Finite axiomatizability of logics of distributive lattices with negation
aa r X i v : . [ m a t h . L O ] F e b FINITE AXIOMATIZABILITY OF LOGICSOF DISTRIBUTIVE LATTICES WITH NEGATION
S´ERGIO MARCELINO AND UMBERTO RIVIECCIO
Abstract.
This paper focuses on order-preserving logics defined from varieties of dis-tributive lattices with negation, and in particular on the problem of whether these canbe axiomatized by means of finite Hilbert calculi. On the side of negative results, we pro-vide a syntactic condition on the equational presentation of a variety that entails failureof finite axiomatizability for the corresponding logic. An application of this result is thatthe logic of all distributive lattices with negation is not finitely axiomatizable; likewise,we establish that the order-preserving logic of the variety of all Ockham algebras is alsonot finitely axiomatizable. On the positive side, we show that an arbitrary subvarietyof semi-De Morgan algebras is axiomatized by a finite number of equations if and onlyif the corresponding order-preserving logic is axiomatized by a finite Hilbert calculus.This equivalence also holds for every subvariety of a Berman variety of Ockham algebras.We obtain, as a corollary, a new proof that the implication-free fragment of intuitionisticlogic is finitely axiomatizable, as well as a new Hilbert calculus for it. Our proofs areconstructive in that they allow us to effectively convert an equational presentation of avariety of algebras into a Hilbert calculus for the corresponding order-preserving logic,and viceversa. We also consider the assertional logics associated to the above-mentionedvarieties, showing in particular that the assertional logics of finitely axiomatizable sub-varieties of semi-De Morgan algebras are finitely axiomatizable as well. Introduction
In the present paper, we study logics associated to subvarieties of the class DN of distribu-tive lattices with negation (Definition 2.2) considered for instance in the papers [11, 12]. DN is a variety that includes many well-known classes of algebras of non-classical logics, suchas (semi-)De Morgan algebras, Stone algebras, pseudo-complemented distributive latticesand Ockham algebras, therefore providing a common semantical framework for the studyof the corresponding logics.We will be mostly concerned with the order-preserving logics associated to the above-mentioned varieties, focusing in particular on the issue of whether they can be axiomatizedor not by means of a Hilbert calculus consisting of finitely many rule schemata; if this isthe case, the logic will be called finitely based .On the side of negative results, we are going to show that the order-preserving logicassociated to the variety DN is not finitely based; the same holds for the order-preservinglogic of all Ockham algebras (Definition 2.3). Indeed, we will give a syntactic criterionregarding the equations that axiomatize (relatively to DN ) a variety V ⊆ DN implyingthat the same holds for the corresponding logic. On the positive side, we will show howto obtain a finite Hilbert calculus that is complete with respect to the logic of semi-DeMorgan algebras, entailing that the latter is finitely based. The same techniques will allow Research funded by FCT/MCTES through national funds and when applicable co-funded by EU un-der the project UIDB/EEA/50008/2020 and by the Conselho Nacional de Desenvolvimento Cient´ıfico eTecnol´ogico (CNPq, Brazil), under the grant 313643/2017-2 (Bolsas de Produtividade em Pesquisa - PQ). us to obtain finite calculi for the logics associated to so-called Berman varieties of Ockhamalgebras [6]. As a corollary of our results, we will also obtain a finite axiomatization forthe logic of pseudo-complemented distributive lattices (i.e. the implication-free fragment ofintuitionistic logic) alternative to the one introduced in [29].Our proof strategies are discussed in more detail in Sections 3 and 4, but we give herean introductory account on the finite axiomatizability problem for order-preserving logicsand the difficulties one faces. First of all, let us clarify the meaning of the terms “order-preserving logic” and “finite Hilbert calculus”.Let K be a class (say, a variety) of algebras such that each algebra A ∈ K has a boundedlattice reduct h A ; ∧ , ∨ , ⊥ , ⊤i One of the standard ways of associating a (finitary) Tarskianlogic ⊢ ≤ K to K is the following. One lets ∅ ⊢ ≤ K ϕ if and only if the equation ϕ ≈ ⊤ isvalid in K and, for all Γ ∪ { ϕ } ⊆ F m such that Γ = ∅ , one lets Γ ⊢ ≤ K ϕ iff there is anatural number n and formulas γ , . . . , γ n ∈ Γ such that the equation γ ∧ . . . ∧ γ n ∧ ϕ ≈ ϕ is valid in K . Thus ⊢ ≤ K is by definition a finitary logic, called the order-preserving logic of the class K . Note that ⊢ ≤ K coincides with the logic defined by the class of matrices {h A , F i : A ∈ K , F ⊆ A is a non-empty lattice filter of A } . Other logics may of course bedefined from K , for instance, the class of matrices {h A , {⊤}i : A ∈ K } also determines a(stronger) logic associated to K . Following [18], we call this the ⊤ -assertional logic of theclass of algebras K (denoted ⊢ ⊤ K ) and will be considered in Section 6.By a Hilbert calculus we mean a logical calculus whose every rule schema is a pair Γϕ where Γ is a finite (possibly empty) set of formulas and ϕ is a formula; we say that sucha calculus is finite when it consists of finitely many rule schemata. Following [27, p. 607],we call a logic that can be axiomatized by a finite Hilbert finitely based . In [15, Sec. 2.1]the authors introduce a finite calculus for the order-preserving logic of the variety SDM ofsemi-De Morgan algebras (Definition 2.3). This, however, is not a Hilbert calculus strictosensu , because it involves sequent-style rule schemata such as the following: from h ϕ, ψ i infer h∼ ψ, ∼ ψ i . The “axioms” of the calculus introduced in [15], on the other hand, areexamples of what are usually called (single-premiss) Hilbert rules. Finite Hilbert calculifor the order-preserving logics of De Morgan algebras ( DM ) and pseudo-complementeddistributive lattices ( PL ) can be found in the papers [13, 29]. We note in this respect that ⊢ ⊤ PL = ⊢ ≤ PL , while ⊢ ⊤ DM is strictly stronger than ⊢ ≤ DM , which is the well-known Belnap-Dunnlogic. ⊢ ⊤ DM is the Exactly True Logic introduced and axiomatized by means of a Hilbertcalculus in [23]; see also [30, 1].A closer look at the order-preserving logic ⊢ ≤ SDM associated to semi-De Morgan algebrasexplains the choice of a hybrid calculus in [15], as well as the challenge one faces when tryingto axiomatize ⊢ ≤ K (for K ⊆ DN ) by means of a calculus that is Hilbert in the strict sense. Infact, the consequence relation of each order-preserving logic ⊢ ≤ K corresponds to the latticeorder on K , in the sense that one has ϕ ⊢ ≤ K ψ if and only if the inequality ϕ ≤ ψ (takingthe latter as a shorthand for the equation ϕ ∧ ψ ≈ ϕ ) is valid in K . Such a partial orderrelation on each A ∈ K enjoys certain (meta-)properties that need to be mirrored by thelogical calculus. Indeed, every order-preserving logic ⊢ ≤ K is selfextensional (see Section 3);moreover, observe that, if K | = ϕ ≤ ψ , then K | = ∼ ψ ≤ ∼ ϕ , but also K | = ϕ ∨ γ ≤ ψ ∨ γ for every γ ∈ F m , and so on.In [15], the above meta-properties are imposed by adding suitable sequent-style ruleschemata such as the one mentioned above (from h ϕ, ψ i infer h∼ ψ, ∼ ϕ i ). As is well known,pure Hilbert calculi ( stricto sensu ) lack the expressive power needed to directly impose such INITE AXIOMATIZABILITY OF LOGICS OF DISTRIBUTIVE LATTICES WITH NEGATION 3 (meta-)properties, which is one of the reasons of interest in more expressive (e.g. sequent-style) calculi. However, Hilbert calculi also allow for more fine-grained analyses of logicsand, being very close to the algebraic semantics, they are generally better suited for thestudy of logics from an algebraic point of view (see e.g. [13, p. 414]).A first approach to the above-mentioned axiomatizability problem suggests the followingstrategy. Take a basic set of Hilbert rule schemata R (which are sound w.r.t. ⊢ ≤ K ) andrecursively close it under the application of rule schemata as follows: whenever h ϕ, ψ i ∈ R ,add to R also h∼ ψ, ∼ ϕ i , h ϕ ∨ γ, ψ ∨ γ i , etc. Such a process is indeed bound to succeed,and allows one to show that the derivability relation ⊢ R thus obtained coincides with ⊢ ≤ K .The non-trivial question is whether some finite subset R ⊆ R also suffices or not. Themain result of the present paper consists in providing a sufficient condition for the negativeresult to hold as well as a few conditions that are sufficient for ensuring a positive answer.As we shall see, the answer relies crucially on the soundness of certain rule schemata.We note for the algebraic logician that the logics considered in the present paper arenot algebraizable in the sense of Blok and Pigozzi, and indeed they are easily shown tobe non protoalgebraic either (see e.g. [14] for the relevant definitions). This is one of thechallenges of our study, for one cannot rely on the existence of the translations betweenequations and formulas that are provided by the general theory of algebraizable logics.Thus, in this setting, there is no standard recipe for obtaining a Hilbert axiomatization ofa given logic from an equational presentation of the corresponding class of algebras. Also,no isomorphism is readily available between (say) the lattice of subquasivarieties of DN andthe lattice of finitary extensions of ⊢ ≤ DN (but see Theorem 2.7 in Section 2).The paper is organized as follows. Section 2 collects the fundamental definitions onalgebras and logics, as well as a few useful lemmas. In Section 3 we give a recipe forobtaining a (potentially infinite) Hilbert axiomatization for the logic ⊢ ≤ K for each class K ⊆ DN . We investigate conditions entailing that the above-mentioned axiomatizationmust be infinite, and in particular we show that ⊢ ≤ DN is not finitely based; the same holdsfor the logic ⊢ ≤ O of the variety of all Ockham algebras (Definition 2.3). By contrast, weshow in Section 4 that, for an arbitrary variety K ⊆ SDM , where
SDM is the class ofsemi-De Morgan algebras (Definition 2.3), the logic ⊢ ≤ K is finitely based if and only if K isaxiomatized by a finite number of equations (in particular, ⊢ ≤ SDM is itself finitely based). InSection 5 we adapt our proof techniques to show that, unlike the whole variety O of Ockhamalgebras, every Berman subvariety O mn ⊆ O determines a logic ⊢ ≤ O mn that is finitely based.In Section 6 we briefly consider ⊤ -assertional logics associated to varieties of distributivelattices with negation, showing in particular that ⊢ ⊤ SDM is finitely based. Lastly, Section 7contains some concluding remarks and suggestions for further research.2.
