aa r X i v : . [ m a t h . L O ] F e b LOGICS OF INVOLUTIVE STONE ALGEBRAS
S´ERGIO MARCELINO
SQIG - Instituto de Telecomunica¸c˜oes, PortugalDepartamento de Matem´atica, Instituto Superior T´ecnico, Universidade de Lisboa
UMBERTO RIVIECCIO
Departamento de Inform´atica e Matem´atica Aplicada, Universidade Federal do RioGrande do Norte, Natal (RN), Brasil
Abstract.
An involutive Stone algebra (IS-algebra) is a structure that is simultaneouslya De Morgan algebra and a Stone algebra (i.e. a pseudo-complemented distributivelattice satisfying the well-known Stone identity, ∼ x ∨ ∼ ∼ x ≈ IS ≤ ) has been introduced only very recently. The logic IS ≤ is thedeparting point for the present study, which we then extend to a wide family of previouslyunknown logics defined from IS-algebras. We show that IS ≤ is a conservative expansionof the Belnap-Dunn four-valued logic (i.e. the order-preserving logic of the variety ofDe Morgan algebras), and we give a finite Hilbert-style axiomatization for it. Moregenerally, we introduce a method for expanding conservatively every super-Belnap logicso as to obtain an extension of IS ≤ . We show that every logic thus defined can beaxiomatized by adding a fixed finite set of rule schemata to the corresponding super-Belnap base logic. We also consider a few sample extensions of IS ≤ that cannot beobtained in the above-described way, but can nevertheless be axiomatized finitely byother methods. Most of our axiomatization results are obtained in two steps: througha multiple-conclusion calculus first, which we then reduce to a traditional one. Themultiple-conclusion axiomatizations introduced in this process, being analytic, are ofindependent interest from a proof-theoretic standpoint. Our results entail that thelattice of super-Belnap logics (which is known to be uncountable) embeds into the latticeof extensions of IS ≤ . Indeed, as in the super-Belnap case, we establish that the finitary extensions of IS ≤ are already uncountably many. Introduction
Involutive Stone algebras (from now on, IS-algebras) were first considered in the papers [8,9] within a study of finite-valued Lukasiewicz logics and, more specifically, in connectionwith the algebraic structures nowadays known as Lukasiewicz-Moisil algebras. The term‘involutive’ is due to the observation that every IS-algebra has a primitive negation operation ∼ that satisfies the involutive law ( ∼ ∼ x ≈ x ), whereas ‘Stone’ refers to the existence of aterm-definable pseudo-complement operation ¬ that satisfies the well-known Stone identity ¬ x ∨ ¬¬ x ≈
1. From an algebraic point of view, IS-algebras are a variety of De Morgan
E-mail addresses : [email protected], [email protected] . algebras endowed with an additional unary operation (here denoted by ∇ ); alternatively, IS-algebras can be viewed as the subclass of De Morgan algebras that satisfy certain structuralproperties ensuring the definability of ∇ . The structural connection between De Morganand IS-algebras will indeed play a prominent role in the present paper.De Morgan algebras form a variety that is well known in non-classical logic as the al-gebraic counterpart of B , the four-valued Belnap-Dunn logic [2, 14]. From a logical pointof view, B can be viewed as a weakening of classical two-valued logic designed to allowfor both paraconsistency (in that B does not validate the rule p ∧ ∼ p ⊢ q , known as excontradictione quodlibet ) and paracompleteness (in that B does not validate the principleof excluded middle, ⊢ p ∨ ∼ p ). De Morgan algebras can thus be viewed as a generalizationof a Boolean algebras on which the operation ∼ (that interprets the negation connective)need not be a Boolean complement, i.e. the classical laws x ∧ ∼ x ≈ x ∨ ∼ x ≈ A having an additionalunary operation ∇ that receives an arbitrary element a ∈ A as argument and outputs acertain ‘classical’ element ∇ a (i.e. an element that possesses a Boolean complement in A ).Just as not every distributive lattice can be equipped with a Boolean complement operation,so not every De Morgan algebra can be endowed with an operation ∇ meeting the aboverequirement. However, if such an operation is definable, then it is unique.One might say that, on every De Morgan algebra A , the behaviour of ∇ provides ameasure of how far A is from being Boolean: the limit cases being, at one end of thespectrum, Boolean algebras themselves (on which ∇ is the identity map) and, at the other,the algebras (such as those depicted in Figure 2) where the only Boolean elements are thetop and the bottom. These structural requirements on ∇ can be completely captured bymeans of identities [9, Thm. 2.1]. Therefore, regardless of the preceding considerations,IS-algebras can be simply introduced as a variety of De Morgan algebras having an extraunary operation ∇ that is required to satisfy four additional identities (see Definition 3.2).A logic associated to IS-algebras has been considered for the first time in [6, 7]. Inthe present paper, we shall denote this logic by IS ≤ , suggesting that IS ≤ is the order-preserving logic canonically associated to the variety of IS-algebras (see Section 2 for therelevant definitions). As we will show, IS ≤ is a conservative expansion of the Belnap-Dunnlogic B , which is itself the order-preserving logic of the variety of De Morgan lattices. Weare moreover going to prove that, between the logics extending B (known as super-Belnaplogics since the paper [21]) and the extensions of IS ≤ , a connection can be established andexploited in order to obtain a number of non-trivial results.The background facts we shall need on the Belnap-Dunn logic can be found in [14],including a complete Hilbert-style axiomatization and a characterization of the reducedmatrix models (see Section 2). For further information on super-Belnap logics, we refer thereader to the papers [21, 1, 20], from which we shall also import a few results as needed.The rest of the paper is organized as follows. Section 2 introduces the generic algebraicand logical notions that will be used in the following ones. Section 3 contains the basicalgebraic results on De Morgan and involutive Stone algebras. In Section 4 we look atthe logic IS ≤ of involutive Stone algebras from a semantical point of view. We observethat IS ≤ non-protoalgebraic (Proposition 4.2) and can be characterized by a single finitematrix (Proposition 4.1). We further introduce a simple operation on logical matrices that OGICS OF INVOLUTIVE STONE ALGEBRAS 3 allows us to associate, to any given super-Belnap logic, a logic extending IS ≤ is such away that the latter is a conservative expansion of the former (Lemma 4.11). This entailsthat the lattice of super-Belnap logics is embeddable into the lattice of extensions of IS ≤ (Corollary 4.13), which in turn tells us that the latter must have at least the cardinality ofthe continuum (Corollary 4.14).In Section 5 we present a uniform method of axiomatizing all the extensions of IS ≤ that are defined from super-Belnap logics via the construction introduced in Section 4(Corollary 5.7, Proposition 5.9); a complete axiomatization for IS ≤ is thus obtained as aspecial application (Example 5.8). In order to achieve these results, we take a little detourthrough the realm of multiple-conclusion logics and the calculi that correspond to them. InSubsection 5.2 a number of logics extending IS ≤ (corresponding to substructures of thematrix that defines IS ≤ ) are axiomatized by a uniform application of the general method;these include logics obtained by adding the ∇ connective to well-known extensions of B ,such as G. Priest’s Logic of Paradox and the strong and weak three-valued logics due toS. C. Kleene. In Subsection 5.3 we axiomatize a few extensions of IS ≤ are not obtainedin this way from a super-Belnap logic, among which we find the three-valued Lukasiewicz(-Moisil) logic. For the latter results we cannot apply the above-mentioned method, sowe need to take a longer detour through multiple-conclusion logics and analytical calculi(Subsection 5.4). Finally, Section 6 contains a few concluding remarks and suggestions forfuture research. 2. Algebraic and logical preliminaries
In this Section we recall the main algebraic and logical notions that will be needed inthe following ones. We assume familiarity with basic results of lattice theory [13], universalalgebra [3] and the general theory of logical calculi [23, 16, 15].We shall denote by A , B etc. algebras over a given algebraic similarity type Σ . Theset of Σ -homomorphisms between two algebras A and B will be denoted by Hom ( A , B ).Given Σ -algebras A , B and a sub-signature Σ ′ ⊆ Σ , we denote by Hom Σ ′ ( A , B ) the set offunctions h : A → B that are only required to preserve the operations in Σ ′ . The algebra offormulas over a signature Σ will be denoted by Fm Σ (or simply by Fm , if Σ is clear fromthe context), and its elements by ϕ, ψ etc. Given a class K of similar algebras, we denoteby I ( K ), H ( K ), S ( K ), P ( K ), P S ( K ) the classes formed by closing K under (respectively),isomorphisms, homomorphisms, subalgebras, direct products and subdirect products. A variety is a class K of algebras that is closed under H , S , P , or, equivalently, that is definableby means of algebraic identities. A quasivariety is a class K of algebras that is definable bymeans of quasi-identities, that is, implications having a finite number of identities as premissand a single identity as conclusion. The variety (resp. quasivariety) generated by K will bedenoted by V ( K ) and Q ( K ). Every variety V is generated by the class V si of its subdirectlyirreducible members, defined as follows: an algebra A is subdirectly irreducible if A has aminimum congruence above the identity relation (as a special case, we say that A is simple if A has exactly two congruences). For every variety V , we have V = V ( V si ) = IP S ( V si ).We view a (propositional, single-conclusion) logic