moreover, it is not hard to see that ∇x=(x→ △x)→ △x. In this setting, the algebraic structures in the signature {→, △} were defined and studied by Canals-Frau and Figallo in [⁶, ⁷]; these structures can be seen as certain {→, △}-fragments of 3-valued Łukasiewicz algebras.

The rest of the paper is organized as follows: in section 2, we introduce and study the class of modal 3-valued Hilbert algebras with supremum and also, as an application of our algebraic work, we present a Hilbert calculus for the fragment with disjunction soundness and completeness, in a strong version, with respect to this class of algebras. In Section 3, we study the first-order logic for the fragment with disjunction by means of an adaptation of the Rasiowa’s technique ([²⁵]) using our algebraic work for the propositional case. In the last Section, we discuss the possibility to applied our proofs to other classes of algebras.

2 Trivalent Modal Hilbert Algebras With Supremum

In this section, we will introduce and study algebraically the trivalent modal Hilbert algebra with supremum that we denote H3∨,△-algebras. From this algebraic work, we present a sound and complete calculus w.r.t. the class of H3∨,△-algebras in propositional case.

For the sake of brevity, we only introduce those essential notions of Hilbert algebras that we need, thought not in full detail. Anyway, for more information about these algebras, the reader can consult the bibliography.

Now, recall that a Hilbert algebra is an algebra (A,→,1) such that for all x,y,z∈A verifies:

Furthermore, we say (A,→,1) is a 3-valued Hilbert algebra if verifies the following equation: (IT3) ((x→y)→z)→(((z→x)→z)→z)=1.

The following lemma is well-known and the proof can be found in [⁸].

Lemma 2.1 Let A be a Hilbert algebra. The following properties are satisfied for every x,y,z∈A:

In the following, we present a definition of the equational class of 3-valued modal Hilbert algebra that was introduced in [⁶].

Definition 2.2 An algebra (A,→, △,1) is said to be a 3-valued modal Hilbert algebra if its reduct (A,→,1) is a 3-valued Hilbert algebra and △ verifies the following identities:

(M1) △x→x=1,

(M2) ((y→ △y)→(x→ △△x))→ △(x→y)= △x→ △△y, and

(M3) (△x→ △y)→ △x= △x.

Moreover, we define a new connective by ∇x=(x→ △x)→ △x.

Now, consider the following Definition that we introduce for the first time.

Definition 2.3 An algebra A=〈A,→,∨, △,1〉 is said to be a trivalent modal Hilbert algebra with supremum if the following properties hold:

From now on, we denote with A the H3∨,△-algebra 〈A,→,∨, △,1〉 and with A its support. Next, we will show some properties that will be very useful for the rest of this section.

Let us notice that there is an H3∨,△-algebra A in which the infimum can not be defined. To see that, take some subalgebras of ℂ3→,∨×ℂ3→,∨ where × is the direct product.

The fragment with infimum has been studied in [²⁴] that will comment in the following Remark.

Remark 2.4 In [²⁴], the class of 3-valued modal Hilbert algebra with imfimum (i△H3-algebra) was defined as follows: An algebra 〈A,→,∧, △,1〉 is said to be an i△H3-algebra if the following conditions hold: (1) the reduct 〈A,→, △,1〉 is a 3-valued modal Hilbert algebra; (2) the following identities hold: (iH1) x∧(y∧z)=(x∧y)∧z, (iH2) x∧x=x, (iH3)x∧(x→y)=x∧y, and (iH4) (x→(y∧z))→((x→z)∧(x→y))=1.

Let us observe that for each i△H3-algebra A and for every x,y∈A, we can define the supremum of {x,y} in the following way:

Indeed, let a,b∈A and put c=((a→b)→b)∧((b→a)→a). Since x≤(x→y)→y and x≤(y→x)→x hold and there exists the infimum ((x→y)→y)∧((y→x)→x), then c is upper bound of the set {a,b}. Now, let us suppose that d is another upper bound of {a,b} such that c≤d. Thus, there exists an irreducible deductive system P such that c∈P and d∉P [⁸, Corolario

1]. Besides, since a,b≤d then a,b∉P. On the other hand, as A is a trivalent Hilbert algebra and according to [¹⁴, Théorème 4.1], we have a→b∈P or b→a∈P. Now, if we suppose that a→b∈P and since c≤(b→a)→a, then we can infer that a∈P, which is a contradiction. If we consider the case b→a∈P, we also obtain a contradiction. Thus, c is the supremum of {a,b}. Therefore, each i△H3-algebra is a relatively pseudocomplemented lattice since x∧z≤y iff x≤z→y, see [²⁵]. From the latter, we have that each i△H3-algebra is a distributive lattice. It is possible to see that every finite and complete i△H3-algebra is a 3-valued Łukasiewicz algebra. To see the details, the reader can consult Section 3 of [²⁴].