Algebraic and logical preliminaries
Algebras.
We adopt the standard conventions and notation of modern universal al-gebra, for which we refer the reader to [7]. All algebras considered in the present paper arebounded (distributive) lattices (Definition 2.1) enriched with a unary negation operation ∼ on which different requirements will be imposed, giving rise to the various classes of interest.The algebraic (as well as the logical) language {∧ , ∨ , ∼ , ⊥ , ⊤} , consisting of a conjunction(interpreted as the lattice meet on algebras), a disjunction (the join), a negation and truthconstants (the top and bottom of the lattice) will stay fixed throughout the paper. Weshall denote by Fm the algebra of formulas over this language, freely generated by a denu-merable set of variables (denoted x, y, z etc.), and by F m the universe of this algebra. We
S´ERGIO MARCELINO AND UMBERTO RIVIECCIO shall mostly be interested in equational classes of algebras, i.e. varieties . An equation is apair of algebraic terms h ϕ, ψ i ∈ F m × F m , and every set E ⊆ P ( F m × F m ) of equationsdetermines a variety which will be denoted by V E . Definition 2.1 ([7]) . A bounded distributive lattice is an algebra A = h A ; ∧ , ∨ , ⊥ , ⊤i oftype h , , , i such that the following equations are satisfied:(L1) x ∨ y ≈ y ∨ x x ∧ y ≈ y ∧ x .(L2) x ∨ ( y ∨ z ) ≈ ( x ∨ y ) ∨ z x ∧ ( y ∧ z ) ≈ ( x ∧ y ) ∧ z .(L3) x ∨ x ≈ x x ∧ x ≈ x .(L4) x ∨ ( x ∧ y ) ≈ x x ∧ ( x ∨ y ) ≈ x .(L5) x ∧ ⊥ ≈ ⊥ x ∨ ⊤ ≈ ⊤ .(L6) x ∧ ( y ∨ z ) ≈ ( x ∧ y ) ∨ ( x ∧ z ). Definition 2.2 ([11, 12]) . A distributive lattice with negation is an algebra A = h A ; ∧ , ∨ , ∼ , ⊥ , ⊤i of type h , , , , i such that h A ; ∧ , ∨ , ⊥ , ⊤i is a bounded distributive lattice (Definition 2.1)and the following equations are satisfied:(N1) ∼ ⊥ ≈ ⊤ .(N2) ∼ ( x ∨ y ) ≈ ∼ x ∧ ∼ y .We shall denote by DN the variety of distributive lattices with negation, and by DN theset of equations axiomatizing this class according to Definition 2.2.The choice of the class of distributive lattices with negation as our base variety is dueto the following reasons. On the one hand, DN is sufficiently general to include many alge-bras of non-classical logics that interest us, in particular pseudo-complemented distributivelattices and semi-De Morgan algebras (our original case study). On the other hand, thetwo items of Definition 2.2 are some minimal equational requirements ensuring that theconnective ∼ indeed behaves like a negation (in particular, ∼ is order-reversing); also, thetheory of DN is sufficiently well developed to allow us to rely on a few algebraic lemmas.Besides DN , we shall be mainly working with the subvarieties introduced below. Definition 2.3 ([31]) . A distributive lattice with negation A = h A ; ∧ , ∨ , ∼ , ⊥ , ⊤i is: • a semi-De Morgan algebra , if A satisfies the following equations:(SDM1) ∼ ⊤ ≈ ⊥ .(SDM2) ∼ ∼ ( x ∧ y ) ≈ ∼ ∼ x ∧ ∼ ∼ y .(SDM3) ∼ x ≈ ∼ ∼ ∼ x . • a De Morgan algebra , if A is a semi-De Morgan algebra satisfying:(DM) ∼ ∼ x ≈ x . • a pseudo-complemented distributive lattice ( p -lattice , for short), if A is a semi-DeMorgan algebra satisfying:(PL) x ∧ ∼ ( x ∧ y ) ≈ x ∧ ∼ y . • an Ockham algebra , if A satisfies (SDM1) plus the following equation:(O) ∼ ( x ∧ y ) ≈ ∼ x ∨ ∼ y .We shall also be interested in the so-called Berman varieties of Ockham algebras [6],defined via the following terms. Let ∼ x := x and ∼ n +1 x := ∼ ∼ n x . For m ≥ n ≥
0, the variety O mn is defined as the subclass of those Ockham algebras that satisfythe equation ∼ m + n x ≈ ∼ n x . The class of Boolean algebras, viewed as a subvariety of DN , will be denoted by B ; also recall from the preceding Section that SDM , DM and PL denote, respectively, the variety of semi-De Morgan algebras, De Morgan algebras and INITE AXIOMATIZABILITY OF LOGICS OF DISTRIBUTIVE LATTICES WITH NEGATION 5 p -lattices. The following inclusions (all proper) hold among the above-defined varieties: B ⊆ DM ⊆ SDM ⊆ DN , B ⊆ DM ⊆ O mn ⊆ O ⊆ DN and B ⊆ PL ⊆ SDM ⊆ DN . DNO SDMO mn DM PLB
Figure 1.
Varieties of distributive lattices with negation, ordered by inclusion.Since its introduction about three decades ago [31], semi-De Morgan algebras have beenstudied especially in the setting of universal algebra [22] and duality theory [17, 11, 12].On the other hand, a logic associated to semi-De Morgan algebras (here denoted ⊢ ≤ SDM )has been first considered in the recent paper [15]. Having been introduced in the late1970’s, Ockham lattices are slightly older than semi-De Morgan algebras; logics associatedto (Berman subvarieties of) Ockham lattices are considered in [20, 21].De Morgan algebras (i.e. involutive semi-De Morgan algebras) are worth mentioning inthe present context especially because of their logical interpretation. In fact, since the1970’s with the seminal papers by N. Belnap [4, 5], the variety DM has been associated toand studied as the standard semantics of the Belnap-Dunn four-valued logic (see e.g. [13]).Indeed, the consequence relation ⊢ ≤ DM is precisely the Belnap-Dunn logic (on the other hand, ⊢ ⊤ DM is strictly stronger than ⊢ ≤ DM ). Sub(quasi)varieties of DM have also been studied froma logical point of view in the more recent papers [30, 24, 1]. From a technical point of view,we shall also be interested in exploiting the structural relation between semi-De Morganand De Morgan algebras stated in Lemma 2.5.The study of p -lattices can be traced back to the 1920’s with V. Glivenko’s classicalwork on intuitionistic logic. From a logical point of view, the importance of p -lattices stemsfrom their relation with intuitionism. In fact, it is well known that p -lattices are preciselythe implication-free subreducts of Heyting algebras: in logical terms, this entails that thelogic ⊢ ≤ PL , or equivalently ⊢ ⊤ PL (both defined as in Section 1), captures the implication-freefragment of intuitionistic logic.We end the Section with a few algebraic lemmas that will be used to make sure thatcertain rules are sound with respect to particular subclasses of DN . Lemma 2.4.
Let A be a semi-De Morgan algebra and a, b, c ∈ A . Then,(i) ∼ ( a ∧ b ) = ∼ ( ∼ ∼ a ∧ b ) = ∼ ( a ∧ ∼ ∼ b ) = ∼ ( ∼ ∼ a ∧ ∼ ∼ b ) .(ii) ∼ ( ∼ ( ∼ a ∧ b ) ∧ c ) ≤ ∼ ( a ∧ c ) .Proof. (i). See [12, Lemma 1.1]. S´ERGIO MARCELINO AND UMBERTO RIVIECCIO (ii). Let a, b, c ∈ A . Observe that, by the preceding item, ∼ ( a ∧ b ) = ∼ ( ∼ ∼ a ∧ b ).Since ∼ is order-reversing, from ∼ a ∧ b ≤ ∼ a we have ∼ ∼ a ∧ c ≤ ∼ ( ∼ a ∧ b ) ∧ c and ∼ ( ∼ ( ∼ a ∧ b ) ∧ c ) ≤ ∼ ( ∼ ∼ a ∧ c ) = ∼ ( a ∧ c ). (cid:3) Let A = h A ; ∧ , ∨ , ∼ , , i be a semi-De Morgan algebra. Defining A ∗ := {∼ a : a ∈ A } and a ∨ ∗ b := ∼ ∼ ( a ∨ b ) for all a, b ∈ A ∗ , we consider the algebra A ∗ = h A ∗ ; ∧ , ∨ ∗ , ∼ , , i .It is easy to show that A ∗ is indeed closed under the operations {∧ , ∨ ∗ , ∼ , , } . Moreover,we have the following result, which may be viewed as a generalization of Glivenko’s theoremrelating Heyting and Boolean algebras. Lemma 2.5 ([31], Thm. 2.4) . If A is a semi-De Morgan algebra, then A ∗ is a De Morganalgebra. The preceding Lemma is interesting for us because of the following logical consequence.Let ϕ be a formula in the language of semi-De Morgan logic. Define the formula ϕ ∗ recur-sively as follows: ϕ ∗ := ∼ ∼ ϕ if ϕ ∈ V ar ∪ {⊤}∼ ϕ ∗ if ϕ = ∼ ϕ ϕ ∗ ∧ ϕ ∗ if ϕ = ϕ ∧ ϕ ∼ ∼ ( ϕ ∗ ∨ ϕ ∗ ) if ϕ = ϕ ∨ ϕ . Lemma 2.6.