as a structural consequence relation on ℘ ( F m ) × F m (see e.g. [15, Def. 1.5]). The symbol ⊢ will be used to denote arbitrary logics.We say that a logic ⊢ is an extension of ⊢ when both logics share the same propositionallanguage Σ and ⊢ ⊆ ⊢ . The family of all extensions of a logic ⊢ forms a complete lattice(in which the meet is the intersection); in this paper we shall be concerned with the latticeof extensions of the logic IS ≤ of involutive Stone algebras, and will relate it to the lattice ofextensions of the Belnap-Dunn logic B (i.e. the super-Belnap logics). We say that a logic ⊢ over a language Σ is an expansion of a logic ⊢ over Σ when Σ ⊆ Σ and ⊢ ⊆ ⊢ . Wespeak of a conservative expansion when both consequence relations coincide on the formulasover Σ .A (logical) matrix is a pair M = h A , D i where A is an algebra and D ⊆ A is a subsetof designated elements . One defines the notions of isomorphism, homomorphism, subma-trix and product of matrices as straightforward extensions of the corresponding universalalgebraic constructions (see [23, 16] for details). Given a matrix M = h A , D i with A a Σ -algebra, we let Val( M ) = Hom ( Fm Σ , A ). We denote by Log the mapping that associates alogic to a class of matrices in the standard fashion. Indeed, each matrix determines a logic(denoted Log M or ⊢ M ) as follows: for all Γ ∪ { ϕ } ⊆ F m , one lets Γ ⊢ M ϕ iff, for everyvaluation v : Fm → A , v [ Γ ] = { v ( γ ) : γ ∈ Γ } ⊆ D entails v ( ϕ ) ∈ D . To a class of matrices M = { M i : i ∈ I } , we associate the logic Log M = ⊢ M := T {⊢ M i : i ∈ I } . We say that amatrix M is a model of a logic ⊢ when ⊢ ⊆ ⊢ M , that is, when Γ ⊢ ϕ entails Γ ⊢ M ϕ , for all Γ ∪ { ϕ } ⊆ F m .Every matrix M = h A , D i has an associated Leibniz congruence Ω A ( D ), which is thegreatest congruence on A that is compatible with D in the following sense: for all a ∈ D and b ∈ A , if h a, b i ∈ Ω A ( D ), then b ∈ D . This property allows one to define the quotientmatrix M ∗ = h A / Ω A ( D ) , D/ Ω A ( D ) i , which is known as the reduction of M . A matrix M is reduced if Ω A ( D ) is the identity relation, so no further reduction is possible. Reducedmatrices are important in the study of algebraic models of logics, because, for every matrix M , one has Log M = Log M ∗ . It follows that every logic coincides with the logic determinedby the class of all its reduced matrix models. In algebraic logic, two classes of algebras, Alg ∗ ( ⊢ ) and Alg ( ⊢ ), are traditionally associated to a given logic ⊢ . The former is definedas follows: Alg ∗ ( ⊢ ) := { A : there is D ⊆ A such that h A , D i is a reduced matrix for ⊢} By the characterization of [16, Thm. 2.23], the latter class can be introduced as follows
Alg ( ⊢ ) := P S ( Alg ∗ ( ⊢ )).Let K be a class of algebras such that each A ∈ K has a semilattice reduct with topelement 1. From K one can obtain a finitary logic ⊢ ≤ K as follows. One lets ∅ ⊢ ≤ K ϕ if andonly if the identity ϕ ≈ K and, for all Γ ∪ { ϕ } ⊆ F m such that Γ = ∅ , onelets Γ ⊢ ≤ K ϕ iff there are γ , . . . , γ n ∈ Γ such that the identity γ ∧ . . . ∧ γ n ∧ ϕ ≈ ϕ isvalid in K . We shall call ⊢ ≤ K the order-preserving logic associated to K . Observe that, bydefinition, K and V ( K ) define the same logic; thus, the order-preserving logics consideredin the literature are usually associated to varieties of semilattice-based algebras. We notethat, if V is a variety of algebras having a bounded distributive lattice reduct (as will alwaysbe the case in the present paper), then ⊢ ≤ V coincides with the logic defined by the class ofmatrices {h A , F i : A ∈ V , F ⊆ A is a (non-empty) lattice filter of A } .3. De Morgan and involutive Stone algebras
In this Section we recall the main definitions and basic results on the classes of algebrasinvolved.
Definition 3.1. A De Morgan lattice is an algebra A = h A ; ∧ , ∨ , ∼i of type h , , i suchthat h A ; ∧ , ∨i is a distributive lattice and the following identities are satisfied:(DM1) ∼ ( x ∨ y ) ≈ ∼ x ∧ ∼ y .(DM2) ∼ ( x ∧ y ) ≈ ∼ x ∨ ∼ y .(DM3) ∼ x ≈ ∼ ∼ x . OGICS OF INVOLUTIVE STONE ALGEBRAS 5 A De Morgan algebra is a De Morgan lattice whose lattice reduct is bounded (thus weinclude constant symbols ⊥ and ⊤ in the algebraic signature) and satisfies the followingidentities: ∼ ⊤ ≈ ⊥ and ∼ ⊥ ≈ ⊤ . Figure 1. (All the) subdirectly irreducible De Morgan algebras.0 b a DM a K B Figure 1 depicts the (only) subdirectly irreducible De Morgan algebras. On each algebra,the lattice operations are determined by the diagram. The negation is defined on DM by ∼ ∼ ∼ a = a and ∼ b = b . These prescriptions apply to K and B as wellviewed as subalgebras of DM . Obviously B is the two-element Boolean algebra, and K is the three-element Kleene algebra associated to the three-valued logics originating fromthe work of S.C. Kleene . Definition 3.2. An involutive Stone algebra (IS-algebra) is an algebra A = h A ; ∧ , ∨ , ∼ , ∇ , , i of type h , , , , , i such that h A ; ∧ , ∨ , ∼ , ⊥ , ⊤i is a De Morgan algebra and the followingidentities are satisfied:(IS1) ∇⊥ ≈ ⊥ .(IS2) x ≈ x ∧ ∇ x .(IS3) ∇ ( x ∧ y ) ≈ ∇ x ∧ ∇ y .(IS4) ∼ ∇ x ∧ ∇ x ≈ ⊥ .The class of IS-algebras will be denoted IS . The name ‘involutive Stone algebras’ ismotivated by the following observation. For every IS-algebra A = h A ; ∧ , ∨ , ∼ , ∇ , ⊥ , ⊤i ,the operation ¬ that realizes the term ¬ x := ∼ ∇ x is a pseudo-complement; moreover, A satisfies the so-called Stone identity ¬ x ∨ ¬¬ x ≈ ⊤ . Hence, h A ; ∧ , ∨ , ¬ , ⊥ , ⊤i is a Stonealgebra . Conversely, given an algebra h A ; ∧ , ∨ , ∼ , ¬ , ⊥ , ⊤i that has both a De Morgannegation and a pseudo-complement operation, upon defining ∇ x := ¬¬ x , one has that h A ; ∧ , ∨ , ∼ , ∇ , , i is an IS-algebra if and only if the following identity is satisfied: ¬ x = ∼ ¬¬ x [9, Remark 2.2].The variety of IS-algebras is generated by the six-element algebra IS , which is shown inFigure 2 together with its subalgebras IS , IS and IS . Our notation reflects the observa-tion that the De Morgan algebra reduct of IS is obtained by adjoining a new top ˆ1 and anew bottom ˆ0 element to the De Morgan algebra DM , and by extending the De Morganoperations in the obvious way (in particular, ∼ ˆ1 = ˆ0 and ∼ ˆ0 = ˆ1); cf. Proposition 3.4. Formally, a
Kleene lattice (algebra) is defined as a De Morgan lattice (algebra) that satisfies x ∧ ∼ x ≤ y ∨ ∼ y . It is well known that the variety of Kleene lattices (algebras) is V ( K ). Formally, a
Stone algebra can be defined as a bounded distributive lattice h A ; ∧ , ∨ , ¬ , ⊥ , ⊤i endowedby an extra unary operation ¬ that satisfies, for all a, b ∈ A , the following requirements: (i) a ∧ b = 0 iff a ≤ ¬ b , and (ii) ¬ a ∨ ¬¬ a = ⊤ . LOGICS OF INVOLUTIVE STONE ALGEBRAS
Figure 2. (All the) subdirectly irreducible IS-algebras.ˆ00 b a IS ˆ001ˆ1 a IS ˆ001ˆ1 IS ˆ00ˆ1 IS Likewise, IS is obtained from K , and IS from B (we can view IS as obtained in thesame way if we start from the one-element trivial De Morgan algebra); this observation willbe central in our approach to IS-algebras (see Section 4). The operation ∇ is given on IS (and on its subalgebras) by ∇ ˆ0 = ˆ0 and ∇ ∇ a = ∇ b = ∇ ∇ ˆ1 = ˆ1. Observe that IS is isomorphic to the two-element Boolean algebra B (on which ∇ is the identity map). It isalso well known that, upon defining x ⇒ y := ( ∇ ∼ x ∨ y ) ∧ ( ∇ y ∨ ∼ x ), the algebra IS canbe endowed with an MV-algebra structure [9, Thm. 2.9]. Conversely, on the three-elementMV-algebra, one obtains an IS-algebra structure by letting ∇ x := ∼ x ⇒ x . Thus IS can be viewed as the three-element MV-algebra. On the other hand, on IS and IS the Lukasiewicz implication is not definable (to see this, it is sufficient to observe that neitherof these algebras is simple, whereas it is known that every finite MV-chain is [10, Cor. 3.5.4].The following result is an easy consequence of the observations in [9], and entails that, toverify the validity on all IS-algebras of not only identities but also quasi-identities, it issufficient to test them on IS . Proposition 3.3. IS = V ( IS ) = Q ( IS ) .Proof. That IS = V ( IS ) is well-known [7, Cor. 3.5]. A sufficient condition for having V ( IS ) = Q ( IS ) = ISP ( IS ) is that all the subdirectly irreducible algebras in V ( IS )be subalgebras of IS (see e.g. [11, Thm. 3.6.ii]). The latter indeed holds, and has beenobserved in [9]; see Thm. 2.8 therein and the subsequent remarks. (cid:3) The following easy observation will be very useful in our study of IS-logics from Section 4on. Given a De Morgan algebra A , let A ∇ = h A ∪{ ˆ0 , ˆ1 } , ∇i be the algebra defined as follows.The lattice reduct of A ∇ is obtained by adjoining a new top element ˆ1 and a new bottomelement ˆ0 to the lattice reduct of A . The De Morgan negation ∼ is extended to A ∇ in theobvious way, i.e. by letting ∼ ˆ1 = ˆ0 and ∼ ˆ0 = ˆ1. Furthermore, the unary operator ∇ isdefined as follows: ∇ ˆ0 = ˆ0 and ∇ a = ∇ ˆ1 = ˆ1 for all a ∈ A . It is then clear that the ∇ -freereduct of A ∇ is a De Morgan algebra, and it is very easy to check that ∇ satisfies all theproperties required by Definition 3.1.The following easy observation will be very useful in our study of IS-logics from Section 4on. Let A = h A ; ∧ , ∨ , ∼ , ⊥ , ⊤i be an algebra in the language of De Morgan algebras. Givenˆ0 , ˆ1 / ∈ A , we define the algebra A ∇ = h A ∪ { ˆ0 , ˆ1 } , ∇i as follows: OGICS OF INVOLUTIVE STONE ALGEBRAS 7 ∇ x := ( ˆ0 if x = ˆ0ˆ1 otherwise ∼ x := ∼ A x if x ∈ A ˆ0 if x = ˆ1ˆ1 if x = ˆ0 x ∧ y := x ∧ A y if x, y ∈ A ˆ1 if x = y = ˆ1ˆ0 if x = ˆ0 or y = ˆ0 x ∨ y := x ∨ A y if x, y ∈ A ˆ0 if x = y = ˆ0ˆ1 if x = ˆ1 or y = ˆ1 . ⊤ := ˆ1 ⊥ := ˆ0 . In case A is a De Morgan algebra, it is clear that the ∇ -free reduct of A ∇ is also aDe Morgan algebra. Moreover, it is very easy to check that the above-defined ∇ operationsatisfies all the properties required by Definition 3.1. Thus, we have the following. Proposition 3.4.