Lemma 2.5 For a given H3∨,△-algebra A and x,y,z∈A, then the following properties hold:

Proof. It is routine.

Definition 2.6 For a given H3∨,△-algebra A and D⊆A. Then, D is said to be a deductive system if (D1) 1∈D, and (D2) if x,x→y∈D imply y∈D. Additionally, we say that D is a modal if: (D3) x∈D implies △x∈D. Moreover, D is said to be maximal if for every modal deductive system M such that D⊆M implies M=A or M=D.

Given a H3∨,△-algebra-algebra A and {Hi}i∈I a family of modal deductive systems of A, then it is easy to see that ∩i∈IHi is a modal deductive system. Thus, we can consider the notion of modal deductive system generated by H, denoted [H)m, as an intersection of all modal deductive system D such that D⊆H. The deductive system generated by H, denoted [H), verify that [H)={x∈A:there exist h1,⋯,hk∈H such as h1→(h2→⋯→(hk→x)⋯)=1} where k is a finite integer, see [⁸]. Now, we will introduce the following notation:

Hence, we can write:

Then, we have the following result:

Proposition 2.7 Let A be a H3∨,△-algebra, suppose that H⊆A and a∈A. Then the following properties hold:

Proof. It is routine.

Lemma 2.8 Given a H3∨,△-algebra A, there exists a lattice-isomorphism between the poset of congruences of A and the poset of the modal deductive systems of A.

Proof. It is well-known that the set of congruences of Hilbert algebra A is lattice-isomorphic to the set of all deductive systems. For each deductive system D we have the relation R(D)={(x,y):x→y,y→x∈D} which is a congruence of A, such that the class of 1 verifies |1|R(D)=D. In addition, for each congruence θ of A, the class of |1|θ is a deductive system and R(|1|θ)=θ. From the latter and Lemma 2.5 (1) and (2), we can infer that every congruence θ for a given A respect △ and |1|θ is a modal deductive system.

For each H3∨,△-algebra A, we can define a new binary operation ↣ named weak implication such that: x↣y= △x→y.

Lemma 2.9 Let A be a H3∨,△-algebra, for any x,y,z∈A the following properties hold:

Proof. The proof immediately follows from the very definitions; and, it can be consulted [²⁴, Lema 2.4.2].

Let A an H3∨,△-algebra and suppose a subset D⊆A, we say that D is a weak deductive system (w.d.s.) if 1∈D, and x,x↣y∈D imply y∈D. It is not hard to see that the set of modal deductive systems is equal to the set of weak deductive systems. We denote by Dw(A) the set of weak deductive systems of a Hilbert algebra.

Now, for a given H3∨,△-algebra A and a (weak) deductive system D of A, D is said to be a maximal if for every (weak) deductive system M such that D⊆M, then M=A or M=D. Besides, let us consider the set of all maximal w.d.s. ℰw(A). A. Monteiro gave the following definition in order to characterize maximal deductive systems:

Definition 2.10 (A. Monteiro) Let A be a H3∨,△-algebra, D∈D(A) and p∈A. We say that D is a weak deductive system tied to p if p∉D and for any D′∈Dw(A) such that D⊊D′, then p∈D′.

The importance for introducing the notion of weak deductive systems is to prove that every maximal weak deductive system is a weak deductive system tied to some element of a given H3∨,△-algebra, A. Conversely, and using (wi6), we can prove every w.d.s is a maximal weak deductive systems. Moreover, from (wi4), (wi5) and (wi1) and using A. Monteiro’s techniques, we also can prove that {1}=∩M∈ℰw(A)M. To see the details of the proof, see Sections 2.4, 2.5 and 2.6 of [²⁴].

In what follows, we will consider the quotient algebra A/M defined by a≡Mb iff a→b, b→a∈M and the canonical projection qM:A→A/M defined by qM=|x|M where |x|M denotes the equivalence class of x generated by M.

Lemma 2.11 Let A be a H3∨,△-algebra. Then, the map Φ:A→∏M∈ℰw(A)A/M defined by Φ(x)(M)=qM(x) is a homomorphism; that is to say, the variety of H3∨,△-algebras is semisimple.