Let h ϕ, ψ i be a rule that is sound w.r.t. ⊢ ≤ DM (i.e. the Belnap-Dunn logic).Then h ϕ ∗ , ψ ∗ i is sound w.r.t. ⊢ ≤ SDM .Proof.
By contraposition, assume h ϕ ∗ , ψ ∗ i is not sound in ⊢ ≤ SDM . Then there is a semi-DeMorgan algebra A that witnesses the failure of the inequality ϕ ∗ ≤ ψ ∗ . It is then easyto check that A ∗ (which is a De Morgan algebra, by Lemma 2.5) witnesses the failure of ϕ ≤ ψ , contradicting the assumption that h ϕ, ψ i is sound w.r.t. the Belnap-Dunn logic. (cid:3) Logics. the propositional language Here, a logic is a structural (Tarskian) consequencerelation on Fm , that is, a subset of P ( F m ) × F m . Logics will be denoted by ⊢ with suitablesubscripts, regardless of the way (syntactical or semantical) they are defined. A logic can, forinstance, be defined through a logical matrix , i.e. a pair M = h A , D i where A is an algebraand D ⊆ A a set of designated elements. One sets Γ ⊢ M ϕ iff for every homomorphism h : F m → A , we have h ( ϕ ) ∈ D whenever h ( Γ ) ⊆ D . Similarly, a class of logical matricesdefines a logic by considering the intersection of the logics defined by each member of theclass. Another way is by considering a class of partially ordered algebras K , giving rise tothe order-preserving logic ⊢ ≤ K defined in the Introduction. Indeed, for a class K of lattice-ordered algebras, ⊢ ≤ K is the logic defined by the class of all matrices h A , D i such that A ∈ K and D is a lattice filter of A .We shall also be interested in logics defined through Hilbert calculi consisting of a finiteor denumerable set of rule schemata. By a Hilbert rule we mean a pair h Γ, ϕ i , usuallydenoted Γψ , where Γ ∪ ψ ⊆ F m . When Γ is a singleton (say, Γ = { ϕ } for some ϕ ∈ F m ),we speak of a formula-to-formula rule, usually written ϕψ . We shall write ϕψ to denote the“bidirectional rule”, which is really just an abbreviation for the pair of formula-to-formularules { ϕψ , ψϕ } . Every set R of Hilbert rules determines a logic ⊢ R in the standard way, andwe write Γ ⊢ R ϕ whenever there is a Hilbert derivation of ϕ from Γ that uses the rules in R .Below we state formally a result that will be central to our study of the relation betweenorder-preserving logics and varieties of distributive lattices with negation. INITE AXIOMATIZABILITY OF LOGICS OF DISTRIBUTIVE LATTICES WITH NEGATION 7
Recall that a logic is said to be non-pseudo-axiomatic if the set of its theorems is the setof formulas that are derivable from every formula [18, p. 78]. Every order-preserving logic ⊢ ≤ K considered in the present paper is non-pseudo-axiomatic. Moreover, since all algebrasin K ⊆ DN have a lattice reduct, ⊢ ≤ K is semilattice-based relative to ∧ and K [18, p.76].Therefore, we can apply [18, Thm. 3.7] to obtain the following. Theorem 2.7.
There is a dual isomorphism between the set of all subvarieties of DN ,ordered by inclusion, and the set of logics ⊢ ≤ K , ordered by extension. The isomorphism isgiven by K
7→ ⊢ ≤ K . In the present paper, we will study the problem of obtaining, from a basis E for theequational theory of K ⊆ DN , a set of rules that form a basis for the logic ⊢ ≤ K ; in particular,we shall be interested in conditions ensuring that the set of rules may be taken to be finite.3. The order-preserving logic of DN In this Section we introduce an infinite Hilbert calculus for the order-preserving logicof the variety DN . Our calculus is obtained by translating the set DN of equations thataxiomatize DN into a set R DN of bidirectional rules, which we then suitably enlarge in orderto ensure that the corresponding inter-derivability relation is a congruence of Fm . Aftershowing that the denumerable set R ω of rules thus obtained axiomatizes ⊢ ≤ DN (Corollary 3.4),we will proceed to show that R ω cannot be replaced by any finite set. This is the main resultof this Section: the order-preserving logic of DN is not finitely based (Theorem 3.8). We notethat most of the results that we proceed to prove below also hold for more general classesthan DN , and thus for logics weaker than ⊢ ≤ DN (for instance, Lemma 3.1 only relies on havingthe set of commutativity rules R C defined below, etc.). In view of future research, thissuggests the project of applying our techniques to more general logics/classes of algebras.Given a set of equations E := { ϕ i ≈ ψ i : i ∈ I } ⊆ F m × F m , we define the following setof bidirectional rules: R E := n ϕ i ψ i : i ∈ I o . Note that every rule in R E is formula-to-formula.Following standard notation, we use x, y, z etc. to denote variables used in equations and p, q, r etc. to denote logical variables. For instance, the equations (L1) in Definition 2.1 giveus p ∨ qq ∨ p and p ∧ qq ∧ p , and so on.Given a set R ⊆
F m × F m of formula-to-formula rules, let { q i : i < ω } be a set of freshvariables such that { q i : i < ω } ∩ var ( R ) = ∅ . Define: R := RR n +1 := n ϕ ∨ q n ψ ∨ q n : ϕψ ∈ R n o ∪ n ϕ ∧ q n ψ ∧ q n : ϕψ ∈ R n o ∪ n ∼ ψ ∼ ϕ : ϕψ ∈ R n o R ω := [ n<ω R n Let us also fix the set R C = { p ∧ qq ∧ p , p ∨ qq ∨ p } and R F = { ⊤ , p , qp ∧ q , pp ∨ q } . As the notation suggests,the set R C ensures that the conjunction and disjunction are commutative, while the rules in R F say that the designated elements are (non-empty) lattice filters of the algebraic modelsof the logic . Observe that R C ⊆ R DN , so we will not need to worry about adding R C when dealing with ⊢ ≤ DN andstronger logics. S´ERGIO MARCELINO AND UMBERTO RIVIECCIO
Recall that a logic ⊢ is said to be selfextensional if the inter-derivability relation ⊣⊢ isa congruence of the formula algebra Fm . Obviously, every order-preserving logic ⊢ ≤ K isselfextensional: thus one needs to ensure that the syntactic counterpart of ⊢ ≤ K also enjoysthis property. Lemma 3.1.
Let
R ⊆
F m × F m be a set of formula-to-formula rules such that R C ⊆ R .Then the inter-derivability relation ⊣⊢ R ω is a congruence of Fm .Proof. By construction, we have that any R ω -derivation of ϕ ⊢ R ω ψ can easily be trans-formed in derivations showing that ϕ ∨ γ ⊢ R ω ψ ∨ γ , ϕ ∧ γ ⊢ R ω ψ ∧ γ and ∼ ψ ⊢ R ω ∼ ϕ (cf. the proof of Lemma 4.2). Hence, if ϕ i ⊣⊢ R ω ψ i we have ϕ ∧ ϕ ⊣⊢ R ω ψ ∧ ψ , ϕ ∧ ϕ ⊣⊢ R ω ψ ∧ ψ and ∼ ψ i ⊣⊢ R ω ∼ ϕ i . (cid:3) The following lemma is an immediate consequence of the definition of R E . Lemma 3.2.
Let ⊢ be a consequence relation over Fm . If R E ⊆ ⊢ and ⊣⊢ is a congruenceof Fm , then the quotient Fm / ⊣⊢ satisfies all the equalities in E . In particular, if DN ⊆ E ,then Fm / ⊣⊢ is a distributive lattice with negation (Definition 2.2) with the order given by ⊢ . Given a set of equations E ⊆ F m × F m , we denote by V E the variety axiomatized by E . Theorem 3.3.
Let E ⊆ F m × F m be a set of equations such that DN ⊆ E . Then R E ω ∪ R F axiomatizes ⊢ ≤ V E .Proof. Let ⊢ := ⊢ R E ω ∪R F . It is clear that ⊢ ⊆ ⊢ ≤ V E . To prove completeness, assume Γ ϕ for some Γ ∪ { ϕ } ⊆ F m . By Lemma 3.1 and the fact that R C ⊆ R DN ⊆ R E , the relation ⊣⊢ R E ω is a congruence of Fm , which in this proof we denote by ≡ . Consider the matrix h Fm / ≡ , F i where F = Γ ⊢ / ≡ (observe that ⊢ R E ω ⊆ ⊢ implies that F is compatible with ≡ ).It follows from R E ⊆ ⊢ and Lemma 3.2 that Fm / ≡ is in V E . In particular, Fm / ≡ is alattice. Thus, to show that F is a non-empty lattice filter, it suffices to use the rules in R F . To conclude the proof, observe that the canonical projection map π : F m → F m/ ≡ isa valuation that satisfies all formulas in Γ but not ϕ . (cid:3) Corollary 3.4. ⊢ R DN ω ∪R F = ⊢ ≤ DN . Recall that an atomic formula is a propositional variable or a constant belonging to ourlanguage ( ⊥ or ⊤ ). Definition 3.5.