For every De Morgan algebra A , the above-defined algebra A ∇ is anIS-algebra. Semantical considerations on IS-logics
It is shown in [7, Thm 5.2] the order-preserving logic of the variety IS (which we denoteby IS ≤ ) coincides with the logic determined by the closure system of all lattice filters onthe generating algebra IS (which are the principal up-sets ↑ , ↑ a, ↑ b, ↑
1, and { ˆ1 } ). Sincethe matrices h IS , ↑ a i and h IS , ↑ b i define the same logic [7, Lemma 5.4], we have that IS ≤ is determined by the following set of matrices: {h IS , ↑ i , h IS , ↑ a i , h IS , ↑ i , h IS , { ˆ1 }i} .This result can be further sharpened, as the following Proposition shows. Proposition 4.1. IS ≤ = Log h IS , ↑ a i .Proof. Observe that the matrices h IS , ↑ i and h IS , ↑ a i are reduced, while h IS , { ˆ1 }i and h IS , ↑ i are not. The reduction of h IS , { ˆ1 }i is isomorphic to h IS , { ˆ1 }i , and istherefore isomorphic to a submatrix of h IS , ↑ b i . The reduction of h IS , ↑ i is isomor-phic to h IS , { , ˆ1 }i , which in turn is isomorphic to a submatrix of h IS , ↑ a i . ThusLog {h IS , ↑ a i , h IS , ↑ i , h IS , { ˆ1 }i , h IS , ↑ i} = Log {h IS , ↑ a i , h IS , ↑ i} . To concludethe proof, it suffices to show that Log h IS , ↑ a i ⊆ Log h IS , ↑ i . To see this, notice that ↑ ↑ a ∩ ↑ b . This easily entails that Log {h IS , ↑ a i , h IS , ↑ b i} ⊆ Log h IS , ↑ i , and wehave seen that Log {h IS , ↑ a i , h IS , ↑ b i} = Log h IS , ↑ a i . (cid:3) In our study, it will be useful to be able to work with reduced matrix models of IS-logics.Proposition 4.2 below suggests that these cannot be characterized by simply applying theBlok-Pigozzi algebraization process, but Proposition 4.4 provides sufficient information forour purposes. (For the definitions of selfextensional, protoalgebraic and algebraizable logic,we refer the reader to [15], respectively, Def. 5.25, 6.1 and 3.11).
Proposition 4.2. IS ≤ is selfextensional and non-protoalgebraic (hence, non-algebraizable).Proof. Selfextensionality simply follows from the observation that two formulas ϕ, ψ areinter-derivable in IS ≤ if and only if the identity ϕ ≈ ψ is valid in the variety of IS-algebras.To show that our logic is not protoalgebraic, we verify that the Leibniz operator Ω is notmonotone on matrix models [15, Thm. 6.13]. To see this, observe that the algebra IS has(exactly) one non-trivial congruence θ , which identifies the elements { , a, b, } . It is theneasy to check that Ω IS ( { ˆ1 } ) = θ . On the other hand, the matrix h IS , ↑ i is reduced.Hence, the Leibniz operator is not monotone on the matrix models based on IS . (cid:3) LOGICS OF INVOLUTIVE STONE ALGEBRAS
Remark 4.3.
Proposition 4.2 can in fact be slightly strengthened. If we consider thematrices h IS , { ˆ1 }i and h IS , ↑ i , where IS is the four-element subalgebra of IS withuniverse { ˆ0 , , , ˆ1 } , we can observe that h IS , ↑ i is reduced while h IS , { ˆ1 }i is not. Hencethe logic determined by these two submatrices (which is obviously stronger than IS ≤ ) isalso non-protoalgebraic. This, in turn, entails that IS ≤ cannot be protoalgebraic.The following result is an instance of a general result on order-preserving logics (seee.g. [1, Thm. 2.13.iii]). Proposition 4.4.
Alg ( IS ≤ ) = IS . The next Proposition we characterizes the logic determined by the class of matrices {h A , { }i : A ∈ IS } . The latter (denoted by ⊢ IS ) is known in the algebraic logic literatureas the of the class IS . Proposition 4.5. ⊢ V ( IS ) = ⊢ IS = Log h IS , { }i .Proof. Obviously ⊢ V ( IS ) ⊆ ⊢ IS ⊆ Log h IS , { }i , so it suffices to verify the inequalityLog h IS , { }i ⊆ ⊢ V ( IS ) . Assume Γ ⊢ Log h IS , { }i ϕ . Observe that, since Log h IS , { }i is finitary, we can assume Γ to be finite. Then γ ⊢ Log h IS , { }i ϕ for γ := V Γ . The latteris equivalent to γ ⊢ Log h IS , { }i ϕ , because, as observed earlier, h A , { }i ∗ = h IS , { }i . Inturn, γ ⊢ Log h A , { }i ϕ entails that IS satisfies the quasi-identity γ ≈ ⊤ ⇒ ϕ ≈ ⊤ . ByProposition 3.3, this entails that γ ≈ ⊤ ⇒ ϕ ≈ ⊤ is satisfied by every A ∈ IS . Hence, γ ⊢ ϕ holds in every matrix in the class {h A , { }i : A ∈ IS } and, a fortiori, in every matrix in theclass {h A , { }i : A ∈ V ( IS ) } . This means that γ ⊢ IS ϕ or, equivalently, Γ ⊢ IS ϕ . (cid:3) Recalling that the algebra IS is isomorphic to to the three-element Lukasiewicz(-Moisil)algebra, Proposition 4.5 tells us that ⊢ IS is (term equivalent) to three-valued Lukasiewiczlogic. This logic is axiomatized, relatively to IS ≤ , in Theorem 5.14 (i).We now return to the construction introduced at the end of Section 3 and illustrate itsremarkable logical consequences. Let M = h A , D i be a matrix, with A an algebra in thelanguage of De Morgan algebras. Then A ∇ is in the language of IS . Denoting by b A the ∇ -free reduct of A ∇ , let b M := h b A , D ∪ { ˆ1 }i . Lemma 4.6.
