Intermediate Logics Admitting a Structural Hypersequent Calculus

We characterise the intermediate logics which admit a cut-free hypersequent calculus of the form HLJ+R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {HLJ} + \mathscr {R}$$\end{document}, where HLJ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {HLJ}$$\end{document} is the hypersequent counterpart of the sequent calculus LJ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {LJ}$$\end{document} for propositional intuitionistic logic, and R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathscr {R}$$\end{document} is a set of so-called structural hypersequent rules, i.e., rules not involving any logical connectives. The characterisation of this class of intermediate logics is presented both in terms of the algebraic and the relational semantics for intermediate logics. We discuss various—positive as well as negative—consequences of this characterisation.


Introduction
Constructing cut-free proof calculi for intermediate logics can be notoriously difficult. In fact, we know only of very few intermediate logics having a cutfree Gentzen-style sequent calculus obtained by adding a finite number of sequent rules to the single-succedent 1 sequent calculus LJ for IPC. 2 On the other hand few decisively negative results have been obtained in this respect. The few such results in the literature are of a rather general nature ruling out sequent systems with rules of a particular syntactic shape for certain (classes of) logics, see, e.g., [22,Cor. 7.2] and [40]. 3 However, by moving to the framework of hypersequent calculi [3,45,47] it is possible to construct cut-free hypersequent calculi for many well-known intermediate logics, see, e.g., [4,20,21,26]. Hypersequents are nothing but finite multisets of sequents. In general adding so-called structural hypersequent rules, viz., rules not involving the logical connectives, usually behaves well with respect to the cut-elimination procedure. In fact, a systematic approach to the problem of constructing cut-free proof calculi has been developed and a class of formulas, called P 3 , has been singled out for which corresponding cut-free structural single-succedent hypersequent calculi may be obtained in a uniform manner [22,24]. However, negative results demarcating the class of intermediate logics admitting cut-free structural hypersequent calculi are still to some extent lacking. Considering substructural logics proper some meaningful necessary conditions for admitting a cut-free structural hypersequent calculus have been provided [22,Cor. 7.3], [25,Thm. 6.8].
Our contribution consists in singling out a purely semantic criterion determining when an intermediate logic can be captured by a cut-free structural single-succedent hypersequent calculus extending the basic single-succedent hypersequent calculus HLJ for IPC. This is done by considering a subclass of the so-called (0, ∧, ∨, 1)-stable logics studied in [7,10,13]. More precisely, we introduce a class of intermediate logics, which we call (0, ∧, 1)-stable, determined by classes of Heyting algebras closed under taking (0, ∧, 1)subalgebras of subdirectly irreducible Heyting algebras. We are also able to show that all such intermediate logics are elementarily determined and we obtain a characterisation of the first-order frame conditions determining intermediate logics with a cut-free structural hypersequent calculus. These frame conditions are analogous to the frame conditions introduced by Lahav [42] for constructing analytic hypersequent calculi for modal logics. Finally, we compare the class of (0, ∧, 1)-stable intermediate logics to the class of (0, ∧, ∨, 1)-stable intermediate logics. We show that there are (0, ∧, ∨, 1)stable intermediate logics which are not (0, ∧, 1)-stable. Furthermore, we show that the (0, ∧, ∨, 1)-stable logics given by the so-called (0, ∧, ∨, 1)stable rules [10,Sec. 5] determined by finite well-connected Heyting algebras which are projective as objects in the category of distributive lattices will necessarily be (0, ∧, 1)-stable. Lastly, we show that the (0, ∧, 1)-stable logics are precisely the cofinal subframe logics which are also (0, ∧, ∨, 1)-stable.
The paper is structured as follows. Section 2 contains a short introduction to hypersequent calculi and their algebraic interpretation. Section 3 contains the algebraic characterisation of intermediate logics with a (cut-free) structural hypersequent calculus. In Section 4 the first-order frame conditions associated with the class of intermediate logics admitting a cut-free structural hypersequent calculi are determined and in Section 5 this class of intermediate logics is compared with the class of (0, ∧, ∨, 1)-stable logics.

Preliminaries on Algebraic Proof Theory
In this section we will briefly review the necessary background on algebraic proof theory [22,24,25] on which the findings of the present paper heavily relies.

Hypersequents
Let P be a set of propositional letters and let Form be the set of propositional formulas in the language of intuitionistic logic given by the following grammar Note that in this language both the connective ↔ and the constant is definable, e.g., as (ϕ → ψ) ∧ (ψ → ϕ) and ⊥ → ⊥, respectively.
By a single-succedent sequent (in the language of propositional intuitionistic logic) we shall understand a pair (Γ, Π), written Γ ⇒ Π, where Γ is a finite multiset of formulas from Form and Π is a stoup, i.e., either empty or a single formula in Form. The sequent system LJ, see, e.g., [31,Chap. 1.3], provides a sequent calculus which is sound and complete with respect to propositional intuitionistic logic IPC. However, when adding additional axioms to LJ the resulting system is no longer guaranteed to enjoy the same proof-theoretic properties as LJ such as cut-elimination.
Nevertheless, many logics for which no cut-free Gentzen-style sequent calculus is available may be captured nicely by the so-called hypersequent calculus formalism. A hypersequent is simply a finite multiset of sequents H written as Γ 1 ⇒ Π 1 | . . . | Γ n ⇒ Π n , where the sequents Γ i ⇒ Π i are called the components of the hypersequent H. One may think of a hypersequent as a "meta-disjunction" of sequents. The hypersequent formalism can therefore be thought of as a proof-theoretic framework that allows for the manipulation of sequents in parallel. In addition to the usual (internal) structural rules such as contraction and weakening the hypersequent framework allows us to consider a wide variety of so-called structural hypersequent rule [22,Sec. 3.1], i.e., hypersequent rules not involving any of the logical connectives, which operates on multiple components at once. For example the structural hypersequent rules However, for what follows it is essential that we consider single-succedent calculi as we will be relying on results from [22,25] which only consider the single-succedent case. In fact, it is not immediately clear if the approach of [22,25] can be successfully adapted to the multi-succedent setting.