The ∼ -depth of an occurrence of an atomic formula ϕ in ψ is the numberof ∼ -headed subformulas of ψ with that occurrence of ϕ . In other words, we consider thetree representation of ψ and a leaf labelled ϕ (representing the ocurence of interest) andcount the number of ∼ -labelled nodes that are ancestors of that leaf. The ∼ -depth of aformula ψ is the maximum ∼ -depth of the atomic subformulas of ψ . The ∼ -depth of aset of rules R is the maximum ∼ -depth among the formulas in R . We say that a rule r is ∼ -balanced if all occurrences of all variables in r have the same the same ∼ -depth. We saythat a set of rules R is ∼ -balanced if every rule r ∈ R is ∼ -balanced.We shall now focus on invariants of logics axiomatized by ∼ -balanced rules having ∼ -depth k < ω . This will allow us to single out certain non-finitely based logics extending ⊢ ≤ DN . To this end, we shall also need the following function. INITE AXIOMATIZABILITY OF LOGICS OF DISTRIBUTIVE LATTICES WITH NEGATION 9
Let { q ϕ : ϕ ∈ F m } be a fresh set of variables. For all k < ω and for all ϕ, ψ ∈ F m , let f k : F m → F m be given by: f k ( ⊤ ) := ⊤ f k ( ⊥ ) := ⊥ f k ( p ) := pf k ( ϕ ∧ ψ ) := f k ( ϕ ) ∧ f k ( ψ ) f k ( ϕ ∨ ψ ) := f k ( ϕ ) ∨ f k ( ψ ) f k ( ∼ ϕ ) := ( q ∼ ϕ if k = 0 ∼ f k − ( ϕ ) if k > f k replaces in ϕ every subformula ψ occurring at ∼ -depth k whose main con-nective is ∼ by a fresh variable indexed by ψ . We extend f k to sets of formulas, rulesand sets of rules in the expected way: f k ( Γ ) = { f k ( ϕ ) : ϕ ∈ Γ } , f k ( Γϕ ) = f k ( Γ ) f k ( ϕ ) and f k ( R ) = { f ( R ) : r ∈ R} . Lemma 3.6.
Let R be a set of rules that is ∼ -balanced and has ∼ -depth k . Then Γ ⊢ R ϕ implies f n ( Γ ) ⊢ R f n ( ϕ ) for every n > k .Proof. Since n > k , for each rule ∆ψ ∈ R , we have f n ( ∆ ) = ∆ and f n ( ψ ) = ψ . Further, forevery substitution σ (i.e. for every endomorphism σ : F m → F m ) there is a substitution σ ′ such that f n ( ∆ σ ) = f n ( ∆ ) σ ′ = ∆ σ ′ and f n ( ψ σ ) = f n ( ψ ) σ ′ = ψ σ ′ , where σ ′ ( p ) = f n − j ( σ ( p ))and j is the ∼ -depth of p in ∆ψ (note that σ ′ is well defined because R is ∼ -balanced). It isthen easy to see (cf. the proof of Lemma 4.2) that every R -derivation of ϕ from Γ can betransformed into a derivation of f n ( ϕ ) from f n ( Γ ). (cid:3) Lemma 3.7.
Let
R ⊆
F m × F m be ∼ -balanced and having ∼ -depth k . If f n + k ( R n ) R ω ∪R F for every n < ω , then the logic ⊢ R ω ∪R F is not finitely based.Proof. Let ⊢ n = ⊢ R n ∪R F and ⊢ ω = ⊢ R ω ∪R F . As R ω = S n<ω R n , it is enough to show that ⊢ n ( ⊢ n +1 . It is clear that R n ∪ R F is ∼ -balanced and with ∼ -depth n + k . Hence, byLemma 3.6, Γ ⊢ n ϕ iff f n + k ( Γ ) ⊢ n f n + k ( ϕ ). Thus, from f n + k ( R n +1 )
6⊆ ⊢ ω and ⊢ n ⊆⊢ ω weconclude that R n +1 ⊆ ⊢ n +1 but R n +1
6⊆ ⊢ n , as was required to prove. (cid:3) Theorem 3.8.
The logic ⊢ ≤ DN of distributive lattices with negation is not finitely based.Proof. Recall that ⊢ R DN ω ∪R F = ⊢ ≤ DN by Corollary 3.4. Then, the result follows directly fromLemma 3.7. Indeed, R DN is ∼ -balanced and has ∼ -depth 1. Moreover, for every n < ω , wehave r = ∼ n +1 ( p ∧ p )) ∼ n +1 p ∈ R DN n and f n ( r ) = f n ( ∼ n +1 ( p ∧ p )) f n ( ∼ n +1 p ) ) = ∼ n ( q ∼ ( p ∧ p ) ) ∼ n ( q ∼ p ) / ∈ ⊢ ≤ DN . (cid:3) Note that the result of Theorem 3.8 holds for every strengthening of ⊢ ≤ DN to whichLemma 3.7 applies. In particular, let E be a set of equations such that R E is ∼ -balancedand has finite ∼ -depth. Then, for Lemma 3.7 to apply, it suffices to have B ⊆ V E ⊆ DN . Forinstance, denoting by ⊢ ≤ O the logic of order of the variety of Ockham algebras (Definition 2.3),it suffices to check that the rule ∼ ( p ∧ q ) ∼ p ∨∼ q is ∼ -balanced to conclude that ⊢ ≤ O is not finitelybased. A similar argument shows that, letting K ⊆ DN be the variety of distributive lattices with negation axiomatized (relatively to DN ) by equations (SDM1) and (SDM2)from Definition 2.3, we have that ⊢ ≤ K is not finitely based.4. The logics of semi-De Morgan algebras and p -lattices In this Section we show that, unlike ⊢ ≤ DN and ⊢ ≤ O , the logic of order of semi-De Morganalgebras ⊢ ≤ SDM is finitely based. In fact, we are going to establish a more general result: everylogic of order extending semi-De Morgan logic is finitely based, provided the correspondingvariety is (Theorem 4.7).Let
R ⊆
F m × F m be a set of rules, and let ( { q } ∪ { q i : i < ω } ) ∩ var ( R ) = ∅ . Given aformula γ , let g ( γ ) = γ ∧ q and g n +1 ( γ ) = ∼ g n ( γ ) ∧ q n . Given a rule ϕψ and n < ω , let r ϕψ n = g n ( ϕ ) ∨ qg n ( ψ ) ∨ q if n = 2 k g n ( ψ ) ∨ qg n ( ϕ ) ∨ q if n = 2 k + 1For n < ω , let R gn = { r ϕψ n : ϕψ ∈ R} , R g ≤ n = R ∪ S k ≤ n R gk and R gω = S n<ω R g ≤ n . Example 4.1.