Proof. Taking ∏α∈ℰw(A)A/Mα={f:A→∪α∈ℰw(A)A/Mα:f(a)∈A/Ma for every α∈ℰw(A)} and ℰw(A) is the set of maximal w.d.s. defined before. Let us define Φ:A→∏α∈ℰw(A)A/Mα such that for every α we have that Φ(α)=fa where fa(α)=qα(a)=|a|α∈A/Mα with a∈A. It is not hard to see that Φ is a homomorphism in view of the fact that ≡Mα is a congruence relation. Now, from the fact that {1}=∩M∈ℰw(A)M , it is possible to see that Φ is one-to-one function which completes the proof.

The construction of the following homomorphism is fundamental to obtaining the generating alge-bras of the variety of H3∨,△-algebra. Moreover, this homomorphism will play a central role in the adequacy theorems in a propositional and first-order version of logic, as we will see later on.

In the next, we consider the algebras ℂ3→,∨=〈{0,12,1},∨,△,1〉 and ℂ2→,∨=〈{0,1},→,∨,△,1〉. We denote ℂ3 and ℂ2 the support of ℂ3→,∨ and ℂ2→,∨, respectively; besides, the operation ∨ is the maximum on the corresponding chain.

Theorem 2.12 Let M be a non-trivial maximal modal deductive system of an H3∨,△-algebra A. Let us consider the sets M0={x∈A:∇x∉M} and M1/2={x∈A:x∉M,∇x∈M}, and the map h:A→ℂ3 defined by

Then, h is a homomorphism from A into ℂ3→,∨ such that h−1({1})=M.

Proof. We shall prove only that h(x∨y)=h(x)∨h(y), for the rest of the proof can be done in a similar manner.

According to Lemma 2.11 and Theorem 2.12, and well-known facts about universal algebra, we have proved the following Corollary.

Corollary 2.13 The variety of H3∨,△-algebras is semisimple. Moreover, the algebras:

and

are the unique simple algebras.

3 Model Theory and First Order version of the logic of ℋ3∨,△ Without Identities

In this section, we will define the first order logic of ℋ3∨,△. First, let Σ={→,∨,△} be the propositional signature of ℋ3∨,△, the symbols ∀ (universal quantifier) and ∃ (existential quantifier), with the punctuation marks (commas and parentheses). Let Var={v1,v2,…} be a numerable set of individual variables. A first order signature Θ is composed of the following elements:

The notions of bound and free variables inside a formula, closed terms, closed formulas (or sentences), and of the term free for a variable in a formula are defined as usual, see [²³]. We will denote by TΘ and FmΘ the sets of all terms and formulas, respectively. Given a formula φ, the formula obtained from φ by substituting every free occurrence of a variable x by a term will be denoted by φ(x/t).

Definition 3.1 Let Θ be a first order signature. The logic Qℋ3∨,△ over Θ is defined by Hilbert calculus obtained by extending ℋ3∨,△ expressed in the language FmΘ by adding the following:

Axioms Schemas

(Ax11)  φ(x/t)→∃xφ, if t is a term free for x in φ,

(Ax12)  ∀xφ→φ(x/t), if t is a term free for x in φ,

(Ax13)  △∃xφ↔∃x△φ,

(Ax14)  △∀xφ↔∀x△φ,

Inferences Rules

(R3)  φ→ψ∃xφ→ψ where x does not occur free in ψ,

(R4)  φ→ψφ→∀xψ where x does not occur free in φ.

We denote by ⊢α the derivation of a formula α in Qℋ3∨,△ and with Γ⊢α the derivation of α from a set of premises Γ. These notions are defined as the usual way. Furthermore, we denote ⊢φ↔ψ as an abbreviation of ⊢φ→ψ and ⊢φ→ψ.

Definition 3.2 Let Θ be a first-order signature. A Θ-structure is a triple S=〈A,S,⋅S〉 such that A is a complete H3∨,△-algebra, and S is a non-empty set and ⋅S is an interpretation mapping defined on Θ as follows:

Given a Θ-structure S=〈A,S,⋅S〉, an S-valuation (or simply valuation) is a function v:Var→S. Given a∈S and S-valuation v, by v[x→a] we denote the following S-valuation, v[x→a](x)=a and v[x→a]=v(y) for any y∈V such that y≠x.

