On a Four-Valued Modal Logicwith Deductive Implication1

Marcelo E. Coniglio Martın FigalloCLE and Department of Philosophy Department of Mathematics

State University of Campinas National University of the South Sur

Campinas, Brazil Bahıa Blanca, Argentina

coniglio@cle.unicamp.br figallomartin@gmail.com

Abstract

In this paper we propose to enrich the four-valued modal logic associ-ated to Monteiro’s Tetravalent modal algebras (TMAs) with a deductiveimplication, that is, such that the Deduction Meta–theorem holds in theresulting logic. All this lead us to establish some new connections betweenTMAs, symmetric (or involutive) Boolean algebras, and modal algebrasfor extensions of S5, as well as their logical counterparts.

1 Introduction

The logic TML was introduced in [6] as a four-valued modal logic naturally as-sociated to Monteiro’s last algebras, namely, tetravalent modal algebras (TMAs).TMAs were first considered by Antonio Monteiro and mainly studied by I.Loureiro (see [8]) and A. V. Figallo et al. (see [5]). A Gentzen style system forTML was presented in [6] in the original signature {∧, ∨, ¬, �, ⊥}.

In [5], it was shown that it is possible to define an implication operator �in TMAs in terms of the original operators. This new connective was namedcontrapositive implication and it was shown that it has some “good” properties.Taking profit of the features of this implication, a Hilbert style system for TMLwas presented in [4] in the signature {�, ⊥} (since the other connectives canbe defined from this). But, the Deduction Meta–theorem does not hold in thislogic with respect to the contrapositive implication. Moreover, it was provedthat TML is not functionally complete and, as a consequence, that in TML isnot possible to define an implication connective such that the deduction meta–theorem holds.

In this paper, we propose to enrich the four-valued modal logic TML witha deductive implication in such a way that in the resulting logic the DeductionMeta–theorem holds. From this, we will find some new connections betweenTMAs, symmetric (or involutive) Boolean algebras, and modal algebras forextensions of S5, as well as their logical counterparts. Specifically, we willprove that tetravalent modal algebras expanded with a deductive implicationare the same as Boolean algebras plus a De Morgan negation, that is, symmetric(or involutive) Boolean algebras. On the other hand, we also shown that thesestructures are modal algebras for modal logic S5 satisfying additional equationssuch that they are generated by the Henle algebra H2.

1This paper constitutes a revised and improved version from the preprint [3].

1

2 Preliminaries

A structure 〈A,∧,∨,∼, T, 0〉 of type (2, 2, 1, 1, 0) is a symmetric Boolean algebraif the reduct 〈A,∧,∨,∼, 0〉 is a Boolean algebra and T is an involutive auto-morphism of A (cf. [10, 11]), that is, T : A → A is an automorphism suchthat T (T (x)) = x for every x ∈ A. On the other hand, an involutive Booleanalgebra is a structure 〈A,∧,∨,∼,¬, 0〉 of type (2, 2, 1, 1, 0) where 〈A,∧,∨,∼, 0〉is a Boolean algebra and ¬ is an involutive dual automorphism of A ([12]), thatis, ¬ : A → Ad is an isomorphism (where Ad denotes the dual of A) such that¬¬x = x for every x ∈ A. It is equivalent to say that ¬ is a De Morgan nega-tion, that is, a negation ¬ defined on a distributive bounded lattice such that¬(x ∨ y) = ¬x ∧ ¬y and ¬¬x = x (and so ¬0 = 1).

As observed in [12], both kinds of structures coincide. Besides, the classof these algebras is a variety which is generated by the two–atoms SymmetricBoolean algebra.

Finally, a tetravalent modal algebra is an algebra U = 〈A,∧,∨,¬,�, 0〉 oftype (2, 2, 1, 1, 0) such that its non-modal reduct U∼ = 〈A,∧,∨,¬, 0〉 is a DeMorgan algebra and the unary operation � satisfies, for all x ∈ A, the following:�x ∧ ¬x = 0, and ¬�x ∧ x = ¬x ∧ x (cf. [6]).

The class of all tetravalent modal algebras constitute a variety which isdenoted by TMA. Besides, TMA is generated by the four-element De Morganalgebra M4m = 〈M4,∧,∨,¬,�, 0〉 where M4 = {1, N,B, 0} is

•

•

•

•

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

1

N B

0

such that ¬N = N , ¬B = B, ¬0 = 1 and ¬1 = 0, and the unary operator � isdefined as follows: �1 = 1, and �x = 0 otherwise. This algebra has two properprime filters, namely, FN = {1, N} and FB = {1, B}.