Let M be a reduced model of B . Then M ∼ = ( b M ) ∗ .Proof. Recall from [14, Thm. 3.14] that all reduced models of B are matrices M = h A , F i ,with A a De Morgan algebra and F a lattice filter. To establish the claim, it suffices toshow that Ω b A ( F ∪ { ˆ1 } ) = Id b A ∪ {h , ˆ0 i , h ˆ0 , i , h , ˆ1 i , h ˆ1 , i} . Let a, b ∈ b A . According to [14,Prop. 3.13], we have h a, b i ∈ Ω b A ( F ∪ { ˆ1 } ) if and only if, for all c ∈ b A , the following hold:( a ∨ c ∈ F ∪ { ˆ1 } iff b ∨ c ∈ F ∪ { ˆ1 } ) and ( ∼ a ∨ c ∈ F ∪ { ˆ1 } iff ∼ b ∨ c ∈ F ∪ { ˆ1 } ). Let usshow that h , ˆ1 i ∈ Ω b A ( F ∪ { ˆ1 } ). Observe that 1 ∨ c, ˆ1 ∨ c ∈ F ∪ { ˆ1 } for all c ∈ b A . The firstcondition is thus obviously satisfied. As to the second, assume ∼ ∨ c = 0 ∨ c ∈ F ∪ { ˆ1 } forsome c ∈ b A . Then c / ∈ { , ˆ0 } , because 0 ∨ ˆ0 = 0 ∨ / ∈ F ∪ { ˆ1 } . This entails 0 < c , so0 ∨ c = c ∈ F ∪ { ˆ1 } . Hence, ∼ ˆ1 ∨ c ∈ F ∪ { ˆ1 } . Conversely, if ∼ ˆ1 ∨ c = ˆ0 ∨ c ∈ F ∪ { ˆ1 } , thenwe immediately have ∼ ∨ c = 0 ∨ c ∈ F ∪ { ˆ1 } because ˆ0 ≤
0. Hence, h , ˆ1 i ∈ Ω b A ( F ∪ { ˆ1 } ).By the congruence properties, this entails h∼ , ∼ ˆ1 i = h , ˆ0 i ∈ Ω b A ( F ∪ { ˆ1 } ), thus also h ˆ0 , i , h ˆ1 , i ∈ Ω b A ( F ∪ { ˆ1 } ). Now let h a, b i ∈ Ω b A ( F ∪ { ˆ1 } ) be such that a, b / ∈ { ˆ0 , ˆ1 } and a = b . The latter assumption entails that h a, b i / ∈ Ω A ( F ), because M was reduced. Then,by [14, Prop. 3.13], there is c ∈ A such that either ( a ∨ c ∈ F and b ∨ c / ∈ F ) or ( ∼ a ∨ c / ∈ F and OGICS OF INVOLUTIVE STONE ALGEBRAS 9 ∼ b ∨ c ∈ F ). In the former case, we have a ∨ c ∈ F ∪{ ˆ1 } and b ∨ c / ∈ F ∪{ ˆ1 } , because b ∨ c = ˆ1for all b, c ∈ A . Hence, we should have h a, b i ∈ Ω b A ( F ∪ { ˆ1 } ), contradicting our assumptions.A similar reasoning shows that the latter case also leads to a contradiction. (cid:3) We note that Lemma 4.6 could be proved in a more general form, that we shall howevernot need for our present purposes. Indeed, given an arbitrary (not necessarily reduced)model M of B , one can show that M ∗ ∼ = ( b M ) ∗ . Corollary 4.7.
Given matrices M and b M as per Lemma 4.6, we have Log M = Log b M . As before, M = h A , F i is a matrix such that A is a De Morgan algebra and F ⊆ A alattice filter on A . Consider the IS-algebra A ∇ defined according to Proposition 3.4, andlet M ∇ = h A ∇ , F ∪ { ˆ1 }i . Corollary 4.8.
Let M = h A , F i be a reduced matrix, with A a De Morgan algebra and F ⊆ A a lattice filter on A . Then Log M ∇ is a conservative expansion of Log M .Proof. Using Corollary 4.7, it suffices to observe that the ∇ -free fragment of Log M ∇ isLog b M . (cid:3) Corollary 4.9. IS ≤ is a conservative expansion of the Belnap-Dunn logic B .Proof. Recall that IS ≤ = Log h IS , ↑ a i (Proposition 4.1), and observe that the matrix h IS , ↑ a i can be obtained as M ∇ from the four-element matrix M = h DM , ↑ a i thatdefines B . Then the result follows from Corollary 4.8. (cid:3) Recall that all reduced matrices for the Belnap-Dunn logic (hence, also all reduced ma-trices for super-Belnap logics) are of the form h A , F i , with A a De Morgan algebra and F a lattice filter [14, Thm. 3.14]. Thus, Lemma 4.6 and Corollaries 4.7 and 4.8 apply, and thelatter gives us the following result. Corollary 4.10.
Let M = h A , F i and M = h A , F i be reduced matrices for the Belnap-Dunn logic. If Log M ∇ = Log M ∇ , then Log M = Log M . Given a super-Belnap logic ⊢ , let ⊢ ∇ = Log { M ∇ : M ∈ Matr ∗ ( ⊢ ) } , where each M ∇ = h A ∇ , F ∪ { ˆ1 }i is defined as before. Since F ∪ { ˆ1 } is a lattice filter of A ∇ , every M ∇ is amodel of IS ≤ . Therefore, each logic ⊢ ∇ is an extension of IS ≤ . Lemma 4.11.
Let ⊢ be a super-Belnap logic. Then ⊢ ∇ is a conservative expansion of ⊢ .Proof. Suppose, in view of a contradiction, that there exist formulas
Γ, ϕ in the ∇ -freelanguage such that Γ ⊢ ϕ holds in ⊢ ∇ but does not hold in ⊢ . Then there is M ∈ Matr ∗ ( ⊢ )such that Γ M ϕ . Then, by Corollary 4.8, we have that Γ M ∇ ϕ . By definition, ⊢ ∇ ⊆ Log M ∇ . Hence, Γ ∇ ϕ . (cid:3) Corollary 4.12.
Let ⊢ , ⊢ be super-Belnap logics. Then ⊢ ⊆ ⊢ if and only if ⊢ ∇ ⊆ ⊢ ∇ .Proof. Assuming ⊢ ⊆ ⊢ , we have Matr ∗ ( ⊢ ) ⊆ Matr ∗ ( ⊢ ). Hence, { M ∇ : M ∈ Matr ∗ ( ⊢ ) } ⊆ { M ∇ : M ∈ Matr ∗ ( ⊢ ) } , which entails ⊢ ∇ ⊆ ⊢ ∇ . Conversely, let ⊢ ∇ ⊆ ⊢ ∇ , and let Γ, ϕ be formulas (in the language of B ) such that Γ ⊢ ϕ . The latter assumption gives usthat Γ ⊢ ∇ ϕ and, therefore, also Γ ⊢ ∇ ϕ . Then, by Lemma 4.11, we conclude Γ ⊢ ϕ . (cid:3) Corollary 4.13.
The map given by ⊢ 7→ ⊢ ∇ is an embedding of the lattice of super-Belnaplogics into the lattice of extensions of IS ≤ . Corollary 4.14.
The lattice of extensions of IS ≤ has (at least) the cardinality of thecontinuum.Proof. By Corollary 4.13 and the observation that the lattice of super-Belnap logics containscontinuum many logics [1, Thm. 4.13]. (cid:3)
In fact, in the light of the results of Section 5, we shall be able to prove that there areat least continuum many finitary extensions of IS ≤ .5. Axiomatizing IS-logics
In [6, 7], the logic IS ≤ is axiomatized by means of a Gentzen calculus. In this Sectionwe tackle the problem of axiomatizing IS ≤ and its extensions by means of Hilbert calculi.From a technical point of view, we shall take profit from the theory of multiple-conclusioncalculi, a generalization of traditional Hilbert-style calculi in which the inference rules canhave more than one conclusion (with a disjunctive reading). In these calculi proofs aretypically ramified instead of sequential. Multiple-conclusion calculi can be used to studysingle conclusion logics, but also correspond to a generalized notion of logic due to D. Scottand developed by D.J. Shoesmith and T.J. Smiley. We recall some of the basic definitionsand results below; for further details see [22, 18].A multiple-conclusion consequence relation (logic) is a relation ⊲ ⊆ ℘F m × ℘F m satis-fying the following conditions. For every Γ, Γ ′ , ∆, ∆ ′ , Λ, T, F ⊆ F m ,(i) Γ ⊲ ∆ whenever Γ ∩ ∆ = ∅ ( overlap ),(ii) Γ, Γ ′ ⊲ ∆, ∆ ′ whenever Γ ⊲ ∆ ( dilution ),(iii) Γ ⊲ ∆ whenever Γ, T ⊲ ∆, F for every partition h T, F i of Λ ( cut for sets ),(iv) Γ σ ⊲ ∆ σ for every substitution σ whenever Γ ⊲ ∆ ( substitution invariance ).Given a set of multiple-conclusion rules R ⊆ ℘F m × ℘F m , we denote by ⊲ R the smallestmultiple-conclusion consequence relation containing R (hence, R axiomatizes ⊲ R ). From aproof-theoretic perspective, we have Γ ⊲ R ∆ whenever there is a labelled tree-proof whoseroot is labelled by Γ and the leaf of every non-discontinued branch is labelled with a formulain ∆ . Every class of matrices M determines a multiple-conclusion logic defined as follows:we let Γ ⊲ M ∆ whenever, for every valuation v ∈ Val( M ) over a matrix M = h A , D i ∈ M ,we have that v ( Γ ) ⊆ D implies v ( ∆ ) ∩ D = ∅ .Multiple-conclusion logics smoothly generalize Tarskian logics and their proof-theoreticand semantical definitions. Indeed, for every multiple-conclusion logic ⊲ , we have that ⊢ ⊲ = ⊲ ∩ ( ℘L × L ) is a Tarskian consequence relation [15, Def. 1.5]. We call ⊢ ⊲ the single-conclusion companion of ⊲ and, given a set of multiple-conclusion rules R , we shallwrite ⊢ R instead of ⊢ ⊲ R . The following remark contains a few useful facts that can be easilydeduced from Sections 5.2 and 17.3 of [22]. Remark 5.1.