Definition 2.5. Given an intermediate logic L we say that a hypersequent calculus HLJ + R determines L if Λ(HLJ + R) = L. Furthermore, given a property P we say that L admits a hypersequent calculus with property P if L is determined by a hypersequent calculus HLJ + R with the property P .
Remark 2.6. Given a sequent S, say, Γ ⇒ Π, there exists a formula ϕ S , namely, Γ → Π, such that S and ϕ S determine the same superintuitionistic logic. We make use of the convention that ∅ = ⊥ and ∅ = . Similarly, given a hypersequent rule (r) there exists a finite set of multi-conclusion rules M r such that (r) and M r determine the same superintuitionistic logic, for details see, e.g., [16,Sec. 2]. Thus for the purpose of axiomatising intermediate logics the two formalisms are equally good. However, when considering properties of formal derivations the hypersequent formalism is arguably more natural. We will come back to multi-conclusion rules, in the form of stable universal clauses, in Section 3.
We would like to know which intermediate logics can be determined by structural intermediate hypersequent calculi as defined in Definition 2.2.
That is, we would like to know which intermediate logics are of the form Λ(HLJ+R), for R a set of structural hypersequent rules. This is interesting to know since such intermediate logics can also be captured by a structural hypersequent calculus without using the cut-rule [22]. 4 Thus answering this question will help us better understand-from a semantical point of viewwhich intermediate logics can be captured by cut-free proof calculi.

Structural Hypersequent Calculi and Universal Clauses
Let HA denote the variety of Heyting algebras. Then for any intermediate logic L ⊇ IPC there is a variety V(L) ⊆ HA such that ϕ ↔ ψ ∈ L if and only if V(L) |= ϕ ≈ ψ, for any pair of formulas ϕ and ψ in the language of propositional intuitionistic logic, see, e.g., [18,Thm. 7 Proof. This is nothing but a modified version of the Lindenbaum-Tarski construction, 5 see, e.g., [15,16,41].
Thus in order to use the algebraic semantics to study structural hypersequent calculi we must identify the classes of Heyting algebras of the form K(R) for R a collection of structural hypersequent rules.
Recall, e.g., [17,Def. V.2.19] that a first-order formula (in a language without relational symbols) in prenex-normal form with all quantifiers universal is called a universal formula or universal clause. Thus, any universal clause may be written 6 as for terms t k ( x) and u k ( x), k ∈ {1, . . . , n}. In the presence of the lattice operation ∧ we will use t ≤ u as an abbreviation of the equation t ≈ t ∧ u. Finally, if it is clear from the context we will drop the universal quantifier, leaving it to be understood that the variables, which may or may not be displayed, are all universally quantified. A class of models of a collection of universal formulas is called a universal class.
The Proof. Given a structural rule (r), say we associate a unique first-order variable x to each multiset variable Γ occurring in (r) and similarly we associate a unique first-order variable y to each stoup variable Π occurring in (r). Then if S i is Γ i1 , . . . , Γ ik ⇒ Π i we define t i to be the term x i1 ∧ . . . ∧ x ik and u i to be the term y i . If S i is Γ i1 , . . . , Γ ik ⇒ we define t i to be the term x i1 ∧ . . . ∧ x ik and u i to be the constant 0. Letting q r be the universal clause it is then straightforward to verify that (r) and q r are equivalent.
Example 2.9. For example the structural hypersequent rules (com) and (lq) correspond to the universal clauses A converse to Proposition 2.8 may be given. Proof. Let q be a universal clause in the {0, ∧, 1}-reduct of the language of Heyting algebras. Any such universal clause will be equivalent to a finite set of clauses of the form where for each i ∈ {1, . . . , n} the term t i is a meet of variables, say x i1 ∧ . . . ∧ x ik , or the constant 1 and the term u i is either a variable, say y i , or the constant 0. Thus without loss of generality we may assume that q is of this form. In fact we may assume that the variables occurring in the terms {t i } n i=1 are disjoint from the variable occurring in the terms {u i } n i=1 , [25,Thm. 4.15]. For each variable x i occurring in the terms {t i } n i=1 we associate a multiset variable Γ i and for each variable y i occurring among the terms {u i } n i=1 we associate a stoup variable Π i . Finally, we let (r q ) be the rule with the left-hand (resp. right-hand) side empty if t i (resp. u i ) is a constant. Again, it is easy to verify that (r q ) and (q) are indeed equivalent.
Thus Propositions 2.8 and 2.10 above establish that for R a collection of structural hypersequent rules the class K(R) of Heyting algebras validating all the rules in R is a universal class of Heyting algebras determined by universal clauses in the {0, ∧, 1}-reduct of the language of Heyting algebras and in fact any such class arises in this way. This characterisation is, however, not very informative and in the following section we shall provide a characterisation which we believe to be more enlightening. Proof. Suppose that L admits a structural intermediate hypersequent calculus, say HLJ + R with R a set of structural hypersequent rules. Then we have that ϕ is a theorem of L iff the sequent ⇒ ϕ is derivable in the hypersequent calculus HLJ + R. By Theorem 2.7 this is the case iff |= K(R) ⇒ ϕ, which in turn happens precisely when the equation 1 ≈ ϕ is valid on every algebra in the class K(R). From this we may deduce that the variety V(L) is indeed generated by the class K(R) which by Proposition 2.8 is a universal class of Heyting algebras axiomatised by universal clauses in the {0, ∧, 1}-reduct as R is a collection of structural hypersequent rules. Conversely, if the variety V(L) is generated by a universal class of Heyting algebras axiomtised by universal clauses in the {0, ∧, 1}-reduct, say U, then by Proposition 2.10 there exists a set R U of structural hypersequent rules such that the class K(R U ) of Heyting algebras validating R U coincides with the class U. Since by assumption U generates V(L) we have that ϕ is a theorem of L iff |= U 1 ≈ ϕ. Therefore, we have that ϕ ∈ L precisely when |= K(R U ) 1 ≈ ϕ, which by Theorem 2.7 is the case exactly when HLJ+R U ⇒ ϕ. Thus we may conclude that HLJ + R U is a structural hypersequent calculus for L.