The Tetravalent Modal Logic, TML is the logic defined by the matrixM =〈M4m, FN 〉 or equivalently by M = 〈M4m, FB〉 (see [4]).

3 A deductive implication on M4m

From a semantic point of view, what we are looking for is a binary operator⇒ defined on M4m in such a way that the following condition holds, for everyx, y, z ∈M4m

2

z ∧ x ≤ y iff z ≤ x⇒ y, (1)

It is easy to see that if an implication as above can be defined in a lattice,it must be unique. A binary operator satisfying (1) is called the residual of ∧.As we prove in [4], the contrapositive implication (see [5]) is not the residualof ∧. Moreover, there is no term definable binary operation in M4m satisfyingcondition (1).

Proposition 3.1 There is no term definable binary operation ⇒ in the algebraM4m such that condition (1) holds.

Proof. Since N ∧B ≤ 0 then N ≤ B ⇒ 0, from (1), and so B ⇒ 0 ∈ {1, N}.But then B ⇒ 0 = 1, since {0, 1, B} is a TMA-subalgebra of M4m, and soB ≤ B ⇒ 0. This implies, by (1), that B = B ∧B ≤ 0, a contradiction. �

Since M4m is a finite distributive lattice, there exists a unique binary oper-ation ⇒ satisfying (1). However, in the light of Proposition 3.1, ⇒ is not termdefinable. It is worth noting that the unary operation ∼ given by ∼x = x⇒ 0for every x is a Boolean complement in M4m, that is: x∧∼x = 0 and x∨∼x = 1for every x. On the other hand, if the Boolean complement ∼ is added to M4m

then ⇒ can be defined as x ⇒ y = ∼ x ∨ y for every x, y. It is easy to see, bya combinatory argument, that the truth-tables for both operators must be thefollowing:

⇒ 0 N B 10 1 1 1 1N B 1 B 1B N N 1 11 0 N B 1

x ∼x0 1N BB N1 0

By the considerations above, it is easy to prove the following:

Proposition 3.2 Adding to M4m the Boolean complement ∼ is equivalent toadding to M4m the deductive implication ⇒.

Let M∼4m be the algebra 〈M4,∧,∨,∼,¬,�, 0〉 of type (2, 2, 1, 1, 1, 0) wherethe reduct 〈M4,∧,∨,¬,�, 0〉 is the four-element TMA-algebra M4m and ∼ isdefined as in the table above.

Proposition 3.3 In M∼4m the following identities hold, for all x ∈M4:

(i) ¬∼x = ∼¬x,

(ii) �x = x ∧ ¬∼x,

(iii) ♦x = x ∨ ¬∼x, where ♦x =def ¬�¬x,

3

(iv) ¬�x = ∼�x,

(v) �¬x = �∼x.

Because of Corollary 3.2, the envisaged extension of the four-valued modallogic M4m by a deductive implication is equivalent to the extension by a classicalnegation. Being so, and by reasons that will be clear in the following sections,we will focus on the logic obtained from M4m by adding the classical negation ∼.As we shall see, the algebraic models will be symmetric (or involutive) Booleanalgebras.

4 De Morgan algebras and Tetravalent modalalgebras extended by a Boolean complement

From now on, an operator∼ : A→ A in a bounded lattice A such that x∧∼x = 0and x ∨ ∼x = 1 for every x ∈ A will be called a Boolean complement in A. Itis easy to see that, if A is also distributive, the Boolean complement ∼x of anelement x of A (if it exists) is unique.

In this section we will see that if we begin by a De Morgan algebra insteadof a tetravalent modal algebra and we extend it by a Boolean complement, theresulting structure can define a unique modality � which satisfies the propertiesof a tetravalent modal algebra. Because of the uniqueness of � it follows thattetravalent modal algebras plus a Boolean complement are the same as DeMorgan algebras plus a Boolean complement. Additionally, as we will see, thesestructures coincide with the symmetric (or involutive) Boolean algebras.

Proposition 4.1 Let 〈A,∧,∨,¬, 0〉 be a De Morgan algebra, and consider aBoolean complement ∼ in A, where 1 = ¬0 is the top element of A. Let �x =def

x∧¬∼x for every x ∈ A. Then 〈A,∧,∨,¬,�, 0〉 is a tetravalent modal algebra.

Proof. Let x ∈ A. Thenx ∨ ¬�x = x ∨ ¬(x ∧ ¬∼x)

= x ∨ (¬x ∨ ¬¬∼x)= x ∨ (¬x ∨ ∼x) = 1.

On the other hand:x ∧ ¬�x = x ∧ ¬(x ∧ ¬∼x)

= x ∧ (¬x ∨ ∼x)= (x ∧ ¬x) ∨ (x ∧ ∼x)= (x ∧ ¬x) ∨ 0 = x ∧ ¬x.