Let S=〈A,S,⋅S〉 be a Θ-structure and v an S-valuation. A Θ-structure S=〈A,S,⋅S〉 and an S-valuation v induce an interpretation map ‖⋅‖vS for terms and formulas that can be defined as follows:

We say that S and v satisfy a formula φ, denoted by S⊨φ‖v‖, if ‖φ‖vS=1Besides, we say that φ is true in S if ‖φ‖vS=1 for each S-valuation v and denoted by S⊨φ. We say that φ is a semantical consequence of Γ in Qℋ3∨,△, if, for any structure S: if S⊨φ for each γ∈Γ, then S⊨φ. For a given set of formulas Γ, we say that the structure S is a model of Γ iff S⊨γ for each γ∈Γγ ∈ Γ.

Now, it is worth mentioning that the following property ‖φ(x/t)‖vS=‖φ‖v[x→‖t‖vS]S holds. Another important aspect of the definition of semantical consequence is that it is different to the propositional case because if we use the definition of valuation for this case, we are unable to prove an important rule as α(x)⊨∀xα(x).

In addition, we need to recall an important property of complete H3∨,△-algebra.

Lemma 3.3 [¹⁶, Lemma 0.1.21] Let A be a complete H3∨,△-algebra and the set {ai}i∈I of element of A for any non-empty set I. Then if there exists ∨i∈Iai(∧i∈Iai), then there exists ∨i∈I△ai(∧i∈I△ai) and also, ∨i∈I△ai=△∨i∈Iai and ∧i∈I△ai=△∧i∈Iai.

This property is useful to prove the following theorem:

Theorem 3.4 (Soundness Theorem) Let Γ∪{φ}⊆FmΘ, if Γ⊢∨φ then Γ⊨φ.

Proof. In what follows we will consider an arbitrary but fixed structure S=〈A,S,⋅S〉. It is clear that the propositional axioms are true in S. Now, we have to prove that the new axioms (Ax11) and (Ax12) are true in S, and the new inference rules (R3) and (R4) preserve trueness in S.

(Ax11) Suppose that φ is α(x/t)→∃xα. Then, ‖φ‖vS=‖α‖v[x→‖t‖vS]S→‖∃xα‖vS. It is clear that ‖α‖v[x→‖t‖vM]S≤∨a∈S‖α‖v[x→α]S and then, ‖α‖v[x→‖t‖vS]S≤‖∃xα‖vS. Therefore ‖α(x/t)→∃xσ‖vS=1 for every S-valuation v. (Ax12) is analogous to (Ax11). Now, according to Lemma 3.3, the axioms (Ax13) (Ax14) are true in S.

(R4) Let α→β such that x is not free in α, and let α→∀xβ. Let us suppose that ‖α→β‖vS=1 for every S-valuation v. Now, consider a fix valuation v, then ‖α→∀xβ‖vS=‖α‖vS→‖∀xβ‖vS=‖α‖vS→∧a∈S‖β‖v[x→a]S. On the other hand, by hypothesis, we know that

‖α‖uS≤‖β‖uS for every S-valuation u. In particular, ‖α‖vS=‖α‖v[x→a]S≤‖β‖v[x→a]S for every S-valuation v. Then, ‖α‖vS≤∧a∈S‖β‖v[x→a]S and so, ‖α‖vS→∧a∈S‖β‖v[x→a]S=1 for every S-valuation v. The proof of preservation of trueness for (R3) is analogous to (R4).

In what follows, we will prove a strong version of Completeness Theorem for Qℋ∨,△3 using the Lindenbaum-Tarski algebra in a similar way to the propositional case. Let us observe that the algebra of formulas is an absolutely free algebra generated by the atomic formulas and its quantified formulas.

Now, let us consider the relation ≡ defined by α≡β iff ⊢α→β and ⊢α→β, then we have the algebra FmΘ/≡ is a ℋ3∨,△-algebra and the proof is exactly the same as in the propositional case (see, for instance, [¹]). On the other hand, it is clear that Qℋ3∨,△ is a Tarskian and finitary logic. So, we can consider the notion of (maximal) consistent and closed theories with respect to some formula in the same way as the propositional case. Therefore, we have that Lindenbaum- Łos’ Theorem holds for Qℋ3∨,△. Then, we have the following:

Theorem 3.5 Let Γ∪{φ}⊆FmΘ, with Γ non-trivial maximal respect to φ in Qℋ3∨,△. Let Γ/≡ ={α¯:α∈Γ} be a subset of FmΘ/≡, then:

Proof. According to the proof of Theorem 2.20, we only have to consider the rules (R3) and (R4). The fact that Γ/≡ is closed follows immediately.