We finish this section with presenting a syntactic characterisation of the intermediate logics admitting a structural intermediate hypersequent calculus [22]. Definition 2.12. (cf. [26]) Let P 0 = N 0 be a (countable) set of propositional variables and define sets of formulas P n , N n in the language of intuitionistic logic by the following grammar The key insight is that using the invertible rules of HLJ, i.e., rules the premises of which are derivable whenever the conclusion is, any P 3 -formula can be transformed into a structural hypersequent rule which preserves the redundancy of the cut-rule when added to HLJ. Thus any intermediate logic axiomatisable by P 3 -formulas, i.e., any logic of the form IPC + {ϕ i } i∈I with ϕ i ∈ P 3 for all i ∈ I, admits a structural hypersequent calculus in which the cut-rule is redundant. 3. The logic L is axiomatisable by P 3 -formulas.
Proof. That items (1) and (2) are equivalent is established in [22], just as the fact that item (3) entails item (1). That item (1) entails item (3) may be seen via an argument analogous to the one used to prove [24,Prop. 7.5].
We supply the details. Given a structural hypersequent rule (r), as in the proof of Proposition 2.8, there is a finite set of equivalent universal clauses such that the variables x and y are disjoint and the terms t are (possible empty) meets of variables (i.e., the constant 1) and the terms u are either a single variable or the constant 0. In fact, we may assume that none of the terms t i are the constant 1 for i ∈ {1, . . . , m}. Let ϕ r be the formula It is straightforward to verify that ϕ r belongs to P 3 . We claim that q r and ϕ r are equivalent on any Heyting algebra with a second greatest element. Therefore, suppose that A is such a Heyting algebra. If A |= ϕ r then for any valuation ν on A we have that Conversely, if A |= q r then A also validates any substitution instance of q r . We construct a substitution σ such that From the existence of such a substitution we may conclude that for any valuation ν on A there exists To construct the substitution σ we let σ( x ∈ x and σ(y) := y, for each variable y ∈ y. Note that since x and y are disjoint this is well-defined. It is now easy to verify that this substitution has the desired properties and therefore that A |= ϕ r . This shows that in fact any Heyting algebra validating the clause q r also validates the formula ϕ r .
Remark 2.14. Note that the proof of Theorem 2.13 presented in [22] is semantic in nature and so does not directly yield an explicit procedure for transforming a derivation using the cut-rule into a cut-free derivation. However, in concrete cases an explicit cut-elimination procedure may be given, see, e.g., [20,21]. We also want to emphasise that it is not the case that the cut-rule is redundant in every structural intermediate hypersequent calculus but only that any such calculus is effectively equivalent a structural intermediate hypersequent calculus in which the cut-rule is redundant.
Example 2.15. For n ≥ 1 let BTW n , BW n and BC n , be the intermediate logics determined by intuitionistic Kripke frames of top width at most n, of width at most n, and of cardinality at most n, respectively. All of these intermediate logics have axiomatisations given by formulas which are ostensibly P 3 , see, e.g., [18,Chap. 2], and so by Theorem 2.13 all of these logics admit a hypersequent calculus of the form HLJ+R, with R a set of structural hypersequent rules, for which the cut-rule is redundant. Concretely, the rules, (com) and (lq) yields cut-free structural intermediate hypersequent calculi for the intermediate logics LC and KC, respectively, when added to HLJ.

Algebraic Characterisation
In this section we provide a semantic characterisation of the intermediate logics admitting a structural-and therefore also a cut-free-intermediate hypersequent calculus in terms of the algebraic semantics. This section builds on the theory of (0, ∧, ∨, 1)-stable intermediate logics as developed in [7,10] where these logics are simply called stable intermediate logics.
In this case we write A → Σ B, and say that the algebra A is a Σ-subalgebra of the algebra B.
1. We say that a class K of Heyting algebras is (finitely) Σ-stable provided that whenever B ∈ K and A → Σ B then A ∈ K for all (finite) Heyting algebras A; 2. We say that an intermediate logic L is (finitely) Σ-stable provided that whenever B ∈ V(L) si and A → Σ B then A ∈ V(L) for all (finite) Heyting algebras A.
Definition 3.6. Let Σ ⊆ {0, ∧, ∨, →, 1}, let A be a finite Heyting algebra and introduce for each element a ∈ A a distinct first-order variable x a . By Remark 3.7. Stable universal clauses may be seen as a propositional version of diagrams as known from classic Robinson-style model theory, see, e.g., [36,Chap. 1.4]. Variants of the Σ-stable clauses defined above have been studied before under the names stable and canonical multi-conclusion rules [8][9][10]41].
The following lemma shows that the Σ-stable clause associated with a finite algebra A encodes the property of not containing A as a Σ-subalgebra.
We then show that a universal class of Heyting algebras is Σ-stable precisely if it is axiomatisable by Σ-stable clauses. 2. The universal class U is finitely Σ-stable; 3. The universal class U is axiomatised by Σ-stable clauses.
Proof. If U is a (universal) class axiomatised by Σ-stable clauses then U must be Σ-stable, since universal clauses in the Σ-reduct of the language of Heyting algebras are preserved by Σ-subalgebras. Moreover, any Σ-stable universal class is evidently finitely Σ-stable.
Thus it remains to be shown that if U is finitely Σ-stable then U is axiomatised by Σ-stable clauses. Therefore, let U be a finitely Σ-stable universal class and let Q = {q Σ (A) : |A| < ℵ 0 , A ∈ U}. We claim that for any Heyting algebra B we have that B ∈ U iff B |= Q. To see this let Th ∀ HA (U) be the universal theory, in the language of Heyting algebras, of U. If B ∈ U then, by the assumption that U is a universal class of Heyting algebras, there exist a universal clause q ∈ Th ∀ HA (U) such that B |= q. Thus, by [10,Lem. 4.3] we must have a finite (0, ∧, ∨, 1)-subalgebra, in particular a Σ-subalgebra, C of B such that C |= q, i.e., C ∈ U whence q Σ (C) ∈ Q. By Lemma 3.8 we must have that B |= q Σ (C) and so B |= Q.