�

Lemma 4.2 ([12]) Let 〈A,∧,∨,¬, 0〉 be a De Morgan algebra, and consider aBoolean complement ∼ in A. Then ¬∼x = ∼¬x for every x ∈ A.

Proof. Let x ∈ A. Then¬x ∧ ¬∼x = ¬(x ∨ ∼x)

= ¬1 = 0,

4

and, dually,¬x ∨ ¬∼x = ¬(x ∧ ∼x)

= ¬0 = 1.From this it follows that ¬∼x is the Boolean complement of ¬x, that is: ¬∼x =∼¬x, for every x ∈ A, by the uniqueness of ∼. �

The following result relates, as we shall see, tetravalent modal algebras ex-tended by a Boolean complement with symmetric (or involutive) Boolean alge-bras.

Proposition 4.3 Let 〈A,∧,∨,¬,�, 0〉 be a tetravalent modal algebra, and con-sider a Boolean complement ∼ in A. Then �x = x ∧ ¬∼x for every x ∈ A.

Proof. Let x ∈ A. Then �x ≤ x, by definition of tetravalent modal algebras.On the other hand x ∨ ¬�x = 1 and so

∼x = ∼x ∧ 1 = ∼x ∧ (x ∨ ¬�x)= (∼x ∧ x) ∨ (∼x ∧ ¬�x)= ∼x ∧ ¬�x

therefore ∼x ≤ ¬�x. From this �x ≤ ¬∼x and so �x ≤ x ∧ ¬∼x. Let y ∈ Asuch that y ≤ x and y ≤ ¬∼x. Then y ≤ ∼¬x, by Lemma 4.2. Thereforey = y ∧ x and y ∧ ¬x = 0. From this, and recalling that x ∨ ¬x = ¬x ∨�x,

y = y ∨ 0 = (y ∧ x) ∨ (y ∧ ¬x) = y ∧ (x ∨ ¬x)= y ∧ (¬x ∨�x) = (y ∧ ¬x) ∨ (y ∧�x)= 0 ∨ (y ∧�x) = y ∧�x.

That is, y ≤ �x. This means that �x = x ∧ ¬∼x. �

Corollary 4.4 Let 〈A,∧,∨,¬, 0〉 be a De Morgan algebra equiped additionallywith a Boolean complement ∼. Then, the operator � defined by �x = x ∧ ¬∼xfor every x ∈ A is the unique unary operator over A such that 〈A,∧,∨,¬,�, 0〉is a tetravalent modal algebra. Thus, De Morgan algebras plus a Boolean com-plement are definitionally equivalent to tetravalent modal algebras plus a Booleancomplement.

It is here where involutive (or symmetric) Boolean algebras appear. Corol-lary 4.4 can be recast as follows:

Corollary 4.5 Let 〈A,∧,∨,∼,¬, 0〉 be an involutive Boolean algebra. Then, theoperator � defined by �x = x∧¬∼x for every x ∈ A is the unique unary operatorover A such that 〈A,∧,∨,¬,�, 0〉 is a tetravalent modal algebra. Thus, involu-tive (or symmetric) Boolean algebras are definitionally equivalent to tetravalentmodal algebras plus a Boolean complement.

From the considerations above, the variety of involutive (or symmetric)Boolean algebras can be defined in several ways. By convenience, we shall adoptthe perspective of seeing such algebras as Boolean algebras extended with a DeMorgan negation. Clearly this variety is generated by the reduct of M∼4m overthe language ∧,∨,∼,¬,⊥, by Proposition 3.3(ii).

Recall that the implication operator ⇒ is defined in any Boolean algebra asfollows: x⇒ y = ∼x∨ y. Since ∧, ∨ and 0 can be defined in terms of ⇒ and ∼

5

then we can consider, from now on, the absolutely free algebra Fm = 〈Fm,⇒,∼,¬〉 of type (2,1,1) generated by some denumerable set V ar of variables asthe formal language for the variety IBA of symmetric (or involutive) Booleanalgebras. A logic preserving degrees of truth (see [2]) associated to IBA can benaturally defined as follows:

Definition 4.6 The logic of involutive Boolean algebras defined over Fm is thepropositional logic IBA = 〈Fm, |=IBA〉 given as follows: for every set Γ∪{α} ⊆Fm, Γ |=IBA α if and only if there exists a finite set Γ0 ⊆ Γ such that, for everyU ∈ IBA and h ∈ Hom(Fm,U),

∧{h(γ) : γ ∈ Γ0} ≤ h(α). In particular,

∅ |=IBA α if and only if h(α) = 1 for every h ∈ Hom(Fm,U).