Given a rule ϕψ , we have( ϕ ∧ q ) ∨ q ( ψ ∧ q ) ∨ q r ϕψ ( ∼ ( ψ ∧ q ) ∧ q ) ∨ q ( ∼ ( ϕ ∧ q ) ∧ q ) ∨ q r ϕψ ( ∼ ( ∼ ( ϕ ∧ q ) ∧ q ) ∧ q ) ∨ q ( ∼ ( ∼ ( ψ ∧ q ) ∧ q ) ∧ q ) ∨ q r ϕψ The general pattern is: ( ∼ . . . ( ∼ ( γ up n ∧ q ) ∧ q ) . . . ∧ q n ) ∨ q ( ∼ . . . ( ∼ ( γ dn n ∧ q ) ∧ q ) . . . ∧ q n ) ∨ q r ϕψ n with γ up n = ( ϕ for even nψ for odd n and γ dn n = ( ψ for even nϕ for odd n. Clearly, R gω ⊆ R ω , where R ω is defined as in the previous Section. Let us fix the set R • consisting of the following rules: pp ∨ ⊥ r ∨⊥ p ∨ r ( p ∧ ⊤ ) ∨ r r ∧⊤ ∼ ( p ∧ ⊤ ) ∨ r ∼ p ∨ r r ∼⊤ ( p ∨ q ) ∧ r ( p ∧ r ) ∨ ( p ∧ r ) r ∨∧ dist (( p ∧ p ) ∧ p ) ∨ q ( p ∧ ( p ∧ p )) ∨ q r ass ∨∧ ∼ ( p ∨ q ) ∼ p ∧ ∼ q r dm ∼∨ Lemma 4.2. If R • ⊆ ⊢ R ω then ⊢ R gω ∪R • = ⊢ R ω .Proof. Let ⊢ = ⊢ R gω . Since R ⊆ R gω ⊆ ⊢ R ω , it is enough to show that if ϕ ⊢ ψ , given afresh variable q , we have:(i) ϕ ∨ q ⊢ ψ ∨ q (ii) ϕ ∧ q ⊢ ψ ∧ q (iii) ∼ ψ ⊢ ∼ ϕ .The proof is by induction on the length of the derivation showing that ϕ ⊢ ψ . In thebase case we have simply ϕ = ψ , in which case (i), (ii) and (iii) follow immediately. For thestep, assume ϕ, γ , . . . , γ k , ψ is an R gω -derivation and by induction hypothesis we have that ϕ ∨ q ⊢ γ k ∨ q , ϕ ∧ q ⊢ γ k ∧ q and ∼ γ k ⊢ ∼ ϕ . To conclude the proof, we consider in each INITE AXIOMATIZABILITY OF LOGICS OF DISTRIBUTIVE LATTICES WITH NEGATION 11 of the cases how to complete the derivations depending on the last rule that was used. Bystructurality, it is enough to show that for each rule ϕψ ∈ R gω we have that (i)–(iii) hold.Concerning the rules ϕψ ∈ R , we have:(i) ϕ ∨ q ⊢ r ∧⊤ ( ϕ ∧ ⊤ ) ∨ q ⊢ r ϕψ ( ψ ∧ ⊤ ) ∨ q ⊢ r ∧⊤ ψ ∨ q (ii) ϕ ∧ q ⊢ r ∨⊥ ( ϕ ∧ q ) ∨ ⊥ ⊢ r ϕψ ( ψ ∧ q ) ∨ ⊥ ⊢ r ∨⊥ ψ ∧ q (iii) ∼ ψ ⊢ r ∼⊤ ∼ ( ψ ∧ ⊤ ) ∨ ⊥ ⊢ r ϕψ ∼ ( ϕ ∧ ⊤ ) ∨ ⊥ ⊢ r ∼⊤ ∼ ϕ ∨ ⊥ ⊢ r ∨⊥ ∼ ϕ .Now, for each ϕψ ∈ R and j < ω , consider r ϕψ j = g j ( ϕ ) ∨ q ′ g j ( ψ ) ∨ q ′ . We have:(i) ( g j ( ϕ ) ∨ q ′ ) ∨ q ⊢ r ∨ ass g j ( ϕ ) ∨ ( q ′ ∨ q ) ⊢ r ϕψj g j ( ψ ) ∨ ( q ′ ∨ q ) ⊢ r ∨ ass ( g j ( ψ ) ∨ q ′ ) ∨ q (ii) For j > j = 0 is analogous)( g j ( ϕ ) ∨ q ′ ) ∧ q = (( ∼ g j − ( ϕ ) ∧ q j ) ∨ q ′ ) ∧ q ⊢ r ∨∧ dist (( ∼ g j − ( ϕ ) ∧ q j ) ∧ q ) ∨ ( q ′ ∧ q ) ⊢ r ass ∨∧ (( ∼ g j − ( ϕ ) ∧ ( q j ∧ q )) ∨ ( q ′ ∧ q ) ⊢ r ϕψj (( ∼ g j − ( ψ ) ∧ ( q j ∧ q )) ∨ ( q ′ ∧ q ) ⊢ r ass ∨∧ (( ∼ g j − ( ψ ) ∧ q j ) ∧ q ) ∨ ( q ′ ∧ q ) ⊢ r ∨∧ dist ( g j ( ψ ) ∨ q ′ ) ∧ q (iii) ∼ ( g j ( ψ ) ∨ q ) ⊢ r dm ∼∨ ∼ g j ( ψ ) ∧ ∼ q ⊢ r ∨⊥ ( ∼ g j ( ψ ) ∧ ∼ q ) ∨ ⊥⊢ r ϕψj +1 ( ∼ g j ( ϕ ) ∧ ∼ q ) ∨ ⊥⊢ r ∨⊥ ∼ ( g j ( ϕ ) ∨ q ) ⊢ r dm ∼∨ ∼ ( g j ( ϕ ) ∨ q ) (cid:3) Since R • ⊆ ⊢ ≤ DN , by Lemma 4.2, we have that ( R DN ) gω ∪ R • ∪ R F provides an alternative(infinite) Hilbert presentation of ⊢ ≤ DN .In order to obtain a finite axiomatization of ⊢ ≤ SDM , let us fix the set S • consisting of thefollowing rules: ∼ ∼ ( p ∧ q ) ∨ r r ∧ ∼ ∼ p ∨ r ∼ ( ∼ ∼ p ∧ q ) r ∼ ∼ ( p ∧ q ) ∼ ( ∼ p ∧ p ) ∼ ( ∼ ( p ∧ p ) ∧ p ) r ∼∧ ∼ ( ∼ ( p ∧ p ) ∧ p ) Proposition 4.3.
Let R be a set of rules, and let R + := R g ≤ ∪ S • . If S • ⊆ ⊢ R gω , then ⊢ R + = ⊢ R gω .Proof. We just need to show that r ϕψ n ∈ ⊢ R + for n >
2. Since r ϕψ , r ϕψ ∈ R + , it suffices toshow that we can derive R ϕψ n +3 , R ϕψ n +4 using the rules in R n + := R + ∪ { r ϕψ n +1 , r ϕψ n +2 } . That is, we need to show that, for every ϕψ ∈ R and n < ω ,(i) g n +3 ( ψ ) ∨ r ⊢ R + n g n +3 ( ϕ ) ∨ r .We have: γ = g n +3 ( ψ ) ∨ r = ∼ ( g n +2 ( ψ ) ∧ q n +3 ) ∨ r = ∼ ( ∼ ( g n +1 ( ψ ) ∧ q n +2 ) ∧ q n +3 ) ∨ r = ∼ ( ∼ ( ∼ ( g n ( ψ ) ∧ q n +1 ) ∧ q n +2 ) ∧ q n +3 ) ∨ r ⊢ r ∧ ∼ ( ∼ ∼ ( g n ( A ) ∧ q n +1 ) ∧ q n +3 ) ∨ r ⊢ r ∼ ∼ ( g n ( ψ ) ∧ ( q n +1 ∧ q n +3 )) ∨ r ⊢ r ϕψ n +1 ∼ ( g n ( ϕ ) ∧ ( q n +1 ∧ q n +3 )) ∨ r ⊢ r ∼ ∼ ( ∼ ∼ ( g n ( ϕ ) ∧ q n +1 ) ∧ q n +3 ) = γ Further, γ , γ ⊢ r ∼∧ ∼ ( ∼ ( ∼ ( g n ( ϕ ) ∧ q n +1 ) ∧ q n +2 ) ∧ q n +3 ) ∨ r = g n +3 ( ϕ ) ∨ r. Hence, g n +3 ( ψ ) ∨ r ⊢ R n + g n +3 ( ϕ ) ∨ r .(ii) g n +4 ( ϕ ) ∨ r ⊢ R + n g n +4 ( ψ ) ∨ r .We have: γ = g n +4 ( ϕ ) ∨ r = ∼ ( g n +3 ( ϕ ) ∧ q n +4 ) ∨ r = ∼ ( ∼ ( g n +2 ( ϕ ) ∧ q n +3 ) ∧ q n +4 ) ∨ r = ∼ ( ∼ ( ∼ ( g n +1 ( ϕ ) ∧ q n +2 ) ∧ q n +3 ) ∧ q n +4 ) ∨ r ⊢ r ∧ ∼ ( ∼ ∼ ( g n +1 ( ϕ ) ∧ q n +2 ) ∧ q n +4 ) ∨ r ⊢ r ∼ ∼ ( g n +1 ( ψ ) ∧ ( q n +2 ∧ q n +4 )) ∨ r = ϕ ⊢ r ϕψ n +2 ∼ ( g n +1 ( ψ ) ∧ ( q n +2 ∧ q n +4 )) ∨ r ⊢ r ∼ ∼ ( ∼ ∼ ( g n +1 ( ψ ) ∧ q n +2 ) ∧ q n +4 ) = γ Further, γ , γ ⊢ r ∼∧ ∼ ( ∼ ( ∼ ( g n +1 ( ψ ) ∧ q n +2 ) ∧ q n +3 ) ∧ q n +4 ) ∨ r = g n +4 ( ψ ) ∨ r. Hence, g n +4 ( ϕ ) ∨ r ⊢ R n + g n +4 ( ψ ) ∨ r . (cid:3) Lemma 4.4.
The rule r ∼∧ is sound in ⊢ ≤ SDM .Proof.
Using the semi-De Morgan equations (SDM1)–(SDM3), it is easy to show that therule ∼ ( ∼ p ∧ p ) ∧ ∼ ( ∼ ( p ∧ p ) ∧ p ) ⊢ ∼ ( ∼ ( p ∧ p ) ∧ p ) is sound in ⊢ ≤ SDM if and only if( ∼ ∼ p ∨ ∗ ∼ p ) ∧ (( ∼ ∼ p ∧ ∼ ∼ p ) ∨ ∗ ∼ p ) ⊢ ( ∼ ∼ p ∧ ∼ ∼ p ) ∨ ∗ ∼ ∼ p is sound in ⊢ ≤ SDM .Letting ϕ := ( p ∨ ∼ p ) ∧ (( p ∧ p ) ∨ ∼ p ) and ψ := ( p ∧ p ) ∨ p , the rule ϕ ⊢ ψ is easilyseen to be sound in ⊢ ≤ DM . Moreover, ϕ ∗ = ( ∼ ∼ p ∨ ∗ ∼ p ) ∧ (( ∼ ∼ p ∧ ∼ ∼ p ) ∨ ∗ ∼ p ) and ψ ∗ = ( ∼ ∼ p ∧ ∼ ∼ p ) ∨ ∗ ∼ ∼ p . The soundness of ϕ ∗ ⊢ ψ ∗ w.r.t. ⊢ ≤ SDM then follows fromLemma 2.6. (cid:3)
Theorem 4.5.
Let E ⊆ F m × F m be a set of equations such that
SDM ⊆ E . Then R E + ∪ R • ∪ R F axiomatizes ⊢ ≤ V E . INITE AXIOMATIZABILITY OF LOGICS OF DISTRIBUTIVE LATTICES WITH NEGATION 13
Proof.