Conversely, if B |= Q then for some finite Heyting algebra A ∈ U we have B |= q Σ (A). By Lemma 3.8 it follows that A is a Σ-subalgebra of B.
Since U is assumed to be finitely Σ-stable we must conclude that B ∈ U since otherwise A ∈ U.
We then obtain the first necessary and sufficient conditions in terms of V(L) for an intermediate logic L to admit a structural intermediate hypersequent calculus. Proof. The equivalence between items (1) and (2) is contained in Theorem 2.13. That items (2) and (3) are equivalent follows from Corollary 2.11 and Lemma 3.9.
In principle Proposition 3.10 gives an algebraic characterisation of the intermediate logics L which admit structural intermediate hypersequent calculi in the sense of Definition 2.2. However, we wish to obtain a characterisation which is local in the sense that it pertains to properties of-individualalgebras in the variety V(L) and not the variety V(L) taken as a whole. We will obtain such a characterisation by showing that the varieties of Heyting algebras generated by (0, ∧, 1)-stable universal classes are precisely the varieties corresponding to (0, ∧, 1)-stable logics.

(0, ∧, 1)-Stable Logics
In this subsection we characterise the (0, ∧, 1)-stable logics in terms of properties of the subdirectly-irreducible algebras in the corresponding variety V(L). As discussed above this will yield a characterisation of the intermediate logics admitting cut-free structural intermediate hypersequent calculi.
Definition 3.11. Let Σ ⊆ {0, ∧, ∨, →, 1}, let A be a finite Heyting algebra and introduce for each element a ∈ A a distinct variable x a . By the Σstable equation ε Σ (A) associated with A we shall understand the equation The Σ-stable equations encode information about finite Heyting algebras in almost the same way as the Σ-stable clauses. However, a version of Lemma 3.8 only obtains for so-called well-connected Heyting algebras, that is, Heyting algebras validating the universal clause ∀x∀y(1 ≤ x ∨ y =⇒ 1 ≤ x or 1 ≤ y). Note that every subdirectly irreducible Heyting algebra will be well-connected and that every finite well-connected Heyting algebra will be subdirectly irreducible, see, e.g., [12,Thm. 2.3.14] and the references therein.
We will need the following lemma showing that homomorphic images of a finite Heyting algebra A must also be (0, ∧, 1)-subalgebras of A. Proof. If h : A B is a surjective Heyting algebra homomorphism then B is isomorphic to A/F , as a Heyting algebra, for some filter F on A. As A is finite the filter F must be a principal filter, say F = ↑a for some a ∈ A, and therefore B ∼ = [0, a]. Evidently, we have a (0, ∧, 1)-embedding f from [0, a] into A, given by showing that B → 0,∧,1 A.
Lemma 3.14. Let U be a universal class axiomatised by a collection {q 0,∧,1 (A i )} i∈I of (0, ∧, 1)-stable universal clauses. Then the variety generated by the class U is axiomatised by the (0, ∧, 1)-stable equations Proof. Let V be the variety determined by the (0, ∧, 1)-stable equations {ε 0,∧,1 (A i )} i∈I . Lemma 3.8 together with item (1) of Lemma 3.13 implies that U ⊆ V. Furthermore, from item (2) of Lemma 3.13 we may conclude that any subdirectly irreducible V-algebra belongs to U. Consequently, being a subclass of the variety V containing all subdirectly irreducible V-algebras, the class U must necessarily generate the variety V. Proof. By [17, Thm. V.2.20] we know that the universal class generated by the class K is given by ISP U (K). Therefore, let {B i } i∈I be a collection of K-algebras, U an ultrafilter on I and A a finite Heyting algebra. If A → Σ B i for all i ∈ I then by Lemma 3.8 we have that B i |= q Σ (A) for all i ∈ I and hence by Los' Theorem we obtain that i∈I B i /U |= q Σ (A) and A is a finite algebra such that A → Σ B then necessarily A → Σ B for some B ∈ P U (K) whence by the above we have that A ∈ K. We have thus shown that ISP U (K) is a finitely Σ-stable universal class and as such it must be Σ-stable by Lemma 3.9. Proof. It follows from Lemma 3.15 that item (2) entails item (3). Furthermore, it follows from [7, Lem. 3.6] that if V(L) is generated by a (0, ∧, 1)stable universal class of Heyting algebras, say U, then V(L) is also generated by the (0, ∧, 1)-stable class of the finite Heyting algebras belonging to U. We proceed to show that the items (1) and (3) are equivalent.
For that purpose, assume that the logic L is (0, ∧, 1)-stable and define and let K be the collection of finite Heyting algebras not belonging to K . Then for any subdirectly irreducible Heyting algebra B we may observe the following: If B ∈ V(L) and A ∈ K then we must have that B |= ε 0,∧,1 (A), since otherwise A → 0,∧,1 B by Lemma 3.13 entailing that A ∈ K , in direct contradiction with the assumption that A ∈ K. Conversely, if B ∈ V(L) then by [7, Lem. 3.6] we have a finite (0, ∧, ∨, 1)-subalgebra A of B such that A ∈ V(L). In particular A is a (0, ∧, 1)-subalgebra of B and hence by Lemma 3.13 B |= ε 0,∧,1 (A), as B, being subdirectly irreducible, is wellconnected. Moreover, we must have that A ∈ K since otherwise, as A is finite, we would have A ∈ K and hence A ∈ V(L) by the assumption that L is (0, ∧, 1)-stable. We have thus shown that for any subdirectly irreducible Heyting algebra B we have Consequently, the variety V(L) is axiomatised by the collection of (0, ∧, 1)stable equations {ε 0,∧,1 (A)} A∈K . From Lemma 3.14 we may then conclude that V(L) is identical to the variety generated by the universal class of Heyting algebras determined by the collection of universal (0, ∧, 1)-stable clauses {q 0,∧,1 (A)} A∈K . Lastly, assume that V(L) is generated by a (0, ∧, 1)-stable universal class say U. By Lemma 3.9 we have that U is determined by a collection of (0, ∧, 1)-stable universal clauses, say {q 0,∧,1 (A i )} i∈I . By Lemma 3.14 it then follows that V(L) is determined by the collection of (0, ∧, 1)-stable equations {ε 0,∧,1 (A i )} i∈I . Consequently, if B is a subdirectly irreducible V(L)-algebra and A is a (0, ∧, 1)-subalgebra of B, then from A ∈ V(L) we can conclude that A |= ε 0,∧,1 (A i ) for some i ∈ I and so by item (1) of Lemma 3.13 it follows that A i → 0,∧,1 A and hence that A i → 0,∧,1 B whence from item (2) of Lemma 3.13 we obtain that B |= ε 0,∧,1 (A i ) in direct contradiction with the assumption that B is a V(L)-algebra.