Remark 4.7 There exist several approaches in the literature to the study oflogics preserving degrees of truth associated to a class of agebras. An interestingprecedent can be found in [7], in which R. Jansana introduces and studies theso-called semilattice-based logics associated to semilattices.

A logic is naturally associated to the algebra M∼4m as in the previous defini-tion:

Definition 4.8 The four-valued modal logic with classical negation M∼4m de-fined over Fm is the propositional logic M∼4m = 〈Fm, |=M∼

4m〉 given as follows:

for every Γ ∪ {α} ⊆ Fm, Γ |=M∼4m

α iff there exists a finite set Γ0 ⊆ Γ suchthat, for every h ∈ Hom(Fm,M∼4m),

∧{h(γ) : γ ∈ Γ0} ≤ h(α). In particular,

∅ |=M∼4m

α if and only if h(α) = 1 for every h ∈ Hom(Fm,M∼4m).

Since IBA is generated by M∼4m, the following result is immediate:

Proposition 4.9 The logics IBA and M∼4m coincide, that is: for every Γ ∪{α} ⊆ Fm, Γ |=IBA α iff Γ |=M∼

4mα.

Proposition 4.10 M∼4m is a strong conservative extension of propositional clas-sical logic CPL: Γ |=M∼

4mβ iff Γ |= β in CPL, for every Γ∪{β} ⊆ Fm without

occurrences of ¬

Proof. It is immediate, from the definition of M∼4m. �

Remark 4.11 M∼4m satisfies the Deduction Metatheorem: Γ, α |=M∼4m

β iffΓ |=M∼

4m(α⇒ β), for every Γ ∪ {α, β} ⊆ Fm.

The next result shows that M∼4m can be seen as a matrix logic. This fol-lows from the definition of the four-valued modal logic TML (see the end ofSection 2). This fact will be useful in the sequel.

Proposition 4.12 Let M∼N = 〈M∼4m, {N, 1}〉 and M∼B = 〈M∼4m, {B, 1}〉 be thelogical matrices obtained from the algebra M∼4m. Then

(i) |=M∼4m

= |=M∼N

,

(ii) |=M∼4m

= |=M∼B

.

Therefore, the logic M∼4m for IBA can be characterized by a single logical matrix.

Proof. The proof is analogous to that for M4m found in [4]. �

6

5 A Hilbert-style presentation for M∼4m

In this section we define a Hilbert calculus for the logic M∼4m of M∼4m which, byProposition 4.9, can be considered as the logic preserving degrees of truth forthe variety IBA of involutive (or symmetric) Boolean algebras. Observe thatin the language Fm the equations characterizing a De Morgan negation ¬ arethe following: x = ¬¬x, and ¬(∼x⇒ y) = ∼(¬x⇒ ∼¬y).

Definition 5.1 Denote by H∼4m = 〈Fm,`H∼4m〉 the propositional logic defined

through the following Hilbert calculus, where α, β, γ ∈ Fm.

Axioms

(A1) α⇒ (β ⇒ α)

(A2) (α⇒ (β ⇒ γ))⇒ ((α⇒ β)⇒ (α⇒ γ))

(A3) (∼β ⇒ ∼α)⇒ ((∼β ⇒ α)⇒ β)

(A4) ¬¬α⇒ α

(A5) α⇒ ¬¬α

(A6) ¬(∼α⇒ β)⇒ ∼(¬α⇒ ∼¬β)

(A7) ∼(¬α⇒ ∼¬β)⇒ ¬(∼α⇒ β)

Inference Rules

(MP)α α⇒ β

β(CP)

α⇒ β

¬β ⇒ ¬α

Remark 5.2 Axioms (A1)-(A3) plus (MP) constitute a sound and completeaxiomatization of propositional classical logic in the language generated by ⇒and ∼ (cf. [9]). On the other hand, axioms (A4)-(A7) describe the conditionsrequired for a De Morgan negation in the given language, as mentioned above.Finally, the rule (CP) is required in order to guarantee a crucial property of aDe Morgan negation, namely: a ≤ b implies that ¬b ≤ ¬a. Additionally, thisaxiom guarantees that the De Morgan negation ¬ preserves logical equivalences.

Definition 5.3 (1) A derivation of a formula α in H∼4m is a finite sequence offormulas α1 . . . αn such that αn is α and every αi is either an instance of anaxiom, or αi is the consequence of αj and αk = (αj ⇒ αi) by (MP) for somej, k ≤ i− 1, or αi = (¬β ⇒ ¬γ) is the consequence of αj = (γ ⇒ β) by (CP)for some j ≤ i− 1. We say that α is derivable in H∼4m, and we write `H∼

4mα, if

there exists a derivation of it in H∼4m.(2) Let Γ ∪ {α} be a set of formulas in Fm. We say that α is derivable in H∼4mfrom Γ, and we write Γ `H∼

4mα, if either α is derivable in H∼4m, or there exists

a finite, non-empty subset {γ1, . . . , γn} of Γ such that (γ1 ⇒ (. . . (γn ⇒ α)) . . .)is derivable in H∼4m.