Since DN ⊆ SDM ⊆ E , we know by Theorem 3.3 that R E ω ∪R F axiomatizes ⊢ ≤ V E . From R • ⊆ ⊢ ≤ SDM ⊆ ⊢ ≤ V E we obtain by Lemma 4.2 that ( R E ) gω ∪ R • axiomatizes ⊢ ≤ V E . Moreover, S • ⊆ ⊢ ≤ SDM ⊆ ⊢ ≤ V E (Lemma 4.4 deals with the less obvious case). Hence, by Proposition 4.3we conclude that R E + ∪ R • ∪ R F axiomatizes ⊢ ≤ V E . (cid:3) Example 4.6.
By Theorem 4.5, the set R SDM + ∪ R • ∪ R F = ( R SDM ) g ≤ ∪ R • ∪ S • ∪ R F axiom-atizes ⊢ ≤ SDM . Since ( R SDM ) g ≤ = R SDM ∪ ( R SDM ) g ∪ ( R SDM ) g ∪ ( R SDM ) g , the axiomatizationthus obtained consists of 4 × | SDM | + |R • ∪ S • ∪ R F | = (4 ×
16) + 11 = 75 rules, many ofwhich bidirectional. However, it is not hard to see that
R ⊆ ⊢ ( R SDM ) g ∪R • , which allows oneto reduce the number of rules to (3 × | E | ) + 11 = 59. Further simplifications are of coursepossible, and in particular cases one may obtain a much more compact axiomatization. Acertain amount of redundancy in the set of rules obtained is the price we have to pay forthe generality and modularity of our approach. Regarding the latter aspect, observe forinstance that PL is axiomatized, relatively to SDM , by adding the pseudo-complement equa-tion x ∧ ∼ ( x ∧ y ) ≈ x ∧ ∼ y . Adding the rule p ∧∼ ( p ∧ q ) p ∧∼ q r P is not sufficient, for we also needto ensure that the resulting logic be self-extensional. To achieve this, by Theorem 4.5, it isenough to add the three rules: g i ( r P ) for 0 ≤ i ≤
2. We then have that ⊢ ≤ PL is axiomatizedover ⊢ ≤ SDM by { g i ( r P ) : 0 ≤ i ≤ } .Theorem 4.5 also provides a means to obtain (alternative) finite axiomatizations of otherlogics of order above ⊢ ≤ SDM . In particular, we can obtain a finite axiomatization of thelogic of p -lattices (Definition 2.3), i.e. the implication-free fragment of intuitionistic logic,that is alternative to the one introduced in [29]. We provide a general formulation of thisobservation below in Theorem 4.7. Theorem 4.7.
Let V ⊆ SDM be a variety. The following are equivalent:(i) V is axiomatized by a finite set of equations.(ii) ⊢ ≤ V is axiomatized by a finite set of finitary rule schemata.Proof. That (i) implies (ii) follows directly from Theorem 4.5.For the other direction, assume (ii) holds, so ⊢ ≤ V is axiomatized by a finite set R offinitary rule schemata. Given a rule r = Γϕ , let E ( r ) be the equation V Γ ∧ ϕ ≈ V Γ . Let E R := SDM ∪ { E ( r ) : r ∈ R} . Observe that the set E R is finite, and let V ′ be the varietydefined by the equations E R . We claim that V ′ = V . Indeed, it is clear that V ⊆ V ′ ⊆ SDM and therefore ⊢ ≤ SDM ⊆ ⊢ ≤ V ′ ⊆ ⊢ ≤ V . For the other direction, we start by observing that for each ϕψ ∈ R we have V Γ ∧ ϕ V Γ ∈ R E R . This, together with the fact that p , qp ∧ q , p ∧ qq ∈ ⊢ ≤ SDM ⊆ ⊢ ≤ V ′ ,implies that R ⊆ ⊢ ≤ V ′ . Hence, ⊢ ≤ V ⊆ ⊢ ≤ V ′ . From, ⊢ ≤ V ′ = ⊢ ≤ V and Theorem 2.7 we conclude that V = V ′ . (cid:3) Order-preserving logics of Berman varieties
We have shown in Section 4 how to obtain a finite axiomatization of the order-preservinglogic ⊢ ≤ K with K ⊆ SDM . Now, suppose K ⊆ O is a variety of Ockham algebras. As observedearlier, ⊢ ≤ O is not finitely based. However, if we restrict our attention to a Berman variety O mn of Ockham algebras, then we can adapt the technique employed in the preceding Sectionto obtain a finite Hilbert axiomatization for ⊢ ≤ O mn (an infinite one being directly given byTheorem 3.3). From now on, let us fix a variety O mn , with m, n < ω , and let E mn be the equationsaxiomatizing O mn . Let k < ω and let t be a fresh variable. Given a rule ϕψ , define s ϕψ k = ∼ k ( ϕ ) ∨ t ∼ k ( ψ ) ∨ t and s ϕψ k +1 = ∼ k +1 ( ψ ) ∨ t ∼ k +1 ( ϕ ) ∨ t . Letting: p ∧ qp r ∧ p ∧ qq r ∧ p , qp ∧ q r in ∧ define R mn ∧ := R E mn ∪ { r ∧ , r ∧ } , and O mn = R mn ∧ ∪ { s i ( r ) : i ≤ m + n, r ∈ R mn ∧ } ∪ { r in ∧ } . Lemma 5.1.
The relation ⊣⊢ O mn is a congruence of Fm .Proof. The key difference with the cases considered in the previous Sections is that ∼ ( p ∧ q ) ∼ p ∨∼ q r ∧∨ dm ∈R mn ∧ , but recall that the following rules are also in R mn ∧ : ( p ∧ q ) ∧ rp ∧ ( q ∧ r ) r ∧ ass ( p ∨ q ) ∨ rp ∨ ( q ∨ r ) r ∨ ass ∼ m + n p ∼ n p r mn p ∨ ( q ∧ r )( p ∨ q ) ∧ ( p ∨ r ) r ∨∧ dist In the presence of r ∧∨ dm , we can show directly that, if Γ ⊢ ϕ , then(i) Γ ∨ u ⊢ ϕ ∨ u (ii) Γ ∧ u ⊢ ϕ ∧ u (iii) ∼ ϕ ⊢ _ γ ∈ Γ ∼ γ where u is a fresh variable. Once more it is enough to show that (i)–(iii) are satisfied when r = Γϕ ∈ O mn .For r = p , qp ∧ q , we have:(i) p ∨ u, q ∨ u ⊢ r ( p ∨ u ) ∧ ( q ∨ u ) ⊢ r ∨∧ dist ( p ∧ q ) ∨ u (ii) p ∧ u, q ∧ u ⊢ r ( p ∧ u ) ∧ ( q ∧ u ) ⊢ r ∧ ass ( p ∧ q ) ∧ u (iii) ∼ ( p ∧ q ) ⊢ r ∧∨ dm ∼ p ∨ ∼ q .For r = ϕψ ∈ R mn ∧ , we have:(i) ϕ ∨ u ⊢ s ϕψ ψ ∨ u (ii) ϕ ∧ u ⊢ r j ∧ ϕ, u ⊢ r ψ, u ⊢ r in ∧ ψ ∧ u (iii) ∼ ψ ⊢ r ∨⊥ ∼ ψ ∨ ⊥ ⊢ s ϕψ ∼ ϕ ∨ ⊥ ⊢ r ∨⊥ ∼ ϕ .For s ϕψ k ∈ { s i ( r ) : i ≤ m + n, r ∈ R mn ∧ } , we let γ up k = ϕ and γ dn k = ψ if k is odd, and γ up k = ϕ and γ dn k = ψ if k is even. We can write s ϕψ k = ∼ k ( γ up k ) ∨ q ∼ k ( γ dn k ) ∨ q , as in Example 4.1.(i) ( ∼ k ( γ up k ) ∨ q ) ∨ r ⊢ r ∨ ass ∼ k ( γ up k ) ∨ ( q ∨ r ) ⊢ s ϕψk ∼ k ( γ dn k ) ∨ ( q ∨ r ) ⊢ ( ∼ k ( γ dn k ) ∨ q ) ∨ r (ii) ( ∼ k ( γ up k ) ∨ q ) ∧ r ⊢ r j ∧ ( ∼ k ( γ up k ) ∨ q ) , r ⊢ s ϕψk ( ∼ k ( γ dn k ) ∨ q ) , r ⊢ ( ∼ k ( γ dn k ) ∨ q ) ∧ r (iii) To show that ∼ ( ∼ k ( γ dn k ) ∨ q ) ⊢ ∼ ( ∼ k ( γ up n ) ∨ q ) we must consider two cases.If k + 1 ≤ m + n , then ∼ ( k ∼ ( γ dn k ) ∨ q ) ⊢ r ∧∨ dm k +1 ∼ ( γ dn k ) ∨ ∼ q ⊢ s ϕψk +1 k +1 ∼ ( γ up k ) ∨ ∼ q ⊢ r ∧∨ dm ∼ ( k ∼ ( γ dn k ) ∨ q ) . Otherwise, let k +1 = n +( i m + j ) for i > ≤ j < m (and thus n + j < m + n ). INITE AXIOMATIZABILITY OF LOGICS OF DISTRIBUTIVE LATTICES WITH NEGATION 15
We have: ∼ ( k ∼ ( γ dn k ) ∨ q ) ⊢ r ∧∨ dm ∼ k +1 ( γ dn k ) ∧ ∼ q ⊢ r mn ∼ n + j ( γ dn k ) ∧ ∼ q ⊢ s ϕψk +1 ∼ n + j ( γ up k ) ∧ ∼ q ⊢ r mn ∼ k +1 ( γ up k ) ∧ ∼ q ⊢ r ∧∨ dm ∼ ( ∼ k ( γ dn k ) ∨ q ) (cid:3) Theorem 5.2.