The sign of a multiple-conclusion relation ⊲ is negative if F m × ∅ ∈ ⊲ ,and is positive otherwise. We denote by ≃ the equivalence relation that identifies twologics ⊲ and ⊲ that may differ only in the sign, that is, we let ⊲ ≃ ⊲ whenever ⊲ ∪ { F m × ∅} = ⊲ ∪ { F m × ∅} . Let ⊲ ≃ ⊲ . Then ⊢ ⊲ = ⊢ ⊲ and also, if ⊲ ⊆ ⊲ ⊆ ⊲ ,then ⊲ ≃ ⊲ ≃ ⊲ . Let P ( M ) be the closure under products of the class M (productsamong matrices are defined as usual for first-order structures; see e.g. [15, p. 225-6]). Thefollowing observations are well known:(i) ⊢ ⊲ M = Log M = Log P ( M ). OGICS OF INVOLUTIVE STONE ALGEBRAS 11 (ii) If Log M = ⊢ R for R ⊆ ℘ ( L ) × L , then ⊲ R ≃ ⊲ P ( M ) . Therefore, if Log M = Log M then ⊲ P ( M ) ≃ ⊲ P ( M ) .Under certain conditions, a (finite) single-conclusion axiomatization can be obtainedalgorithmically from a (finite) multiple-conclusion axiomatization. The following resultcovers the case of some of the logics that interest us here.Given a finite set Φ = { ϕ , . . . , ϕ n } ⊆ F m and ψ ∈ F m , let W Φ := (( ϕ ∨ ϕ ) ∨ . . . ) ∨ ϕ n ,and let Φ ∨ ψ = { ϕ ∨ ψ : ϕ ∈ Φ } . Theorem 5.2. [22, Thm. 5.37]
Let R be a set of multiple-conclusion rules. Suppose ⊢ ⊲ R satisfies, for all Γ ∪ { ϕ, ψ, ξ } ⊆ F m , the following property:
Γ, ϕ ∨ ψ ⊢ ⊲ R ξ if and only if Γ, ϕ ⊢ ⊲ R ξ and Γ, ψ ⊢ ⊲ R ξ . Then ⊢ ⊲ R is axiomatized by the set R ∨ consisting of the followingrules:(i) r ∨ = ϕ for each r = ϕ ∈ R ,(ii) r ∨ = Γ ∨ p ( W ∆ ) ∨ p for each r = Γ∆ ∈ R ,(iii) pp ∨ q , p ∨ qq ∨ p , p ∨ pp and p ∨ ( q ∨ r )( p ∨ q ) ∨ r where p is a variable not occurring in R . We now proceed to explain how the results of Section 4 together with the general consid-erations on multiple-conclusion logics introduced above are going to be help us deal withextensions of IS ≤ .5.1. Adding ∇ to the Belnap-Dunn logic. Let Σ = {∧ , ∨ , ∼ , ⊥ , ⊤} be the languageof B , and let Σ ∇ be the expansion of Σ with the unary connective ∇ (i.e. the language of IS ≤ ). Given a Σ -matrix M = h A , D i , let M ∇ = h A ∇ , D ∪ { ˆ1 }i be the Σ ∇ -matrix withunderlying algebra A ∇ defined as in Section 3 (cf. Propositiion 3.4). Let us denote by b M the Σ -fragment of M ∇ . Observe that, if M = h A , D i with A a De Morgan algebra, then b M is precisely the matrix considered in Corollary 4.7. Given a class of Σ -matrices M , we let M ∇ := { M ∇ : M ∈ M} and c M := { b M : M ∈ M} .The following Theorem contains a generic recipe for axiomatizing the multiple-conclusionlogic determined by the class M ∇ , assuming we have a set of rule R that axiomatizes themultiple-conclusion logic determined by c M . Theorem 5.3.
Let M be a class of Σ -matrices. If ⊲ c M ≃ ⊲ R , then ⊲ M ∇ = ⊲ R ∪ R ∇ , where R ∇ consists of the following rules: ∇ p , ∼ ∇ p r ∇ p ∼ ∇ ∼ ∇ p r ∇∇ p ∇ p r ∇ p , ∼ ∇ p r ∼ ∇ p ∇ ∼ p r ∇ ∼ ∼ p ∇ p r ∇ p ∇ ∼ ∼ p r ∇ ( p ∧ q ) , ∇ ∼ ( p ∧ q ) ∇ p r ∇ ( p ∧ q ) , ∇ ∼ ( p ∧ q ) ∇ q r ∇ ( p ∧ q ) , ∇ ∼ ( p ∧ q ) ∇ ∼ p r ∇ ( p ∧ q ) , ∇ ∼ ( p ∧ q ) ∇ ∼ q r ∇ ∼ p ∇ ∼ ( p ∧ q ) r ∇ ∼ q ∇ ∼ ( p ∧ q ) r ∇ p , ∇ q ∇ ( p ∧ q ) r
142 LOGICS OF INVOLUTIVE STONE ALGEBRAS ∇ ( p ∨ q ) , ∇ ∼ ( p ∨ q ) ∇ p r ∇ ( p ∨ q ) , ∇ ∼ ( p ∨ q ) ∇ q r ∇ ( p ∨ q ) , ∇ ∼ ( p ∨ q ) ∇ ∼ p r ∇ ( p ∨ q ) , ∇ ∼ ( p ∨ q ) ∇ ∼ q r ∇ p ∇ ( p ∨ q ) r ∇ q ∇ ( p ∨ q ) r ∇ ∼ p , ∇ ∼ q ∇ ∼ ( p ∨ q ) r ∼ ∇⊥ r ∼ ∇ ∼ ⊤ r Proof.
Checking the soundness of the new rules is routine. We give only a couple of ex-amples. Let v be a valuation over a matrix M ∇ . The rule r is sound in M ∇ , for either v ( ∇ ϕ ) = ˆ1 (if v ( ϕ ) = ˆ0) or v ( ∼ ∇ ϕ ) = ˆ1 (if v ( ϕ ) = ˆ0). Regarding rule r , we have that, if v ( ∇ ϕ ) = ˆ1 then v ( ∼ ∇ ϕ ) = v ( ∇ ∼ ∇ ϕ ) = ˆ0, so v ( ∼ ∇ ∼ ∇ ϕ ) = ˆ1.For completeness, assume Γ ⊲ R ∇ ∆ . Then, by cut for sets, there is a partition h T, F i of L Σ ∇ such that Γ ⊆ T and ∆ ⊆ F and T ⊲ R ∇ F . Note that (by r and r ) for each ϕ , wehave either ∇ ϕ ∈ T or ∼ ∇ ϕ ∈ T , but never both. In particular, F is never empty. Also,by r we must have either ∇ ϕ ∈ T or ∇ ∼ ϕ ∈ T . Hence, each ϕ must be exactly in one ofthree cases: (i) ∇ ϕ, ∇ ∼ ϕ ∈ T , (ii) ∼ ∇ ∼ ϕ ∈ T , or (iii) ∼ ∇ ϕ ∈ T .Since R ⊆ R ∪ R ∇ , we also have T ⊲ R F . From the fact that ⊲ R ≃ ⊲ b M and F = ∅ weknow that T ⊲ b M F . We can therefore pick v ∈ Hom Σ ( L Σ ∇ , b M ), for some M ∈ M such that v ( T ) ⊆ D and v ( F ) ∩ D = ∅ . Consider v ′ : L Σ ∇ → M ∇ defined by: v ′ ( ϕ ) := v ( ϕ ) if ∇ ϕ, ∇ ∼ ϕ ∈ T ˆ1 if ∼ ∇ ∼ ϕ ∈ T ˆ0 if ∼ ∇ ϕ ∈ T. We will show that v ′ ∈ Val( M ∇ ) = Hom Σ ∇ ( L Σ ∇ , b M ).1. v ′ ( ∇ ϕ ) = ∇ v ′ ( ϕ )From r we have that either (ii) ∇ ϕ ∈ T or (iii) ∼ ∇ ϕ ∈ T .If (ii) ∇ ϕ ∈ T , by r we have that ∼ ∇ ϕ / ∈ T and so v ′ ( ϕ ) = ˆ0. Further, by r weobtain that ∼ ∇ ∼ ∇ ϕ ∈ T , hence v ′ ( ∇ ϕ ) = ∇ ( v ′ ( ϕ )) = ˆ1.If instead iii) ∼ ∇ ϕ ∈ T then v ′ ( ∇ ϕ ) = ∇ ( v ′ ( ϕ )) = v ′ ( ϕ ) = ˆ0 = as by r , ∼ ∇∇ ϕ ∈ T .2. v ′ ( ∼ ϕ ) = ∼ v ′ ( ϕ )If (i) ∇ ∼ ϕ, ∇ ∼ ∼ ϕ ∈ T then by r , ∇ ϕ ∈ T (so v ′ ( ϕ ) = v ( ϕ )) and therefore v ′ ( ∼ ϕ ) = v ( ∼ ϕ ) = ∼ M ( v ( ϕ )) = ˜ ∼ ( v ′ ( ϕ )).If (ii) ∼ ∇ ∼ ∼ ϕ ∈ T then by r and r we have that ∼ ∇ ϕ ∈ T (so v ′ ( ϕ ) = ˆ0) hence v ′ ( ∼ ϕ ) = ˆ1 = ˜ ∼ ( v ′ ( ϕ )).If (iii) ∼ ∇ ∼ ϕ ∈ T then v ′ ( ∼ ϕ ) = ˆ0 and v ′ ( ϕ ) = ˆ1, thus we immediately obtain v ′ ( ∼ ϕ ) = ∼ v ′ ( ϕ ).3. v ′ ( ϕ ∧ ψ ) = v ′ ( ϕ ) ∧ v ′ ( ϕ )If (i) ∇ ( ϕ ∧ ψ ) , ∇ ∼ ( ϕ ∧ ψ ) ∈ T then by r − r we obtain that ∇ ϕ, ∇ ψ, ∇ ∼ ϕ, ∇ ∼ ψ ∈ T . Hence v ′ ( ϕ ∧ ψ ) = v ( ϕ ∧ ψ ) = v ( ϕ ) ∧ M v ( ψ ) = v ′ ( ϕ ) ∧ v ′ ( ϕ ).If (ii) ∼ ∇ ∼ ( ϕ ∧ ψ ) ∈ T then by r and r we have that ∼ ∇ ∼ ϕ, ∼ ∇ ∼ ψ ∈ T (so v ′ ( ϕ ) = v ′ ( ψ ) = ˆ1) hence v ′ ( ϕ ∧ ψ ) = ˆ1 = v ′ ( ϕ )˜ ∧ v ′ ( ψ ).If iii) ∼ ∇ ( ϕ ∧ ψ ) ∈ T then by r we have that either ∼ ∇ ϕ ∈ T or ∼ ∇ ψ ∈ T (soeither v ′ ( ϕ ) = ˆ0 or v ′ ( ψ ) = ˆ0) hence v ′ ( ϕ ∧ ψ ) = ˆ0 = v ′ ( ϕ )˜ ∧ v ′ ( ψ ). OGICS OF INVOLUTIVE STONE ALGEBRAS 13 v ′ ( ϕ ∨ ψ ) = v ′ ( ϕ ) ∨ v ′ ( ψ )If (i) ∇ ( ϕ ∨ ψ ) , ∇ ∼ ( ϕ ∨ ψ ) ∈ T then by r − r we obtain that ∇ ϕ, ∇ ψ, ∇ ∼ ϕ, ∇ ∼ ψ ∈ T . Hence v ′ ( ϕ ∨ ψ ) = v ( ϕ ∨ ψ ) = v ( ϕ ) ∨ M v ( ψ ) = v ′ ( ϕ ) ∨ v ′ ( ϕ ).If (ii) ∼ ∇ ∼ ( ϕ ∨ ψ ) ∈ T then by r we have that either ∼ ∇ ∼ ϕ ∈ T or ∼ ∇ ∼ ψ ∈ T (so either v ′ ( ϕ ) = ˆ1 or v ′ ( ψ ) = ˆ1) hence v ′ ( ϕ ∧ ψ ) = ˆ1 = v ′ ( ϕ )˜ ∧ v ′ ( ψ ).If (iii) ∼ ∇ ( ϕ ∨ ψ ) ∈ T then by r and r we have that ∼ ∇ ϕ, ∼ ∇ ψ ∈ T (so v ′ ( ϕ ) = v ′ ( ψ ) = ˆ0) hence v ′ ( ϕ ∧ ψ ) = ˆ1 = v ′ ( ϕ )˜ ∧ v ′ ( ψ ).5. v ′ ( ⊥ ) = ˆ0 and v ′ ( ⊤ ) = ˆ1Directly from rules r and r . (cid:3) Regarding the preceding Proposition, note that, in order to show that v ′ ( ξ ) is welldefined for every ξ ∈ F m Σ ∇ (and that v ′ ∈ Val( M ∇ )), we only need to consider rulesin R ∇ instantiated with formulas in sub ( ξ ). This observation will be crucial in the proof ofTheorem 5.15.We now proceed to explain how a single-conclusion axiomatization (for a logic extending IS ≤ ) can be extracted from the multiple-conclusion rules of Theorem 5.3. We shall need afew technical lemmas. In the next one, Q i M i denotes the product of a family of matrices { M i : i ∈ I } . Lemma 5.4.
Given a class { M i : i ∈ I } of Σ -matrices, we have the following embeddings: Y i M i ֒ → ( Y i M i ) ∇ ֒ → Y i ( M i ∇ ) . Proof.
The fist embedding is simply the identity function. The second one is also theidentity for the elements in Q i ( M i ), whereas ˆ1 is sent to Q i { ˆ1 } and ˆ0 to Q i { ˆ0 } . (cid:3) The following result is an immediate consequence of Lemma 5.4.
Lemma 5.5.
For every class M of Σ -matrices, we have:(i) ⊲ P ( c M ) ⊆ ⊲ \P ( M ) ⊆ ⊲ P ( M ) .(ii) ⊲ P ( M ∇ ) ⊆ ⊲ ( P ( M )) ∇ ⊆ ⊲ P ( M ) . Let R be a a set of single-conclusion rules. Recall that R ∇ is a set of multiple-conclusionrules, and we abbreviate ⊢ R ∪ R ∇ = ⊢ ⊲ R ∪ R ∇ . Corollary 5.6.
Let M be a class of Σ -matrices, and let R be a set of single-conclusionrules. If ⊢ R = Log M = Log c M , then ⊢ R ∪ R ∇ = Log M ∇ .Proof. From Log M = Log c M = ⊢ R we have ⊲ R ≃ ⊲ P ( c M ) ≃ ⊲ P ( M ) by Remark 5.1 (ii). FromLemma 5.5 (i) and Remark 5.1 we obtain that ⊲ R ≃ ⊲ \P ( M ) . By Theorem 5.3 we concludethat ⊲ ( P ( M )) ∇ = ⊲ R ∪ R ∇ . Finally, from Lemma 5.5 (ii) and Remark 5.1 we have Log M ∇ =Log P ( M ) ∇ . Hence ⊢ M ∇ is axiomatized by R ∪ R ∇ . (cid:3) Joining Theorem 5.2 and Corollary 5.6, we obtain the following recipe for capturing theeffect of adding ∇ to single-conclusion axiomatizations. Corollary 5.7.
Let M be a class of Σ -matrices and let R be a set of single-conclusionrules. If ⊢ R = Log M = Log c M , then ⊢ ( R ∪ R ∇ ) ∨ = Log M ∇ . Example 5.8.
Let M = h DM , ↑ a i be the four-element matrix that defines the Belnap-Dunn logic B . By Corollary 4.6, we know that B = Log M = Log b M . Hence, from Corollar-ies 4.7 and 5.7 we can obtain a Hilbert axiomatization for IS ≤ = Log M ∇ = Log h IS , ↑ a i .Let R B be the Hilbert-style calculus used in [14] to axiomatize Log B (expanded with therules introduced in [1, p. 1065] to account for the constants). p ∧ qp p ∧ qq p qp ∧ qpp ∨ q p ∨ qq ∨ p p ∨ ppp ∨ ( q ∨ r )( p ∨ q ) ∨ r p ∨ ( q ∧ r )( p ∨ q ) ∧ ( p ∨ r ) ( p ∨ q ) ∧ ( p ∨ r ) p ∨ ( q ∧ r ) p ∨ r ¬¬ p ∨ r ¬¬ p ∨ rp ∨ r ¬ ( p ∨ q ) ∨ r ( ¬ p ∧ ¬ q ) ∨ r ( ¬ p ∧ ¬ q ) ∨ r ¬ ( p ∨ q ) ∨ r ¬ ( p ∧ q ) ∨ r ( ¬ p ∨ ¬ q ) ∨ r ( ¬ p ∨ ¬ q ) ∨ r ¬ ( p ∧ q ) ∨ r ⊤ ∼ ⊥ ⊥ ∨ pp ∼ ⊤ ∨ pp Then IS ≤ is axiomatized by ( R B ∪ R ∇ ) ∨ , which is the result of adding to R B the followingrules: ∇ p ∨ ∼ ∇ p r ∨ ∇ p ∨ r ∼ ∇ ∼ ∇ p ∨ r r ∨ ∇∇ p ∨ r ∇ p ∨ r r ∨ ∇ p ∨ r , ∼ ∇ p ∨ rr r ∨ ∼ ∇ p ∨ r ∇ ∼ p ∨ r r ∨ ∇ ∼ ∼ p ∨ r ∇ p ∨ r r ∨ ∇ p ∨ r ∇ ∼ ∼ p ∨ r r ∨ ∇ ( p ∧ q ) ∨ r , ∇ ∼ ( p ∧ q ) ∨ r ∇ p ∨ r r ∨ ∇ ( p ∧ q ) ∨ r , ∇ ∼ ( p ∧ q ) ∨ r ∇ q ∨ r r ∨ ∇ ( p ∧ q ) ∨ r , ∇ ∼ ( p ∧ q ) ∨ r ∇ ∼ p ∨ r r ∨ ∇ ( p ∧ q ) ∨ r , ∇ ∼ ( p ∧ q ) ∨ r ∇ ∼ q ∨ r r ∨ ∇ ∼ p ∨ r ∇ ∼ ( p ∧ q ) ∨ r r ∨ ∇ ∼ q ∨ r ∇ ∼ ( p ∧ q ) ∨ r r ∨ ∇ p ∨ r , ∇ q ∨ r ∇ ( p ∧ q ) ∨ r r ∨ ∇ ( p ∨ q ) ∨ r , ∇ ∼ ( p ∨ q ) ∨ r ∇ p ∨ r r ∨ ∇ ( p ∨ q ) ∨ r , ∇ ∼ ( p ∨ q ) ∨ r ∇ q ∨ r r ∨ ∇ ( p ∨ q ) ∨ r , ∇ ∼ ( p ∨ q ) ∨ r ∇ ∼ p ∨ r r ∨ ∇ ( p ∨ q ) ∨ r , ∇ ∼ ( p ∨ q ) ∨ r ∇ ∼ q ∨ r r ∨ ∇ p ∨ r ∇ ( p ∨ q ) ∨ r r ∨ ∇ q ∨ r ∇ ( p ∨ q ) ∨ r r ∨ ∇ ∼ p ∨ r , ∇ ∼ q ∨ r ∇ ∼ ( p ∨ q ) ∨ r r ∨ In the next Subsection we are going to apply Corollary 5.7 to axiomatize (relatively to B ∇ ), some extensions of IS ≤ that are characterized by M ∇ for some matrix M that is amodel of B .5.2. Adding ∇ to super-Belnap logics. As observed earlier, IS ≤ = ⊢ M , where M isthe following class of matrices: M := {h A , F i : A ∈ IS , F ⊆ A is a (non-empty) lattice filter } . Thus, each subclass M ′ ⊆ M (it suffices to consider those M ′ consisting of reduced matri-ces) defines a logic ⊢ M ′ which is an extension of IS ≤ . We have seen with Corollary 4.14 thatthere are at least continuum many of these, and Corollary 4.13 suggests that the structureof the lattice of extensions of IS ≤ is quite complex (see [1, 20] for analogous considerationson the lattice of super-Belnap logics). A systematic study of this lattice lies outside thescope of the present paper and even beyond our present grasp on IS-logics; however, inthis Subsection we consider a few extensions of IS ≤ that are defined by substructures of h IS , ↑ a i , illustrating how our methods can be used to axiomatize them.The following result, which is an immediate consequence of Corollary 5.7, shows thatExample 5.8 smoothly generalizes to all super-Belnap logics. Proposition 5.9.