7

Remark 5.4 H∼4m satisfies, by the very definition, the Deduction Metatheorem:Γ, α `H∼

4mβ iff Γ `H∼

4m(α ⇒ β), for every Γ ∪ {α, β} ⊆ Fm. Additionally,

`H∼4m

α iff ∅ `H∼4m

α.

Let ≡ ⊆ Fm× Fm be the relation defined by

≡ =def {(α, β) : `H∼4m

α⇒ β and `H∼4m

β ⇒ α}.

Lemma 5.5 The relation ≡ is a congruence on Fm.

Proof. Clearly, ≡ is compatible with ⇒ and ∼ since (A1)–(A3) plus (MP) isan axiomatization of CPL. That is, ≡ is a Boolean congruence. On the otherhand, by rule (CP), it is immediate that ≡ is compatible with ¬. �

Theorem 5.6 The Lindenbaum algebra Fm/≡ of H∼4m is an involutive Booleanalgebra with: |α| ⇒ |β| =def |α ⇒ β|, ∼|α| =def |∼α| and ¬|α| =def |¬α|,where |γ| denotes the equivalence class of the formula γ.

Proof. The operations are well-defined, by Lemma 5.5. It is clear that Fm/≡is a Boolean algebra with an additional operation ¬. On the other hand, byaxioms (A4)–(A7) and by rule (CP), we have that ¬ is a De Morgan negation.�

Theorem 5.7 (Soundness and Completeness of H∼4m) The following conditionsare equivalent, for every subset Γ ∪ {β} of Fm:

(i) Γ `H∼4m

β,

(ii) Γ |=M∼4m

β.

Proof. (i) ⇒ (ii) (Soundness): It is easy to see that every axiom of H∼4m isvalid in M∼4m. On the other hand, if an instance of the premises of (MP) is validin M∼4m then the respective conclusion is also valid in M∼4m. The same holds for(CP). From this it follows that the theorems of H∼4m are valid in M∼4m.

Suppose now that Γ `H∼4m

β. If `H∼4m

β then β is valid in M∼4 , by theobservation above. Thus, Γ |=M∼

4mβ. Otherwise, there exists a finite, non-empty

set {γ1, . . . , γn} ⊆ Γ such that `H∼4m

(γ1 ⇒ (. . .⇒ (γn ⇒ β) . . .)). Then |=M∼4m

(γ1 ⇒ (. . . ⇒ (γn ⇒ β) . . .)) (as observed above) and so {γ1, . . . , γn} |=H∼4m

β.From this, Γ |=M∼

4mβ.

(ii) ⇒ (i) (Completeness): Suppose that Γ |=M∼4m

β. By Definition 4.6, thereexists a finite subset {γ1, . . . , γn} of Γ such that

∧{h(γ) : γ ∈ Γ0} ≤ h(β). If

n = 0, that is, if |=M∼4m

β, then h(β) = 1 for every h ∈ Hom(Fm,U) for all U ∈IBA, by Definition 4.6. In particular, h(β) = 1 for every h ∈ Hom(Fm,Fm/≡),by Theorem 5.6. Let h : Fm→ Fm/≡ be the canonical map given by h(δ) = |δ|,for every δ. Then h ∈ Hom(Fm,Fm/≡) and so |β| = 1. It follows that `H∼

4mβ.

Thus, Γ `H∼4m

β. On the other hand, if n > 0, γ1, . . . , γn |=M∼4m

β. Then, byRemark 4.11, |=M∼

4m(γ1 ⇒ (. . . (γn ⇒ β)) . . .). By the former case, `H∼

4m

(γ1 ⇒ (. . . (γn ⇒ β)) . . .). From this it follows that Γ `H∼4m

β as desired. �

Corollary 5.8 The consequence relation `H∼4m

introduced in Definition 5.3(2)is Tarskian, finitary and structural.

8

6 M∼4m as a normal extension of S5

In this section, we will prove that there is still one more perspective to under-stand involutive Boolean algebras: they are modal algebras for S5 satisfyingadditional modal axioms and so they are generated by an algebra belonging tothe hierarchy of Henle algebras (cf. [13]).