The set of rules O mn ∪ { ⊤ } axiomatizes ⊢ ≤ O mn Proof.
It is clear that O mn ∪ { ⊤ } ⊆ ⊢ ≤ O mn . Completeness follows by a similar reasoning as inTheorem 3.3 from Lemma 5.1. (cid:3) We note that, given E such that V E ⊆ O mn , it is easy to see that ⊢ ≤ V E is axiomatized,relatively to ⊢ ≤ O mn , by the set R E ∪ { s i ( r ) : i ≤ m + n, r ∈ R E } . Hence, if E is finite then ⊢ ≤ V E is finitely based. In fact, one can easily adapt the argument of Theorem 4.7 to obtainthe following: Corollary 5.3.
Let V ⊆ O mn be a variety. The following are equivalent:(i) V is axiomatized by a finite set of equations.(ii) ⊢ ≤ V is axiomatized by a finite set of finitary rule schemata. ⊤ -assertional logics As mentioned earlier, another logic (alternative to ⊢ ≤ K ) canonically associated to a givenclass K of algebras having a constant ⊤ is the so-called ⊤ -assertional logic ⊢ ⊤ K determinedby the class of all matrices {h A , {⊤}i : A ∈ K } . By definition, ⊢ ⊤ K is stronger than ⊢ ≤ K , butit is well known that ⊢ ⊤ K = ⊢ ≤ K for K = B or K = PL . On the other hand, it is easy to checkthat ⊢ ⊤ DN = ⊢ ≤ DN . For this, it suffices to observe that the rule p ∧ ∼ p ∼ q r wxc is sound w.r.t. ⊢ ⊤ DN but not w.r.t. ⊢ ≤ DN . The same example witnesses ⊢ ⊤ SDM = ⊢ ≤ SDM and ⊢ ⊤ O = ⊢ ≤ O .In this Section we take a closer look at the assertional logic ⊢ ⊤ SDM from an algebraic logicpoint of view. This perspective will allow us to obtain further information on the poset offinitary selfextensional extensions of ⊢ ≤ SDM , as well as to provide a Hilbert calculus for ⊢ ⊤ SDM .For all unexplained terminology used in this Section, we refer the reader to [14].As mentioned in the Introduction, all logics considered in this paper are non-protoalgebraic.We state this formally below.
Theorem 6.1.
Let K ⊆ DN . If PL ⊆ K or DM ⊆ K , then ⊢ ⊤ K (and, a fortiori, ⊢ ≤ K ) is notprotoalgebraic.Proof. Observe that both ⊢ ⊤ PL and ⊢ ⊤ DM are non-protoalgebraic. The former was remarkedin [28, p. 320], while the latter is proved in [1, Thm. 5.1]. The result then follows from the observation that the property of being protoalgebraic is preserved by extensions. (Indeed,we notice that [1, Thm. 5.1] even entails that ⊢ ⊤ K is not protoalgebraic for every K with B K ⊆ DM .) (cid:3) We next provide a better description of reduced matrix models of ⊢ ≤ SDM . Recall that amatrix M is a model of a logic ⊢ when ⊢ ⊆ ⊢ M . The Leibniz congruence Ω A ( D ) of a matrix M = h A , D i is the largest congruence of A that is compatible with D in the following sense:for all a, b ∈ A , if a ∈ D and h a, b i ∈ Ω A ( D ), then b ∈ D . A matrix M = h A , D i is reduced when Ω A ( D ) is the identity relation. Proposition 6.2.
Let M = h A , D i be a model of ⊢ ≤ SDM with A ∈ SDM , and let a, b ∈ A .Then h a, b i ∈ Ω A ( D ) if and only if, for all c , c , c ∈ A , the following conditions hold:(i) a ∨ c ∈ D iff b ∨ c ∈ D ,(ii) ∼ ( a ∧ c ) ∨ c ∈ D iff ∼ ( b ∧ c ) ∨ c ∈ D ,(iii) ∼ ( ∼ ( a ∧ c ) ∧ c ) ∨ c ∈ D iff ∼ ( ∼ ( b ∧ c ) ∧ c ) ∨ c ∈ D .Proof. Let θ be the relation defined by items (i)–(iii). Let us check that θ is compatiblewith the algebraic operations of A .( ∼ ). Assume h a, b i ∈ θ . That ∼ a ∨ c ∈ D iff ∼ b ∨ c ∈ D follows from (ii): observethat, taking c = ⊤ , we have ∼ a ∨ c = ∼ ( a ∧ ⊤ ) ∨ c and ∼ b ∨ c = ∼ ( b ∧ ⊤ ) ∨ c . Asimilar reasoning, taking c = ⊤ in (iii), shows that ∼ ( ∼ a ∧ c ) ∨ c ∈ D iff ∼ ( ∼ b ∧ c ) ∨ c ∈ D . Now, assume ∼ ( ∼ ( ∼ a ∧ c ) ∧ c ) ∨ c ∈ D . By Lemma 2.4.ii, we have ∼ ( ∼ ( ∼ a ∧ c ) ∧ c ) ∨ c ≤ ∼ ( ∼ ∼ a ∧ c ) ∨ c = ∼ ( a ∧ c ) ∨ c . Hence, ∼ ( a ∧ c ) ∨ c ∈ D ,and we can apply (ii) to obtain ∼ ( b ∧ c ) ∨ c = ∼ ( ∼ ∼ b ∧ c ) ∨ c ∈ D . Thus we have( ∼ ( ∼ ∼ b ∧ c ) ∨ c ) ∧ ( ∼ ( ∼ ( ∼ a ∧ c ) ∧ c ) ∨ c ) ∈ D , because D is closed under ∧ . Then, taking p = ∼ b , p = c , q = c , p = ∼ a , p = c in r ∼∧ , we have ∼ ( ∼ ( ∼ b ∧ c ) ∧ c ) ∨ c ∈ D .To check that θ is compatible with the binary operations, assume h a , b i , h a , b i ∈ θ .Relying on completeness (Theorem 4.5), we can use any logical rule ϕψ such that ϕ ≤ ψ is aninequality valid in SDM . In particular, in the proof below, by (e.g.) ‘commutativity’ for ∧ we shall refer not only to the rule p ∧ qq ∧ p , but also ∼ ( p ∧ q ) ∼ ( q ∧ p ) , ∼ ( p ∧ q ) ∨ r ∼ ( q ∧ p ) ∨ r , etc. In the computationsthat follow, we shall skip the steps that follow trivially (by symmetry) from the precedingones; the dots (. . . ) will be used to indicate the passages that have been omitted.( ∧ ). We have:(i) ( a ∧ a ) ∨ c ∈ D iff ( a ∨ c ) ∧ ( a ∨ c ) ∈ D by distributivityiff a ∨ c , a ∨ c ∈ D by p ∧ qp iff b ∨ c , b ∨ c ∈ D by (i)( . . . ) iff ( b ∧ b ) ∨ c ∈ D. INITE AXIOMATIZABILITY OF LOGICS OF DISTRIBUTIVE LATTICES WITH NEGATION 17 (ii) ∼ (( a ∧ a ) ∧ c ) ∨ c ∈ D iff ∼ ( a ∧ ( a ∧ c )) ∨ c ∈ D by ∧ -associativityiff ∼ ( b ∧ ( a ∧ c )) ∨ c ∈ D by (ii)iff ∼ (( b ∧ a ) ∧ c ) ∨ c ∈ D by ∧ -associativityiff ∼ (( a ∧ b ) ∧ c ) ∨ c ∈ D by ∧ -commutativityiff ∼ ( a ∧ ( b ∧ c )) ∨ c ∈ D by ∧ -associativityiff ∼ ( b ∧ ( b ∧ c )) ∨ c ∈ D by (ii)( . . . ) iff ∼ (( b ∧ b ) ∧ c ) ∨ c ∈ D. (iii) ∼ ( ∼ (( a ∧ a ) ∧ c ) ∧ c ) ∨ c ∈ D iff ∼ ( ∼ (( a ∧ ( a ∧ c )) ∧ c ) ∨ c ∈ D by ∧ -associativityiff ∼ ( ∼ (( b ∧ ( a ∧ c )) ∧ c ) ∨ c ∈ D by (iii)( . . . ) iff ∼ ( ∼ (( a ∧ ( b ∧ c )) ∧ c ) ∨ c ∈ D iff ∼ ( ∼ (( b ∧ ( b ∧ c )) ∧ c ) ∨ c ∈ D by (iii)( . . . ) iff ∼ ( ∼ (( b ∧ b ) ∧ c ) ∧ c ) ∨ c ∈ D. ( ∨ ). We have:(i) ( a ∨ a ) ∨ c ∈ D iff a ∨ ( a ∨ c ) ∈ D by ∨ -associativityiff b ∨ ( a ∨ c ) ∈ D by (i)iff ( b ∨ a ) ∨ c ∈ D by ∨ -associativityiff ( a ∨ b ) ∨ c ∈ D by ∨ -commutativityiff a ∨ ( b ∨ c ) ∈ D by ∨ -associativityiff b ∨ ( b ∨ c ) ∈ D by (i)iff ( b ∨ b ) ∨ c ∈ D by ∨ -associativity.