We may then obtain the following algebraic characterisation of the intermediate logics admitting a structural intermediate hypersequent calculus and therefore by Theorem 2.13 also a cut-free structural intermediate hypersequent calculus. Proof. This follows directly from Theorem 3.16 and Proposition 3.10.
Remark 3.18. Note that if L is a finitely axiomatisable (0, ∧, 1)-stable intermediate logic then L admits a structural intermediate hypersequent calculus given by only finitely many structural hypersequent rules. To see this simply note that being (0, ∧, 1)-stable V(L) is axiomatised by a collection of (0, ∧, 1)-equations and and since L is finitely axiomatisable we may conclude that only finitely many of the (0, ∧, 1)-stable equations are needed to axiomatise V(L). Hence by Lemma 3.14 V(L) is determined by a finite number of (0, ∧, 1)-stable clauses. Thus from the correspondence between (0, ∧, 1)-stable clauses and structural hypersequent rules we obtain that L indeed admits a structural intermediate hypersequent calculus given by only finitely many structural hypersequent rules.

Applications
We here present some consequences of Corollary 3.17.
Remark 3.21. Note that Proposition 3.20 gives an alternative way of seeing that every finitely axiomatisable intermediate logic admitting a structural hypersequent calculus is decidable. This of course also already follows from the fact that such logics admit a cut-free intermediate hypersequent calculus given by finitely many rules. Remark 3.23. As we will see in Section 4 any (0, ∧, 1)-stable logic is in fact elementary from which the canonicity of such logics may also be inferred by Fine's Theorem, see, e.g., [18,Thm. 10.22]. Furthermore, every (0, ∧, 1)stable logic must also be a (locally) cofinal subframe logic. This can be seen either by considering the frame characterisation presented in Section 4 or by an argument similar to the one presented in Section 3.1. From this fact it can also be inferred that (0, ∧, 1)-stable logics must be both canonical and elementary [53].
We conclude this section by drawing attention to some negative consequence of Corollary 3.17. Proof. We know that for n ≥ 2 the logic BD n is not (0, ∧, ∨, 1)-stable [7,Thm. 7.4(2)] and so in particular it cannot be (0, ∧, 1)stable. Knowing this the proposition is an immediate consequence of Corollary 3.17.
Remark 3.25. That this was the case had been expected in the literature, see, e.g., [26,27]. However, we have not been able to find any proof of this fact before. The logic BD 2 does, however, admit an analytic hypersequent calculus obtained by adding an additional logical hypersequent rule for the introduction of the implication to the multi-succedent hypersequent calculus HLJ [26]. Furthermore, the logics BD n , for n ≥ 2, do admit analytic display calculi [27], analytic labelled sequent calculi [30] as well as so-called pathhypertableau and path-hypersequent calculi [19].
As a final application of the algebraic characterisation of the intermediate logics admitting structural intermediate hypersequent calculi we give a semantic proof of [22,Cor. 7.2].
Proposition 3.26. Let (r) be a structural sequent rule. Then either the calculus LJ + (r) is inconsistent or the rule (r) is derivable in LJ.
Proof. Let (r) be a structural sequent rule, i.e., a structural hypersequent rule the premises and conclusion of which only consists of single component hypersequents. Let K(r) be the class of Heyting algebras validating (r). If LJ + (r) is consistent then the class K(r) is non-trivial. In particular the two element Boolean algebra 2 will belong to K(r), as K(r) is closed under subalgebras and 2 is a subalgebra of every non-trivial Heyting algebra. Because (r) is a sequent rule the class K(r) will not only be a universal class but in fact a quasi-variety and as such closed under the formation of direct products. Since any bounded distributive lattice can be realised as a subdirect product of the lattice 2 [5, Thm. II.10.1] it follows that any Heyting algebra A will be a (0, ∧, ∨, 1)-subalgebra of some member of K(r). Therefore, since (r) is structural and so the class K(r) is (0, ∧, 1)-stable, we may conclude that A ∈ K(r), showing that every Heyting algebra validates the rule (r). Given this the proposition then follows from Theorem 2.7.
Remark 3.27. Note that the proof of Proposition 3.26 shows the stronger claim that any structural multi-succedent sequent rule is either derivable in LJ or derives every formula in LJ , where LJ is the multi-succedent version of LJ.

Frame Based Characterisation
In this section we identify the first-order frame conditions which determine (0, ∧, 1)-stable logics. This is done using the duality theory for (0, ∧, 1)homomorphism between Heyting algebras developed in [11].
Given a Heyting algebra A we let A + denote the underlying intuitionistic Kripke frame of the Esakia space A * dual to A, i.e., the Kripke frames consisting of the set of prime filters of A ordered by set-theoretic inclusion. Similarly, given an intuitionistic Kripke frame F (Esakia space X) we let F + (X * ) denote the Heyting algebra of (clopen) upsets of F (X * ). In the following we will use that A ∼ = (A + ) + for every finite Heyting algebra A, see, e.g., [ where ClpUp(X) denotes the clopen upset of X and R (U ) := {x ∈ X : Moreover if R −1 [Y ] = X we say that R is total and if for every y ∈ Y there is x ∈ X such that R[x] = ↑y then we say that R is onto.
Remark 4.2. Finite intuitionistic Kripke frames may be identified with finite Priestley spaces and so, forgetting the topology, we will also speak about generalised Priestley morphisms between finite intuitionistic Kripke frames.
for all x, x ∈ X such x ≤ X x . This observation will be useful when proving Proposition 4.8 and Lemmas 5.5 and 5.6 below.