In [12] it was observed that, given an involutive Boolean algebra A, theoperator ∃x = x ∨ ¬∼x defines a possibility operator on A, in the sense of [10,11]. It was also obtained a characterization of involutive Boolean algebras as aparticular case of monadic Boolean algebras. Moreover, it was proved that theDe Morgan negation ¬ can be defined in terms of ∃ and the other operators ofA as

¬x = (x ∧ ∃∼x) ∨ ∼∃x. (2)

It should be noted that Monteiro’s definition of the possibility operator ∃(or, in the notation of modal algebras, ♦, see Proposition 3.3(iii)) coincides, upto duality, with the necessity operator � we found in Proposition 4.1 which, byProposition 4.3, is unique. By adapting equation (2), it follows that

¬x = (x ∧ ∼�x) ∨�∼x. (3)

From this, we can enrich a Boolean algebra with a modal operator � sat-isfying certain properties, instead of adding a De Morgan negation. In bothcases it is obtained the same class of structures, namely the involutive Booleanalgebras, since the De Morgan negation can then be defined by equation (3).The key is to translate the equations which characterize ¬ into the language ofBoolean algebras plus �. Thus:

Proposition 6.1 Involutive Boolean algebras are definitionally equivalent toBoolean algebras equipped with a unary operator � satisfying the following equa-tions:

¬′(x ∨ y) = ¬′x ∧ ¬′y (4)

¬′¬′x = x (5)

where ¬′x is an abbreviation for (x ∧ ∼�x) ∨�∼x, for every x.

Proof. As it was shown, given an involutive Boolean algebra then, by defining�x = x ∧ ¬∼x, the obtained structure is a tetravalent modal algebra. Now,define ¬′x = (x ∧ ∼�x) ∨�∼x. Then it is easy to see that ¬′x = ¬x for everyx. Since ¬ is a De Morgan negation and ¬′x = ¬x for every x, the equations (4)and (5) hold.

Conversely, if a Boolean algebra is equipped with a monadic operator �satisfying the equations (4) and (5) for ¬′x given by (x ∧ ∼�x) ∨ �∼x thenclearly ¬′x is a De Morgan negation and so the induced structure is an involutiveBoolean algebra. �

9

Proposition 6.2 Let 〈A,∧,∨,∼,�, 0〉 be an involutive Boolean algebra seen asa Boolean algebra together with an operator � as in Proposition 6.1. Then, theoperator ¬′ defined by ¬′x = (x ∧ ∼�x) ∨�∼x, for every x ∈ A, is the uniqueunary operator over A such that 〈A,∧,∨,¬′,�, 0〉 is a tetravalent modal algebra.

Proof. Clearly ¬′ is a De Morgan negation and so 〈A,∧,∨,¬′,∼, 0〉 is aDe Morgan algebra enriched by a Boolean complement ∼. By Corollary 4.4,the operator �′x = x ∧ ¬′∼x is such that 〈A,∧,∨,¬′,�′, 0〉 is a tetravalentmodal algebra. It is easy to prove that �′x = �x and then it follows that〈A,∧,∨,¬′,�, 0〉 is a tetravalent modal algebra.

Now, suppose that ¬ is a De Morgan negation in A such that the structure〈A,∧,∨,¬,�, 0〉 is a tetravalent modal algebra. By Proposition 4.3, �x =x ∧ ¬∼x. From this, it follows easily that ¬′x = (x ∧ ∼�x) ∨ �∼x = ¬x forevery x. �

Recall that a modal algebra is a structure 〈A,∧,∨,∼,�, 0, 1〉 of type (2, 2, 1, 1, 0, 0)where the reduct 〈A,∧,∨,∼, 0, 1〉 is a Boolean algebra and � is an unary op-erator on A such that �1 = 1 and �(x ∧ y) = �x ∧ �y (cf. [1]). By its turn,a modal algebra for S5 is a modal algebra satisfying, additionally, the followingrelations, for every x:

(T) �x ≤ x;

(4) �x ≤ ��x;

(B) x ≤ �∼�∼x.

We thus obtain the following result:

Proposition 6.3 Involutive Boolean algebras are definitionally equivalent tomodal algebras for modal logic S5 satisfying additionally the equations (4) and(5) of Proposition 6.1.

Proof. Let A be an involutive algebra A seen as a tetravalent modal al-gebra plus a Boolean complement ∼, by Corollary 4.5. Then, A is a modalalgebra which satisfies the properties (T), (4), (B) above (cf. [6]). On the otherhand, by Proposition 6.1 the equations (4) and (5) also hold. The converse is aconsequence of Proposition 6.1. �

Recall that a sentencial calculus L′ is a (weak) extension of another sentencialcalculus L if they are defined in the same language and every formula provablein L is provable in L′. In [13], it was introduced the notion of normal extensionof S5 as any extension S of S5 that is closed under substitution, modus ponensand, if α is provable in S, then �α is provable in S.