(ii) ∼ (( a ∨ a ) ∧ c ) ∨ c ∈ D iff ∼ (( a ∧ c ) ∨ ( a ∧ c )) ∨ c ∈ D by distributivityiff ( ∼ ( a ∧ c ) ∧ ∼ ( a ∧ c )) ∨ c ∈ D by (SDM1)iff ( ∼ ( a ∧ c ) ∨ c ) ∧ ( ∼ ( a ∧ c ) ∨ c ) ∈ D by distributivityiff ∼ ( a ∧ c ) ∨ c , ∼ ( a ∧ c ) ∨ c ∈ D by p ∧ qp iff ∼ ( b ∧ c ) ∨ c , ∼ ( b ∧ c ) ∨ c ∈ D by (ii)( . . . ) iff ∼ (( b ∨ b ) ∧ c ) ∨ c ∈ D. (iii) ∼ ( ∼ (( a ∨ a ) ∧ c ) ∧ c ) ∨ c ∈ D iff ∼ ( ∼ (( a ∧ c ) ∨ ( a ∧ c )) ∧ c ) ∨ c ∈ D by distributivityiff ∼ (( ∼ ( a ∧ c ) ∧ ∼ ( a ∧ c )) ∧ c ) ∨ c ∈ D by (SDM1)iff ∼ ( ∼ ( a ∧ c ) ∧ ( ∼ ( a ∧ c ) ∧ c )) ∨ c ∈ D by ∧ -associativityiff ∼ ( ∼ ( b ∧ c ) ∧ ( ∼ ( a ∧ c ) ∧ c )) ∨ c ∈ D by (iii)iff ∼ (( ∼ ( b ∧ c ) ∧ ( ∼ ( a ∧ c )) ∧ c ) ∨ c ∈ D by ∧ -associativityiff ∼ ((( ∼ ( a ∧ c ) ∧ ∼ ( b ∧ c )) ∧ c ) ∨ c ∈ D by ∧ -commutativityiff ∼ ((( ∼ ( b ∧ c ) ∧ ∼ ( b ∧ c )) ∧ c ) ∨ c ∈ D by (iii)iff ∼ ((( ∼ ( b ∧ c ) ∧ ∼ ( b ∧ c )) ∧ c ) ∨ c ∈ D by ∧ -commutativity( . . . ) iff ∼ ( ∼ (( b ∨ b ) ∧ c ) ∧ c ) ∨ c ∈ D. Hence, θ is a congruence of A . Also, θ is obviously compatible with D . Indeed, if a ∈ D and h a, b i ∈ θ , then we can use pp ∨ q to conclude a ∨ b ∈ D . Then we have b ∨ b ∈ D by (i),which gives us b ∈ D using the rule of ∨ -idempotency. Lastly, if θ ′ is a congruence of A thatis compatible with D , then it is easy to show that θ ′ ⊆ θ . Indeed, if h a, b i ∈ θ ′ , then we alsohave, for instance, h a ∧ c , b ∧ c i , h∼ ( a ∧ c ) , ∼ ( b ∧ c ) i , h∼ ( a ∧ c ) ∨ c , ∼ ( b ∧ c ) ∨ c i ∈ θ ′ and so on. Thus, assuming ∼ ( a ∧ c ) ∨ c ∈ D , we have ∼ ( b ∧ c ) ∨ c ∈ D because θ ′ iscompatible with D . Hence, h a, b i ∈ θ . Thus, θ is the largest congruence compatible with D , as required. (cid:3) The following auxiliary result is well known to hold for semilattice-based logics (see e.g. [1,Thm. 2.13.iii]; for a definition of the classes
Alg ∗ ( ⊢ ) and Alg ( ⊢ ), see see [14]). Proposition 6.3.
Alg ( ⊢ ≤ SDM ) =
SDM . Table 1 introduces the two extra rules that will permit us to axiomatize ⊢ ⊤ SDM . Observethat r WP is a weaker form of the pseudo-complement rule r P introduced in Example 4.6.Note also that none of the rules in R ⊤ corresponds to an (in)equality: their role is toensure that reduced models satisfy F = {⊤} , rather than to restrict the underlying class ofalgebras. p ∧ ( ∼ ( p ∧ q ) ∨ r ) r WP ∼ q ∨ r p ∧ ( ∼ ( ∼ q ∧ r ) ∨ s ) r Q ∼ ( ∼ ( p ∧ q ) ∧ r ) ∨ s Table 1.
The set of rules R ⊤ . Lemma 6.4.
Let h A , F i be a reduced matrix for the strengthening of ⊢ ≤ SDM with R ⊤ . Then F = {⊤} .Proof. By Proposition 6.3 (and the well-known fact that
Alg ∗ ( ⊢ ) ⊆ Alg ( ⊢ ) holds for anylogic ⊢ [14, Thm. 2.23]), we have that every reduced matrix for ⊢ ≤ SDM is of the form h A , F i with A ∈ SDM and F a lattice filter [18, Lemma 3.8].Suppose, by way of contradiction, that there is a ∈ F such that a = ⊤ . Then h a, ⊤i / ∈ Ω A ( F ). This means that there are c , c , c ∈ A such that at least one of the three items ofProposition 6.2 fails. Clearly, item (i) cannot fail, because a, ⊤ ∈ F . Thus, suppose item (ii)fails. Then there are c , c ∈ A such that ∼ ( a ∧ c ) ∨ c ∈ F and ∼ ( ⊤∧ c ) ∨ c = ∼ c ∨ c / ∈ F . INITE AXIOMATIZABILITY OF LOGICS OF DISTRIBUTIVE LATTICES WITH NEGATION 19
But, since a ∈ F , the latter cannot happen because of the rule r WP . Now, assume item (iii)fails. Then there are c , c , c ∈ A such that ∼ ( ∼ ( ⊤ ∧ c ) ∧ c ) ∨ c = ∼ ( ∼ c ∧ c ) ∨ c ∈ F but ∼ ( ∼ ( a ∧ c ) ∧ c ) ∨ c / ∈ F . But since a ∈ F , this cannot happen because of rule r Q . (cid:3) Theorem 6.5.
For every K ⊆ SDM , the logic ⊢ ⊤ K is axiomatized, relatively to ⊢ ≤ K , by R ⊤ .Proof. Soundness is clear. For completeness, assume Γ ϕ where ⊢ is the strengtheningof ⊢ ≤ K with r WP and r Q . Then there is a reduced matrix model h A , F i of ⊢ witnessing this.Moreover, A ∈ Alg ( ⊢ ≤ K ) = V ( K ) ⊆ SDM (cf. Proposition 6.3). So we can invoke Lemma 6.4to obtain F = {⊤} . Hence, Γ ⊤ K ϕ , as required. (cid:3) Taking into account Theorem 4.5, the preceding Theorem immediately gives us the fol-lowing.
Corollary 6.6.
For every K ⊆ SDM , if ⊢ ≤ K is finitely based, then so is ⊢ ⊤ K . Concluding remarks
The present paper has been a contribution to improving our current understanding ofthe expressivity of Hilbert calculi. As observed earlier, Gentzen calculi allow one to imposedirectly the meta-properties needed to ensure that the inter-derivability relation is a con-gruence of the formula algebra. By contrast, we have shown that under certain conditionsthis is beyond what Hilbert calculi can capture finitely. Our main results are displayed inTable 2 below. Conditions on E ⊢ ≤ V E ⊢ ⊤ V E Examples ∼ -balanced, V E ⊆ DN and ∼ k p ≤ V E ∼ k q N ? DN , O finite and V E ⊆ SDM
Y Y
SDM , PL finite and V E ⊆ O mn Y ? O mn , DM Table 2.
Finite axiomatizability results.On the front of positive results, we have identified certain subvarieties of DN for whichHilbert calculi are indeed able to reflect finitely the effect of imposing extra equations onthe algebras. The well-known result that finitely-generated varieties of lattices are finitelybased [7, Cor. V.4.18] implies that our methods may be successfully applied to every finite-valued order-preserving logic that extends ⊢ ≤ SDM . We believe it would be interesting to takea closer look at the conditions that characterize this divide.Yet another approach to the axiomatization of logics, which is intermediate betweenHilbert and Gentzen, is provided by multiple-conclusion calculi. These are an extensionof traditional (single-conclusion) Hilbert calculi where rules may have non-singleton setsof conclusions (which are read disjunctively). With multiple-conclusion calculi one gains aconsiderably greater expressive power without expanding the signature with metalinguisticsymbols as happens with Gentzen systems. For instance, it is known that every finite-valued logic is finitely axiomatizable by multiple-conclusion calculi, and desirable proof-theoretical properties (e.g. analiticity, effective proof search) are more easily establishedfor the latter than for the their single-conclusion counterparts (see e.g. [32, 25, 26]). Wespeculate whether the logics we have shown to be non-finitely based (by means of single-conclusion Hilbert calculi) might be axiomatizable by means of a finite multiple-conclusion calculus (as happens, for instance, with the logic defined by Wro´nski’s three-element matrix:see [33, 26]).A related question is whether logics of distributive lattices with negation that are notgiven by any finite set of finite matrices may admit some finite non-deterministic partialmatrix semantics (see [2, 3, 10, 8, 9]).A last research direction worth mentioning is the study of logics defined from classes ofdistributive lattices with negation through different choices of the designated elements. Aswe have seen earlier, one such choice yields ⊤ -assertional logics associated to subvarietiesof DN . In this respect, we speculate whether the finite axiomatizability result obtained inSection 6 for ⊢ ⊤ SDM might be extended to other logics (e.g. ⊢ ⊤ DN , ⊢ ⊤ O , ⊢ ⊤ O mn ). References [1] H. Albuquerque, A. Prenosil, and U. Rivieccio. An algebraic view of super-Belnap logics.
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