We are interested in total generalised Priestley morphisms because they are the duals of (0, ∧, 1)-homomorphisms. To be precise we have the following theorem. This allows us to translate questions about (0, ∧, 1)-homomorphisms between Heyting algebras into questions about total generalised Priestley morphism between their dual spaces.
Recall [46] that a geometric axiom is a first-order sentence of the form with ϕ, ψ 1 , . . . , ψ m conjunctions of atomic formulas and the variable v not occurring free in ϕ. A geometric implication is then taken to be a finite conjunction of geometric axioms.
Definition 4.5. (cf. [42]) We say that a geometric axiom ∀ w (ϕ( w) =⇒ ∃v OR m j=1 ψ j ( w, v)) is simple if 1. There exists w 0 ∈ w such that ϕ( w) is the conjunction of the atomic formulas {w 0 ≤ w} w∈ w ; 2. Every atomic subformula of ψ j ( w, v) is of the form w ≤ w or w ≤ v for w, w ∈ w.
A simple geometric implication is then a conjunction of simple geometric axioms.
Example 4.6. The intermediate logics BTW n , BW n , BC n , for n ≥ 1, are all complete with respect to an elementary class of intuitionistic Kripke frames determined by simple geometric implications. Furthermore the logics BD n , for n ≥ 2, are all complete with respect to an elementary class of intuitionistic Kripke frames determined by geometric implications, namely , which are ostensibly not simple.
Remark 4.7. Intermediate (and modal) logics determined by a class of Kripke frame defined by geometric implications have been shown to admit so-called labelled sequent calculi [30,46,49]. Thus as a consequence of Proposition 4.13 below we obtain that any (0, ∧, 1)-stable logic admits a cut-free labelled sequent calculus. This is consistent with the existence of a translation of hypersequents into labelled sequents, see, e.g., [48] for an overview.
Finally, a variant of the simple geometric implications appears in the work of Lahav [42] where these are used to construct analytic hypersequent calculi for modal logics which are sound and complete with respect to a class of Kripke frames determined by such simple geometric implications.
Proof. It suffices to show that ( †) holds for an arbitrary simple geometric axiom θ, say, , v)). Therefore, assume that (X, ≤ X ) |= θ. Suppose that y 0 , . . . , y k−1 ∈ Y are such that ϕ(y 0 , . . . , y k−1 ) holds in (Y, ≤ Y ), then by the assumption that R is an onto generalised Priestley morphism there are x 0 , . . . , that ϕ(x 0 , x 1 , . . . , x k−1 ) and so since (X, ≤ X ) |= θ there is z ∈ X such that ψ l ( x, z) holds in (X, ≤ X ) for some l ∈ {1, . . . , m}. Furthermore, since R is total we have z ∈ Y such that zRz . We claim that ψ l ( y, z ) holds in (Y, ≤ Y ). If x t ≤ X x t in (X, ≤ X ) for some t, t ∈ {0, . . . , k − 1} then we have that ↑y t = R[x t ] ⊆ R[x t ] = ↑y t and thus y t ≤ Y y t . Similarly if x t ≤ X z for some t ∈ {0, . . . , k − 1}, then we have that z ∈ R[z] ⊆ R[x t ] = ↑y t and hence that y t ≤ z . This shows that (Y, ≤ Y ) satisfies θ. Definition 4.9. Recall that a variety V of Heyting algebras is elementarily determined if there exists an elementary class of intuitionistic Kripke frames F such that the variety is generated by the class of complex algebras F + := {F + : F ∈ F}, with F + denoting the Heyting algebra of upsets of F.  Proof. Let L be an intermediate logic characterised by a class of intuitionistic Kripke frames, say F, defined by simple geometric implications. Note that if ϕ is a formula in the language of propositional intuitionistic logic such that F + |= ϕ then there exists some point-generated subframe G of F such that G + |= ϕ. Therefore, since simple geometric implications are evidently preserved by taking generated subframes, we obtain that V(L) is in fact determined by the class of rooted intuitionistic Kripke frames belonging to F. Furthermore, any filtration F of an intuitionistic Kripke frame F induces an order preserving surjection f : F F . Therefore, since on rooted frames any (simple) geometric implication is equivalent to a positive first-order formulas and such formulas are preserved by order-preserving surjections we see that simple geometric implications will be preserved under taking filtrations of rooted intuitionistic Kripke frames. Consequently, we obtain that V(L) is in fact determined by the finite rooted members of F. In particular V(L) will be generated by the class of complex algebras obtain from the set G := {F ∈ F : |F| < ℵ 0 , F rooted}. Finally, letting K := {A : ∃B ∈ G + (A → 0,∧,1 B)} it follows from Proposition 4.8 together with Theorem 4.4 that K is a (0, ∧, 1)-stable class of Heyting algebras generating V(L) and therefore by Theorem 3.16 that the logic L is (0, ∧, 1)-stable.
To establish the converse of Corollary 4.11 we need the following lemma. Proof. Let a universal (0, ∧, 1)-clause q be given. As before q is equivalent to a finite conjunction of universal clauses of the form u 1 ( y) and . . . and t m such that every term t k is a {∧, 1}-term and every term u k is either 0 or a single variable. Thus, by [25,Thm. 4.15] we may without loss of generality assume that (i) x and y are disjoint, (ii) every variable in q occurs exactly once on the right-hand side of q . First we will show that for every such clause q there exists a simple geometric axiom θ q such that for every rooted intuitionistic Kripke frame F = (W, ≤) we have that F |= θ q iff F + |= q . From which we obtain a simple geometric implication θ q such that for every rooted intuitionistic Kripke frame F we have that F |= θ q iff F + |= q.