On the other hand, recall that a Henle algebra is an algebra 〈A,∧,∨,∼, i, 0, 1〉of type (2, 2, 1, 1, 0, 0) such that its reduct 〈A,∧,∨,∼, 0, 1〉 is a Boolean Algebra,i(1) = 1 and i(x) = 0 for all x ∈ A, x 6= 1 (cf. [13]). A Henle matrix is thelogical matrix 〈H, {1}〉 such that H is a Henle algebra. Hn is the Henle algebrawith n atoms (2n elements). Besides, if H is a finite Henle matrix, then thereexists n ∈ N such that H and Hn are isomorphic.

10

Theorem 6.4 ([13]) The following conditions are equivalent:

(i) S is a proper normal extension of S5;

(ii) S is the logic given by the matrix 〈Hn, {1}〉, for some n ∈ N.

By the results of Section 4, it is clear that M∼4m is a Henle algebra whenit is presented in the language ∧,∨,∼, 0, 1 and �. Then, it is immediate thatM∼4m = H2.

Taking into account that M∼4m can be seen as a modal logic and that modallogics are mainly concerned with theorems (or valid formulas), we may considerthe matrix logic given by the matrix M1 = 〈M∼4m, {1}〉 over the signature∧,∨,∼, 0, 1, �. That is: Γ |=M1 α iff there exists a finite subset Γ0 of Γ suchthat, for every homomorphism h over M∼4m, if h(γ) = 1 for every γ ∈ Γ0 thenh(α) = 1. Then, we have that M1 = 〈H2, {1}〉 and so, by Theorem 6.4:

Theorem 6.5 The matrix logic given by M1 is a proper normal extension ofS5.

Remark 6.6 Observe that, in the terminology of [2],M1 is the truth-preservinglogic associated to IBAs. The fact that the (global) logic of involutive Booleanalgebras is an extension of S5 is an interesting discovery: it means that thevariety generated by the Henle algebra H2 is IBA.

Finally, a Hilbert-style axiomatization for M1 in the language Fm can beeasily obtained from H∼4m, following the technique used in [4] for the logics ofTMAs. Indeed, let (H∼4m)N be the Hilbert system identical to H∼4m, but wherethe notion of derivation is the usual one in Hilbert calculi. That is, instead ofDefinition 5.3 we consider the following:

Definition 6.7 Let Γ ∪ {α} be a set of formulas in Fm. We say that α isderivable in (H∼4m)N from Γ, and we write Γ `(H∼

4m)N α, if there exists a finitesequence of formulas α1 . . . αn such that αn is α and every αi is either an instanceof an axiom, or αi ∈ Γ, or αi is the consequence of αj and αk = (αj ⇒ αi)by (MP) for some j, k ≤ i− 1, or αi = (¬β ⇒ ¬γ) is the consequence ofαj = (γ ⇒ β) by (CP) for some j ≤ i− 1.

Theorem 6.8 (Soundness) If Γ `(H∼4m)N α then Γ |=M1

α.

Proof. Every axiom is valid, and the inference rules preserve validity. �In order to prove completeness, it will be useful to consider the modality

�α =def α ∧ ¬∼α, where α ∧ β =def ∼(α ⇒ ∼β) (recall Section 4). Byadapting a result by [6], it is immediate to show the following:

Proposition 6.9 Γ |=M1α iff �Γ |=M∼

4mα.

On the other hand, it is easy to prove the following:

Proposition 6.10 For every formula α in Fm: α `(H∼4m)N �α.

11

Proof. Let γ be a theorem of (H∼4m)N . Then `(H∼4m)N ¬¬γ, by (A5) and

(MP). Consider now the following sketch of derivation in (H∼4m)N :

1. α (Hyp.)

2. α⇒ (∼α⇒ ¬γ) (by classical logic)

3. ∼α⇒ ¬γ (MP, 1,2)

4. ¬¬γ ⇒ ¬∼α (CP, 3)

5. ¬∼α (MP with 4 and the theorem ¬¬γ)

6. α ∧ ¬∼α (by 1,5 and classical logic)

�

Proposition 6.11 If �Γ `H∼4m

α then Γ `(H∼4m)N α.