In order to complete the proof, we have to consider two new cases 5 and 6. It is clear that Γ/≡ is a subset of D¯. Now, let us consider ϕ¯∈D¯ then ϕ¯∉Γ/≡ and remember D={α:α¯∈D¯}.

Case 5: There exists {j,t1,…,tm}⊆{1,…,k−1} such that αt1,…,αtm is a derivation of αj=θ→β. Let us suppose that αn=∃xθ→β is obtained by αj applying (R3). From induction hypothesis, we have that θ→β¯∈D¯. From the latter, we obtain ∃xθ→β¯∈D¯.

Case 6: There exists {j,t1,…,tm}⊆{1,…,k−1} such that αt1,…,αtm is a derivation of aj=θ→β. Let us suppose that αn=θ→∀xβ is obtained by αj applying (R4). From induction hypothesis, we have θ→β¯∈D¯ and then, θ→∀xβ¯∈D¯.

We note that for a given maximal consistent theory Γ of FmΘ we have Γ/≡ is a maximal modal deductive system of FmΘ/≡. By well-known results of Universal Algebras, if we denote A: =FmΘ/≡ and θ: =Γ/≡, we have the quotient algebra A/θ is a simple algebra, see Corollary 2.13. From the latter and by adapting the first isomorphism theorem for Universal Algebras, we have that A/θ is isomorphic to FmΘ/Γ which is defined by the congruence α≡Γβ iff α→β,β→α∈Γ.

Theorem 3.6 (Completeness Theorem) Let Γ∪{φ} be a set of sentences, then Γ⊨φ then Γ⊢φ.

Proof. Let us suppose Γ⊨φ and Γ⊢γ. Then, by Lindenbaum- Los’ Lemma, there exists Δ maximal consistent theory such that Γ⊆Δ. Now, consider the algebra FmΘ/Δ defined by the congruence α≡Δβ iff α→β,β→α∈Δ. In view of the above observations, we know that FmΘ/Δ is isomorphic to a subalgebra of ℂ3→,∨ and so, complete as lattice.

Now, let us take the canonical projection πΔ:Fm→FmΘ/Δ defined by πΔ(α)=|α| where |α| denotes the equivalence class of α∈Fm. In this sense, consider the structure M=〈FmΘ/Δ,TΘ,⋅TΘ〉 where TΘ is a set of terms. It is clear that for every t∈TΘ we have an associated constant t^ of Θ. Now, let us take a function μ:Var→TΘ defined by μ(x)=x. Then, we have the interpretation ‖⋅‖μM:Fm→FmΘ/Δ defined by if t^ is a constant, then ‖t^‖μM: =t; if f∈ℱ, then ‖f(t1,⋯,tn)‖μM=f(t1,⋯,tn); if P∈P, then ‖P(t1,⋯,tn)‖μM=πΔ(P(t1,⋯,tn)). Our interpretation is defined for atomic formulas but it is easy to see that ‖α‖μM=πΔ(α) for every quantifier-free formula α. Moreover, it is easy to see that for every formula ϕ(x) and every term t, we have ‖ϕ(x/t^)‖μM=‖ϕ(x/t)‖μM. Therefore, from the latter property and by (Ax12) and (R4), we have ‖∀xα‖μM=∧a∈TΘ‖α‖μ[x→a]M and now using (Ax11) and (R3), we obtain ‖∃xα‖μM=∨a∈TΘ‖α‖μ[x→a]M. So, ‖⋅‖μM is an interpretation map such that ‖α‖μM=1 iff α∈Δ. On the other hand, it is not hard to see for every closed formula (sentence) β, we have ‖β‖μM=‖β‖vM for every M-valuation v. Therefore, M⊨γ every γ∈Γ but M⊨φ which is a contradiction.

Given a formula φ and suppose {x1,⋯,xn} is the set of variables of φ, the universal closure of φ is defined by ∀x1…∀xnφ. Thus, it is clear that if φ is a sentence, then the universal closure of φ is itself. Now, we are in condition to prove the following Completeness Theorem for formulas:

Theorem 3.7 Let Γ∪{φ} be a set of formulas, then Γ⊨φ then Γ⊢φ.

Proof. Let us suppose Γ⊨φ and consider the set ∀Γ all universal closure of Γ. From the latter and definition of ⊨, we have ∀Γ⊨∀x1⋯Αxnφ. Then, according to Theorem 3.6, ∀Γ⊢∀x1⋯∀xnφ. Now, from the latter, (Ax12) and (R4), we have Γ⊢φ as desired.