In the following we write P ( x, y) for the left-hand side of the clause q , and for k ∈ {1, . . . , n} we let x k 1 , . . . , x k m k denote the variables occurring in term t k ( x), if any, and let y k 0 denote the variable occurring in the term u k ( y), if any. Thus we then have that and w j ∈ u j ( V )) The special syntactic shape of the clause q ensures that that the secondorder variables among U only occur negatively in P ( U, V ) and that the second-order variables among V only occur positively in P ( U, V ). Moreover, every second-order variable among U, V occurs exactly once somewhere on the right-hand side. This allows us to eliminate all the second-order variables via a standard and straightforward application of the Ackermann Lemma [1], see, e.g., [28,Lem. 0.1], to obtain that for some collection w of first-order variables. Thus we see that ). This shows that q is equivalent to a formula in the first-order language of intuitionistic Kripke frames.
To see that q is equivalent to a simple geometric implication on rooted intuitionistic Kripke frames simply note that if for some i ≤ m we have that the variable w i 0 occurs as one of the variables w i k , say w i k , then it must be the case that On the other hand if for some i ≤ m we have that the variable w i 0 does not occur as one of the variables w i k then we must have that . Thus we obtain a formula ψ( w, v), which is a disjunction of conjunctions of atomic formulas of the form w ≤ w and w ≤ v, such that Finally, letting θ q be the formula ∀w 0 ∀ w(AND w∈ w (w 0 ≤ w)=⇒∃vψ( w, v)), for w 0 some fresh first-order variable, we obtain a simple geometric axiom such that θ q is equivalent to q on rooted intuitionistic Kripke frames. Proof. Given a variety V of Heyting algebras generated by a (0, ∧, 1)-stable universal class, say U, axiomatised by (0, ∧, 1)-stable clauses, say {q i } i∈I , we see, by an argument completely similar to the one found in the proof of Corollary 4.11, that V will be generated by the class F + := {F + : ∀i ∈ I (F |= θ q i )}, where θ i is the simply geometric implication corresponding to q i obtain from Lemma 4.12.

A B
We easily see that A is a (0, ∧, 1)-subalgebra of B but not a (0 Despite the fact that there are (0, ∧, ∨, 1)-stable logics which are not (0, ∧, 1)-stable all the examples of (0, ∧, ∨, 1)-stable logics considered so far [7,Sec. 7] are in fact (0, ∧, 1)-stable. The following theorem may be seen as explaining why this indeed the case. Furthermore, this also provide us with examples of (0, ∧, ∨, 1)-stable logics which are not (0, ∧, 1)-stable. 3. The Heyting algebra A is weakly projective as an object in the category DL of distributive lattices and lattice homomorphism.
Proof. Evidently item 2 entails item 1. Conversely, to see that item 1 entails item 2, it suffices, due to Lemma 3.8, to show for any Heyting algebra B that Since and so since, by assumption, q 0,∧,∨,1 (A) is equivalent to collection of universal (0, ∧, 1)-clauses and such clauses are preserved by (0, ∧, 1)-embeddings we must have that A |= q 0,∧,∨,1 (A) which is absurd as every Heyting algebra refutes all of the stable clauses associated with it. To see that item 3 entails item 2 suppose that A is weakly projective as an object in the category DL. We claim that the (0, ∧, ∨, 1)-stable clause q 0,∧,∨,1 (A) associated with A is equivalent to the (0, ∧, 1)-stable clause q 0,∧,1 (A) associated with A. As before it suffices to show that A → 0,∧,1 B =⇒ A → 0,∧,∨,1 B. Therefore, suppose that A → 0,∧,1 B, say via h : A → B. Since A is assumed to be weakly projective as an object in the category DL it follows from a well-known result [6] that the poset J 0 (A) of join-irreducibles 7 of A including the element 0 is a (0, ∧)subalgebra of A. Moreover, by the assumption that A is well-connected J 0 (A) will in fact be a (0, ∧, 1)-subalgebra of A. Consequently, restricting h to J 0 (A) we obtain a (0, ∧, 1)-homomorphism h 0 : J 0 (A) → 0,∧,1 B. Because A is weakly projective we have by [6,Thm. 4] that h 0 has a unique extension to a (0, ∧, ∨)-homomorphism h 0 : A → 0,∧,∨ B. In fact, since h 0 is an extension of h 0 and 1 ∈ J 0 (A) we obtain that h 0 (1) = h 0 (1) = h(1) = 1. Thus, h 0 : A → 0,∧,∨,1 B leaving us with the task of proving that h 0 is injective. By direct inspection of the construction of the map h 0 it may easily be verified that h 0 (a) ≤ h(a) for all a ∈ A. As a consequence of this we see that if h 0 (a 1 ) = h 0 (a 2 ) for some a 1 , a 2 ∈ A then for each a 1 ∈ J 0 (A) ∩ ↓a 1 we must have From this and the fact that h is a (0, ∧, 1)-embedding we may conclude that a 1 ≤ a 2 for all a 1 ∈ J 0 (A) ∩ ↓a 1 . By a completely analogous argument we may deduce that a 2 ≤ a 1 for all a 2 ∈ J 0 (A) ∩ ↓a 2 . Since A is finite every element is uniquely determined by the set of join-irreducible elements below it and so we must have that a 1 = a 2 and therefore that h : A → 0,∧,∨,1 B, as desired.
Conversely, to see that item 2 entails item 3 suppose that A is not weakly projective as an object in the category DL. We exhibit a (finite) Heyting algebra B such that A → 0,∧,1 B but A → 0,∧,∨,1 B, showing that the universal clauses q 0,∧,1 (A) and q 0,∧,∨,1 (A) are not equivalent. To this effect let P := J(A) ∂ , be the order dual of J(A). Note that the Heyting algebra Up(P ) of upsets of P is isomorphic to A, as A is finite. Again, by the characterisation of finite weakly projective distributive lattices [6], A not being weakly projective entails the existence of a 1 , a 2 ∈ J(A) such that a 1 ∧a 2 ∈ J 0 (A), in particular a 1 and a 2 must be incomparable. Let b 1 , . . . , b n be the set of join-irreducibles which are below a 1 ∧a 2 in A. Necessarily, n ≥ 2. Given this, let P be the poset obtained from P by adding a new element a 0 covering a 1 , a 2 and covered by b 1 , . . . , b n . Thus |P | = |P | + 1. Evidently there can be no order-preserving surjection from P onto P , since this would entail that a 1 and a 2 are comparable. Consequently, letting B denote the dual Heyting algebra Up(P ) of P , this shows that A → 0,∧,∨,1 B. We claim, however, that A → 0,∧,1 B. To establish this it suffice by Theorem 4.4 to exhibit a total and onto generalised Priestley morphism R ⊆ P × P . We claim that letting R ⊆ P × P be given by is such a generalised Priestley morphism. It may readily be verified that R is a generalised Priestley morphisms. Moreover, that R is total and onto is evident from the definition.