Proof. Assume that �Γ `H∼4m

α. If `H∼4m

α then the result is obvious, sinceboth calculi have the same theorems. If 0H∼

4mα, there exists a finite, non-

empty subset {γ1, . . . , γn} of Γ such that `H∼4m

(�γ1 ⇒ (. . . (�γn ⇒ α)) . . .),and so `(H∼

4m)N (�γ1 ⇒ (. . . (�γn ⇒ α)) . . .). Consider now a derivation in

(H∼4m)N as follows: from the hypothesis γ1, . . . , γn derive �γ1, . . . ,�γn, byProposition 6.10. Using (MP) n times with the theorem (�γ1 ⇒ (. . . (�γn ⇒α)) . . .) it follows α. This shows that {γ1, . . . , γn} `(H∼

4m)N α, and so Γ `(H∼4m)N

α. �

Theorem 6.12 (Completeness) If Γ |=M1α then Γ `(H∼

4m)N α.

Proof. If Γ |=M1α then �Γ |=M∼

4mα, by Proposition 6.9. Using Theorem 5.7

it follows that �Γ `H∼4m

α and so Γ `(H∼4m)N α, by Proposition 6.11. �

7 Concluding Remarks

This paper shows that tetravalent modal algebras expanded with a deductiveimplication (or, equivalently, with a Boolean complement) are definitionallyequivalent to De Morgan algebras expanded with a Boolean complement, thatis, Boolean algebras plus a De Morgan negation. The latter are known assymmetric (or involutive) Boolean algebras. On the other hand, we also shownthat these structures are modal algebras for modal logic S5 satisfying additionalequations such that they are generated by the Henle algebra H2. Thus, one ofthe two logics that can be naturally associated to these algebraic structuresis a proper normal extension of S5. Finally, a simple Hilbert presentation forthese two logics was also proposed, proving the corresponding soundness andcompleteness theorems. In particular, the Hilbert calculus we propose for thenormal extension of S5 is interesting, since it does not include any axiom orinference rule using the modal operators.

12

Acknowledgements: We thank the anonymous referees for their helpful com-ments and suggestions, which allow us to improve the paper. The first authorwas financed by FAPESP (Brazil), Thematic Project LogCons 2010/51038-0and by an individual research grant from The National Council for Scientificand Technological Development (CNPq), Brazil. The second author was par-tially financed by UNS (Argentina), PGI Project 24/L090. This research is beingconducted within the IRSES Project MaToMuVi, PIRSES-GA-2009-247584.

References

[1] P. Blackburn, M. de Rijke and Y. Venema, Modal Logic. Cambridge Uni-versity Press, 2001.

[2] F. Bou, F. Esteva, J.M. Font, A. Gil, L. Godo, A. Torrens and V. Verdu,Logics preserving degrees of truth from varieties of residuated lattices. Jour-nal of Logic and Computation 19 (2009), no. 6, 1031–1069.

[3] M. E. Coniglio and M. Figallo. On the relationship between tetravalentmodal algebras, symmetric Boolean algebras and modal algebras for S5.CLE e-Prints, v. 13, n. 3, 2013.URL = http://www.cle.unicamp.br/e-prints/vol 13,n 3,2013.html

[4] M.E. Coniglio and M. Figallo, Hilbert-style Presentations of Two LogicsAssociated to Tetravalent Modal Algebras. Studia Logica , to appear. Firstpublished online: July 16, 2013. DOI: 10.1007/s11225-013-9489-0.

[5] A. V. Figallo and P. Landini, On generalized I-algebras and 4-valued modalalgebras. Reports on Mathematical Logic 29 (1995), 3–18.

[6] J. M. Font and M. Rius, An abstract algebraic logic approach to tetravalentmodal logics.J. Symbolic Logic 65 (2000), no. 2, 481–518.

[7] R. Jansana, Selfextensional Logics with a Conjunction. Studia Logica 84(2006), no. 1, 63–104.

[8] I. Loureiro, Algebras Modais Tetravalentes. PhD thesis, Faculdade deCiencias de Lisboa, 1983.

[9] E. Mendelson, Introduction to Mathematical Logic. Springer; 4th edition,1997.

[10] Gr.C. Moisil, Algebra schemelor cu elemente ventil. Revista UniversitatiiC.I. Parhon si a Politechnicci Bucaresti Seria St. Nat. 4-5 (1954), pp. 9–42.

[11] Gr.C. Moisil, Essais sur les logiques non chrysippiennes. Editions del’Academie de la Republique Socialiste de Roumanie, 1972.

13

[12] A. Monteiro, Notas del curso Algebras de Boole Involutivas. Instituto deMatematica, Universidad Nacional del Sur, Bahia Blanca, 1969. Reprintedas Algebras de Boole Involutivas, Informe Tecnico Interno No. 78, Univer-sidad Nacional del Sur, Bahia Blanca, 2002.

[13] S.J. Scroggs, Extensions of The Lewis System S5. The Journal of SymbolicLogic, 16 (1951), no. 2, 112–120.

14