Let M be a class of models of B . If Log M is axiomatized relative to B by a set of single conclusion rules R , then Log M ∇ is also axiomatized by R relative to B ∇ . Let M and M be classes of models of B such that Log M = Log M . Then Log M and Log M are axiomatized by the same set R of single-conclusion rules. Hence, Log M ∇ =Log M ∇ is axiomatized by the set R ∨∇ defined above. This entails, in particular, that,if a super-Belnap logic ⊢ is finitary (resp. finitely axiomatized), then ⊢ ∇ (defined as inCorollary 4.13) is also finitary (resp. finitely axiomatized). Since the lattice of super-Belnaplogics contains continuum many finitary logics [20, Cor. 8.17], the above considerations allowus to obtain the following sharpening of Corollary 4.14. Proposition 5.10.
There are (at least) continuum many finitary extensions of IS ≤ . The super-Belnap logics considered below are the so-called
Exactly True Logic
ET L of [19] (which is the 1-assertional logic of the variety of De Morgan algebras), G. Priest’s
Logic of Paradox LP , the two logics K ≤ and K named after S. C. Kleene, and classicallogic CL . K ≤ is the order-preserving logic of the variety of Kleene algebras, and K is the 1-assertional logic associated to the same variety (see [1] for further details). Proposition 5.11below shows that each of these logics can be axiomatized, relative to B , by a combinationof the following rules: p ∧ ( ∼ p ∨ q ) ⊢ q (DS) ( p ∧ ∼ p ) ∨ q ⊢ q ( K ) ( p ∧ ∼ p ) ∨ r ⊢ q ∨ ∼ q ∨ r ( K ≤ ) ∅ ⊢ p ∨ ∼ p (EM)(Regarding the names of the above rules, the K ’s are suggestive of Kleene’s logics, (DS)stands for Disjunctive Syllogism and (EM) for
Excluded Middle .) Proposition 5.11 ([1], Thm. 3.4) . (i) ET L = Log h DM , { }i = B +(DS) .(ii) LP = Log h K , ↑ a i = B +(EM) .(iii) K = Log h K , { }i = B +( K ) . (iv) K ≤ = Log {h K , ↑ a i , h K , { }i} = B +( K ≤ ) .(v) CL = Log h B , { }i = B +(DS)+(EM) . Theorem 5.12.
For logics above IS ≤ we have the following relative axiomatizations:(i) Log h IS , ↑ i = ET L ∇ = IS ≤ +(DS) .(ii) Log h IS , ↑ a i = LP ∇ = IS ≤ +(EM) .(iii) Log h IS , ↑ i = K ∇ = IS ≤ +( K ) .(iv) Log {h IS , ↑ a i , h IS , ↑ i} = K ∇≤ = IS ≤ +( K ≤ ) .(v) Log h IS , ↑ i = CL ∇ = IS ≤ +(DS)+(EM) .Proof. The statement follows directly from Proposition 5.9 and Proposition 5.11, havingnoticed that, for x ∈ { , a } , we have h DM , ↑ x i ∇ = h IS , ↑ x i , h K , ↑ x i ∇ = h IS , ↑ x i and h B , ↑ i ∇ = h IS , ↑ i . (cid:3) Other extensions of IS ≤ . In this Subsection we consider a few examples of exten-sions of IS ≤ (defined by substructures of the matrix h IS , ↑ a i )Given a Σ -matrix M = h A , D i , a set of axioms Ax ⊆ F m Σ and a set of rules R ⊆ ℘ ( F m Σ ) × F m Σ , we write Val Ax M for the set of valuations on M such that v ( ϕ σ ) ⊆ D forevery ϕ ∈ Ax substitution σ , and Val R M for the set of valuations on M such that v ( Γ σ ) ⊆ D implies ϕ σ ∈ D for every Γϕ ∈ R and substitution σ .The following result (whose simple proof we omit) is a corollary of [4, Lemma 2.7] andwill be very useful to show relative axiomatization results (this technique is used in [5] toobtain general modular semantics for axiomatic extensions of a given logic). In item (ii), M ω is a shorthand for Q i<ω M . Proposition 5.13.
Let M = h A , D i be a Σ -matrix. Given Ax ⊆ F m Σ and R ⊆ ℘ ( F m Σ ) × F m Σ . We have that:(i) Val Ax M is a complete semantics for Log( M ) + Ax .(ii) Val R M ω is a complete semantics for Log( M ) + R . We are now ready to give an axiomatization relative to IS ≤ of ⊢ IS , the 1-assertionallogic of the class IS (i.e. three-valued Lukasiewicz(-Moisil) logic; cf. Proposition 4.5). Thisis the first item of Theorem 5.14 below. The logic axiomatized by the second item is theorder-preserving logic of the variety V ( IS ) of three-valued Lukasiewicz(-Moisil) algebras,which is also Log {h IS , { ˆ1 }i , h IS , ↑ i} . Theorem 5.14. (i)
Log h IS , { ˆ1 }i = Log h IS , { ˆ1 }i = Log h IS , { ˆ1 }i = Log h IS , { ˆ1 }i = ⊢ IS = IS ≤ + p ⊢∼ ∇ ∼ p .(ii) Log {h IS , { ˆ1 }i , h IS , ↑ i} = ⊢ ≤ V ( IS ) = IS ≤ +( K ≤ ) + ∼ p ∨ r, ∇ p ∨ r ⊢ p ∨ r + ∼ ∇ p ∨ r ⊢∼ p ∨ r .(iii) Log( h IS , ↑ i ) = Log( h IS , ↑ i ) = Log( h IS , ↑ i ) = Log( h IS , ↑ i ) = IS ≤ + p ∨∼ ∇ p .Proof. (i). For the equalities Log( h IS , { ˆ1 }i ) = Log( h IS , { ˆ1 }i ) = Log( h IS , { ˆ1 }i ) =Log( h IS , { ˆ1 }i ), it suffices to observe that h IS , { ˆ1 }i ∗ = h IS , { ˆ1 }i ∗ = h IS , { ˆ1 }i ∗ = h IS , { ˆ1 }i .Consider M = h IS , ↑ b i ω and R = { p ⊢ ∼ ∇ ∼ p } . Note for every v ∈ Val M ω we have that v ( ∼ ∇ ∼ p ) ∈ D iff v ( p ) = { ˆ1 } ω . Hence, from v ∈ Val R M ω we have that v ( A ) ∈ D ω iff OGICS OF INVOLUTIVE STONE ALGEBRAS 17 v ( A ) = { } ω . As R is sound in h IS , i ω we obtain the equality Val R M ω = Val h IS , i ω , thus IS ≤ + p ⊢ ∼ ∇ ∼ p = Log( h IS , i ω ) = Log( h IS , i ).(ii). That Log( {h IS , { ˆ1 }i , h IS , ↑ i} ) is the order-preserving logic of the variety V ( IS )follows from the observation that ⊢ ≤ V ( K ) = ⊢ ≤ K holds for any class K . Applying Theorem 5.2to the multiple-conclusion axiomatization for Log( {h IS , { ˆ1 }i , h IS , ↑ i} ) presented in Ex-ample 5.17 in the following Section, we conclude that collecting all the s ∨ i for 1 ≤ i ≤ {h IS , { ˆ1 }i , h IS , ↑ i} ). The latter equal-ity in the statement follows as the new rules, corresponding to s ∨ , s ∨ and s ∨ , are exactlyones that fail in h IS , ↑ a i .(iii). For the equalities Log( h IS , ↑ i ) = Log( h IS , ↑ i ) = Log( h IS , ↑ i ) = Log( h IS , ↑ i ), it suffices to observe that h IS , ↑ i ∗ = h IS , ↑ i ∗ = h IS , ↑ i ∗ = h IS , ↑ i . Let R = { p ∨ ∼ ∇ p } . In order to obtain the last equality, in the light of Proposition 5.13 (i),it is enough to show that Val h IS , ↑ i = Val R h IS , ↑ a i . This follows from the fact that given v ∈ Val R h IS , ↑ a i we must have that for every formula ϕ both v ( ϕ ) = b and v ( ϕ ) = 0 (thereforealso v ( ϕ ) = 1), and the fact that R is sound w.r.t. h IS , ↑ i . (cid:3) Analtytic calculi.