Remark 5.3. We note that the use of generalised Priestley morphisms in the proof of Theorem 5.2 can be avoided. To see this observe, with the notation of the proof of Theorem 5.2, that P is, in fact, a subposet of P , whence by Priestley duality we have a surjective map h : B 0,∧,∨,1 A. Moreover, we may easily verify that h −1 (0) = {0} and h −1 (1) = {1}. From [37,Cor. 5.4] we know that A, being a finite distributive lattice, is projective as a meetsemilattice and therefore we obtain a map h : A → ∧ B such that h • h is the identity on A. In particular, h must be injective and as h(h(0)) = 0 and h(h(1)) = 1 it follows that h(0) = 0 and h(1) = 1. Thus we have h : A → 0,∧,1 B.
For this we need two simple lemmas.  1 ] and consequently that f is order-preserving when considering W 1 as a subframe of W 1 . We then note that for any w 1 ∈ max(W 1 ) since S is total we have w 2 ∈ W 2 such that w 1 Sw 2 . Moreover, if for some v 1 ∈ W 1 we have v 1 ≤ 1 w 1 then S[w 1 ] ⊆ S[v 1 ] = ↑f (v 1 ) as S is a generalised Priestley morphism. Thus we may define g : W 1 ∪ max(W 1 ) → W 2 by letting g(w 1 ) = f (w 1 ) if w 1 ∈ W 1 and letting g(w 1 ) be some element of S[w 1 ] if w 1 ∈ max(W 1 )\W 1 . Since S is total this is a well-defined order-preserving map. Evidently W 1 ∪ max(W 1 ) is a cofinal subframe of W 1 and so F 2 is the image of a cofinal subframe of F 1 under an order-preserving map.
Proof. From Lemma 5.5 we know that F 2 is the image of a cofinal subframe F 1 := (W 1 , ≤ 1 ) of F 1 , under an order-preserving map, say f : W 1 → W 2 . Let w 0 be the root of F 1 . If w 0 ∈ W 1 then there is nothing to show. If w 0 is not in W 1 then letting W 1 := W 1 ∪ {w 0 } we obtain a rooted cofinal subframe F 1 of F 1 . Similarly, by adjoining a new root w 0 to W 2 we obtain a rooted frame F 2 of which F 2 is a cofinal subframe. Finally, the map f extends to a surjective order-preserving map from F 1 to F 1 by mapping w 0 to w 0 . Proof. Every (0, ∧, 1)-stable logic is evidently (0, ∧, ∨, 1)-stable. Furthermore, by Proposition 4.13 every (0, ∧, 1)-stable logic is sound and complete with respect to a class of intuitionistic Kripke frames determined by simple geometric implications. It is straightforward to verify that such first-order formulas are preserved by taking cofinal subframes. Consequently, being generated by a class of Kripke frames closed under cofinal subframes it follows that any (0, ∧, 1)-stable logic is indeed a cofinal subframe logic.
Conversely, suppose that L is a cofinal subframe logic which is (0, ∧, ∨, 1)stable. We show that if A and B are finite Heyting algebras with B subdirectly irreducible such that B ∈ V(L) and A → 0,∧,1 B then A ∈ V(L). Therefore, let F = (W, ≤) be the dual intuitionistic Kripke frame of B and let F = (W , ≤ ) be the dual intuitionistic Kripke frame of A. By the assumption that A → 0,∧,1 B we have a total and onto generalised Priestley morphism S ⊆ W × W . Moreover, since B is subdirectly irreducible we have that F is rooted and hence by Lemma 5.6 that F is a cofinal subframe of an image G under an order-preserving map of a rooted cofinal subframe G of F. Since by assumption L is a cofinal subframe logic we must have that G is also an L-frame. Moreover, since G is rooted and L is (0, ∧, ∨, 1)-stable we obtain that G is an L-frame [7,Thm. 6.7], and so, again using the fact that L is a cofinal subframe logic, we see that F is an L-frame. We may therefore conclude that A ∈ V(L) as desired. It follows that V(L) is generated by a (0, ∧, 1)-stable class of Heyting algebras and therefore that L is (0, ∧, 1)-stable.
Remark 5.8. It is known that there are continuum-many (0, ∧, ∨, 1)-stable logics [7, Thm. 6.13] just as it is known that there are continuum-many cofinal subframe logics [18,Thm. 11.19]; however, we have not been able to determine how many (0, ∧, 1)-stable logics there are. The problem is that, using duality, we cannot work with order-preserving surjections which are the duals of (0, ∧, ∨, 1)-embeddings, but we must work with total and onto generalised Priestley morphism which are more complicated. In this context it is no longer clear if an argument similar to the one presented in [7] will work.

Future Work
We conclude by mentioning a number of open questions arising in the context of the present work.
It is not clear if the property of being (0, ∧, 1)-stable can be effectively verified for finitely axiomatisable intermediate logics.  8 Much of the original work on algebraic proof theory has been done in the context of substructural logic. Consequently, we find it worth investigating if a similar characterisation of substructural logics admitting structural hypersequent calculi can be given. However, since the subdirectly irreducible residuated lattices are more complicated than their Heyting algebra counterparts it is not immediately clear if all the necessary results transfer to the setting of substructural logic. Here the work of Bezhanishvili et al. [14], investigating canonical formulas in the context of certain substructural logics might be helpful.
Finally, it is our hope that the findings in Section 4 will help to bridge the gap between the different approaches to systematic proof theory regarding substructural, intermediate and modal logics. Some of these approaches are primarily based on the algebraic semantics [22,24,25,35], while others primarily make use of the relational semantics [30,42,43].