Let Λ ⊆ F m and let R be a set of multiple-conclusion rules. Wewrite Γ ⊲ Λ R ∆ when there exists an R -proof of ∆ from Γ where only formulas in Λ occur. LetΦ ⊆ F m . We say that R is Φ -analytic if when Γ ⊲ R ∆ then Γ ⊲ Υ Φ R ∆ with Υ = sub ( Γ ∪ ∆ )and Υ Φ = Υ ∪ { A σ : A ∈ Φ , σ : P → Υ } . Intuitively, this means that an R -proof of ∆ from Γ needs only to use formulas which are subformulas of Γ ∪ ∆ , or instances of Φ with suchsubformulas. Hence, formulas in Υ Φ can be seen as ‘generalized subformulas’.Given Φ ⊆ F m , let Φ ∇ := Φ ∪ {∇ p, ∼ ∇ p, ∇ ∼ p, ∼ ∇ ∼ p } . The Theorem below isa refinement of Theorem 5.3 that applies when we depart from calculus that is analytic,entailing that the operation described in Theorem 5.3 preserves analyticity. Theorem 5.15.
Let M be a class of Σ -matrices. If R is an Φ -analytic axiomatization of ⊲ c M then R ∪ R ∇ is an Φ ∇ -analytic axiomatization of ⊲ M ∇ .Proof. The proof can be easily obtained by adapting the proof of Theorem 5.3. Let Υ = sub ( Γ ∪ ∆ ) and Λ = Υ Φ ∇ . Assume Γ ⊲ Λ R ∪ R ∇ ∆ . Then, by cut for sets, there is a partition h T, F i of Λ such that Γ ⊂ T , ∆ ⊂ F and T ⊲ Λ R ∪ R ∇ F . Since R ⊆ R ∪ R ∇ , we know that T ⊲ Λ R ∪ R ∇ F . Therefore, since Υ Φ ⊆ Υ Φ ∇ , by Φ-analyticity of R we have that T ⊲ c M F andwe can pick v ∈ Hom Σ ( L Σ ∇ , b M ) for some M ∈ M such that v ( T ) ⊆ D and v ( F ) ∩ D = ∅ .Noting that for every A ∈ Υ we have ∇ A, ∇ ∼ A, ∼ ∇ A, ∼ ∇ ∼ A ∈ Υ S ∇ = Λ , we can define v ′ : Υ → M ∇ as in Theorem 5.3. That v ′ respects all the connectives (and is thereforea partial M ∇ valuation) follows from the fact that in the proof of Theorem 5.3 we onlyused instances of the rules using formulas in Υ yielding formulas in Υ S ∇ = Λ . As M is amatrix, v ′ can be extended to a total valuation and therefore Γ ⊲ M ∇ ∆ , thus concludingthe proof. (cid:3) The papers [17, 18] introduced a general method for obtaining analytic calculi for logicsgiven by (partial non-deterministic) matrices whenever a certain expressiveness requirementis met. In particular, for the logic determined by a single matrix M = h A , D i , it sufficesthat M be monadic [22, p. 265]. This means that, for all x, y ∈ A with x = y , there is a Note that in general ⊲ Λ R is not a multiple-conclusion consequence relation. It still satisfies dilution andcut for set properties, but only weaker versions of overlap and substitution invariance. one-variable separating formula, that is, a formula ϕ ( p ) such that ϕ ( x ) ∈ D and ϕ ( y ) / ∈ D (or vice versa).From now on, let us fix the separating set S := { p, ∼ p } . Applying the above-describedmethod, we obtain the following axiomatization for B . Example 5.16.
The matrix h DM , ↑ a i is monadic with set of separators S . We cantherefore apply the method introduced in [18] to we obtain the following S -analtytic axiom-atization for B = Log h DM , ↑ a i : p ∼ ∼ p ∼ ∼ ppp ∧ qp p ∧ qq p, qp ∧ q ∼ p ∼ ( p ∧ q ) ∼ q ∼ ( p ∧ q ) ∼ ( p ∧ q ) ∼ p, ∼ qpp ∨ q qp ∨ q p ∨ qp , q ∼ p , ∼ q ∼ ( p ∨ q ) ∼ ( p ∨ q ) ∼ p ∼ ( p ∨ q ) ∼ q ⊤ ∼ ⊤ ∼ ⊥ ⊥ Note that this axiomatization coincides with the one presented in [20, Section 9]. Theo-rem 5.15 then tells us that we can obtain an S ∇ -analtytic axiomatization of IS ≤ by adding R ∇ to the above rules. Example 5.17.
In [17, Example 5] we showed that the following rules provide an S -analtytic axiomatization of Kleene’s logic of order K ≤ = Log {h K , { }i , h K , ↑ a i} . p , qp ∧ q s p ∧ qp s p ∧ qq s ∼ p ∼ ( p ∧ q ) s ∼ q ∼ ( p ∧ q ) s ∼ ( p ∧ q ) ∼ p , ∼ q s pp ∨ q s qp ∨ q s ∼ ( p ∨ q ) ∼ p s ∼ ( p ∨ q ) ∼ q s ∼ p , ∼ q ∼ ( p ∨ q ) s p ∨ qp , q s p ∼ ∼ p s ∼ ∼ pp s p , ∼ pq , ∼ q s Since {h IS , ↑ i , h IS , ↑ a i} = {h K , { }i , h K , ↑ a i} ∇ , by Theorem 5.15, we have thatadding R ∇ to the above rules gives us a S ∇ -analtytic axiomatization of Log {h IS , ↑ i , h IS , ↑ a i} .The method of [18] can also be applied directly to obtain an S -analtytic axiomatization of ⊢ ≤ V ( IS ) = Log {h IS , { ˆ1 }i , h IS , ↑ i} . Indeed, the ∇ -free fragment of Log {h IS , { ˆ1 }i , h IS , ↑ a i} is Log {h K , { }i , h K , ↑ a i} . The latter set of matrices can be viewed as a partialmatrix [18, Section 2.2] and is monadic with separating set S (in which ∇ does not occur).Therefore, the modularity of the method of [18] tells us we just have to add the rulescorresponding to ∇ . Hence, it suffices to add the rules: ∼ p , ∇ pp s ∼ pp , ∼ ∇ p s p ∇ p s p , ∼ ∇ p s ∼ ∇ p ∼ p s ∼ p, ∇ p s which give us the single-conclusion axiomatization relative to IS ≤ mentioned in Theo-rem 5.14 (ii). OGICS OF INVOLUTIVE STONE ALGEBRAS 19 Conclusions and future work
As we have shown, the lattice of super-Belnap logics is embeddable in the lattice ofextensions of IS ≤ . This connection provides significant insight, but it also suggests thatfully describing the latter is at least as complex as describing the former, whose structureis still not completely understood (see [1]). Obviously, in the present study we have onlyscratched the surface of the general problem. A reasonable starting point for a systematicaccount of the extensions of IS ≤ is to adapt the various results and strategies in [21, 1, 20]to the richer setting of involutive Stone algebras. We mention, in particular, the issues ofcharacterizing the reduced models of extensions of IS ≤ , and that of providing a generalsemantical description of the explosive extensions (in the sense of [1, 20]) of logics over IS ≤ .An altogether different perspective on extensions of IS ≤ , which has not been consideredin the present paper, comes from the observation made in [7, Sec. 6] that IS ≤ may be viewedas a paraconsistent logic, more precisely as a Logic of Formal Inconsistency (LFIs) in thesense of N. da Costa [12]. Indeed, IS ≤ (and so its extensions) can be equivalently presentedin a language that replaces the ∇ connective with either the consistency operator ( ◦ ) orthe inconsistency operator ( • ) that are usually considered in the literature on LFIs. Onepossible definition is ∇ ϕ := ∼ ◦ ϕ ∨ ϕ , and, conversely, one may define ◦ ϕ := ∼ ∇ ( ϕ ∧ ∼ ϕ )and • ϕ := ∇ ( ϕ ∧ ∼ ϕ ).From a philosophical logic point of view, the advantage of the latter presentation is thatthe operators ◦ and • have a clearer logical interpretation than ∇ , namely, ◦ ϕ means ‘ ϕ isconsistent’ and • ϕ means ‘ ϕ is inconsistent’; on the other hand, ∇ behaves very well fromthe points of view of algebraic logic and duality theory, for it satisfies the usual axiomsfor modal operators. A more interesting observation is that, in the setting of LFIs, the(in)consistency operators are usually required to satisfy much weaker axioms than thosethat result from the definitions ◦ ϕ := ∼ ∇ ( ϕ ∧ ∼ ϕ ) and • ϕ := ∇ ( ϕ ∧ ∼ ϕ ) within IS ≤ .This suggests a potentially fruitful project for future research: namely, a systematic studyof more general algebraic structures (e.g. De Morgan algebras endowed with a consistencyoperator) corresponding to weaker logics (viewed as LFIs) that approximate IS ≤ frombelow. Acknowledgements
Research funded by FCT/MCTES through national funds and when applicable co-fundedby EU under the project UIDB/EEA/50008/2020 and by the Conselho Nacional de Desen-volvimento Cient´ıfico e Tecnol´ogico (CNPq, Brazil), under the grant 313643/2017-2 (Bolsasde Produtividade em Pesquisa - PQ).
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