Admissible Bases Via Stable Canonical Rules

We establish the dichotomy property for stable canonical multi-conclusion rules for IPC, K4, and S4. This yields an alternative proof of existence of explicit bases of admissible rules for these logics.


Introduction
An inference rule is admissible in a given logical system L if no new theorems are derived by adding this rule to the rules of inference of L. Friedman [10] raised the question whether admissibility of rules in the intuitionistic propositiolculus (IPC) is decidable. A solution to this problem for IPC, as well as for well-known systems of modal logic such as K4 and S4, was first given by Rybakov ([26,27], see also the comprehensive book [24] and the references therein). An alternative solution via projectivity and unification was supplied in [11,12]. Explicit bases for admissible rules were built in [15,17,22,23,25]. We refer to Goudsmit [14] for a modern historic account of the admissibility problem.
Recently Jeřábek [18] developed a new technique for building bases for admissible rules by generalizing Zakharyaschev's canonical formulas [29] to multi-conclusion canonical rules, and by developing the dichotomy property for canonical rules. This property states that a canonical multi-conclusion rule is either admissible or equivalent to an assumption-free rule. Our goal is to establish the same property for stable multi-conclusion canonical rules for IPC, K4, and S4. These rules were recently introduced in [1], where it was shown that each normal modal multi-conclusion consequence relation is axiomatizable by stable multi-conclusion canonical rules. The same result for intuitionistic multi-conclusion consequence relations was established in [2].
The proof methodology we follow is similar to [18] and goes through a semantic characterization of non-admissible stable canonical rules in terms of the finite domains they are built from. In spite of the similarities, the semantic characterization we obtain is different than the one given in [18]. As a simple corollary of our main theorem, similarly to [18], we obtain decidability of the admissibility problem for IPC, K4 and S4. Finally, we note that admissibility for the basic modal logic K is a long standing open problem. While the proofs of this paper do not directly apply to K, we observe that the method of stable canonical rules, unlike that of canonical rules of [18], is not limited to the transitive case. Therefore, our method is potentially applicable to non-transitive logics such as K.
The paper is organised as follows: In Section 2 we recall Esakia duality for Heyting algebras, multi-conclusion consequence relations and stable canonical rules for IPC. In Section 3 we obtain an explicit basis of admissible rules for IPC via stable canonical rules and prove that the latter have the dichotomy property. In Section 4 we recall duality for modal algebras, modal multi-conclusion consequence relations and stable canonical rules for modal logic. Finally, in Section 5 we obtain explicit bases of admissible rules for K4 and S4 via stable canonical rules and prove their dichotomy property.

Esakia Duality for Heyting Algebras
We recall that a Heyting algebra is a bounded distributive lattice with an additional binary operation → that is the residual of ∧. For Heyting algebras A and B, a Heyting homomorphism is a bounded lattice homomorphism h : A → B such that h(a → b) = h(a) → h(b) for each a, b ∈ A. Let Heyt be the category of Heyting algebras and Heyting homomorphisms. It is well known (see, e.g., [21,Chap. IX] or [6,Chap. 7]) that Heyting algebras provide an adequate algebraic semantics for superintuitionistic logics. In fact, there is a dual isomorphism between the (complete) lattice of superintuitionistic logics and the (complete) lattice of varieties of Heyting algebras.
In order to introduce topological duality for Heyt, we need to fix some notation for posets. If X is a poset (partially ordered set), we denote the partial order on X by . For Y ⊆ X, we recall that the down-set of Y is the set ↓Y = {x ∈ X : ∃y ∈ Y with x ≤ y}. The up-set of Y is defined dually and is denoted by ↑Y . If Y is a singleton set {y}, then we use ↓y and ↑y instead of ↓{y} and ↑{y}, respectively. We call U ⊆ X an up-set if x ∈ U and x ≤ y imply y ∈ U . A down-set of X is defined dually. For Y ⊆ X we denote by max Y , resp. min Y the set of its maximal, resp. minimal points. That An Esakia space is a Priestley space X such that ↓U is clopen for each clopen U of X; recall that a poset X is a Priestley space if X is a compact space and for each x, y ∈ X, from x y it follows that there is a clopen (closed and open) up-set U of X such that x ∈ U and y / ∈ U . It follows easily from e.g. [8, 11.15(i)] that for any Priestley space (X, ), any closed subset Y ⊆ X and any y ∈ Y there are y 1 ∈ min Y , y 2 ∈ max Y with y 1 y y 2 .
For posets X and Y , a map f : for all x, y ∈ X; an order-preserving f is said to be a bounded morphism (or p-morphism) iff for each x ∈ X and y ∈ Y , from f (x) ≤ y it follows that there exists z ∈ X such that x ≤ z and f (z) = y.
For Esakia spaces X and Y , a map f is an Esakia morphism if it is a bounded morphism which is also continuous. Let Esa be the category of Esakia spaces and Esakia morphisms. By Esakia duality [9], Heyt is dually equivalent to Esa (the dual of a Heyting algebra A is indicated with A * ). The functors (−) * : Heyt → Esa and (−) * : Esa → Heyt that establish this dual equivalence are constructed as follows. For a Heyting algebra A, let A * = (X, ), where X is the space of all prime filters of A (topologized by the subbasis {α(a), X \ α(a) : a ∈ A}, where α(a) = {x ∈ X : a ∈ x}) and x y iff x ⊆ y. For a Heyting algebra homomorphism h, let h * = h −1 . For an Esakia space (X, ), let (X, ) * = A, where A is the Heyting algebra of clopen up-sets of X, with meet and join given by intersection and union respectively and with implication given It follows from Esakia duality that onto Heyting homomorphisms dually correspond to 1-1 Esakia morphisms, and 1-1 Heyting homomorphisms to onto Esakia morphisms. In particular, homomorphic images of A ∈ Heyt correspond to closed up-sets of the Esakia dual of A.

Intuitionistic Multi-conclusion Consequence Relations
We use greek letters γ, δ, . . . , ϕ, ψ, . . . to denote formulas built up from propositional variables using the connectives ¬, ∧, ∨, →, ⊥, . A valuation on a Heyting algebra A is a map associating an element of A with every propositional variable. It is then extended to all formulas in a standard way. An intuitionistic Kripke model is a triple (X, , V ) where (X, ) is a poset and V is a valuation on the Heyting algebra of its up-sets. We use letters M, N, . . . for Kripke models and the notation M, x |= ϕ to mean that x belongs to V (ϕ), where V is the valuation on the Kripke model M. The notation M |= ϕ means that M, x |= ϕ holds for all x from the underlying poset of M.
We denote the smallest intuitionistic multi-conclusion consequence relation by S IPC . For a set R of multi-conclusion rules, let S IPC + R be the smallest intuitionistic multi-conclusion consequence relation containing R. If S = S IPC +R, then we say that S is axiomatized by R or that R is a basis for S. Whenever Γ/Δ belongs to S IPC + R we say that Γ/Δ is derivable from R.
A Heyting algebra A validates a multi-conclusion rule Γ/Δ provided for every valuation v on A, if v(γ) = 1 for all γ ∈ Γ, then v(δ) = 1 for some δ ∈ Δ. If A validates Γ/Δ, we write A |= Γ/Δ. The following result is proved in [4,18]: Theorem 2.2. Γ/Δ is derivable from R iff every Heyting algebra validating all rules in R validates also Γ/Δ. We will say that rules ρ 1 and ρ 2 are equivalent if ρ 1 is derivable from {ρ 2 } and ρ 2 is derivable from {ρ 1 }. By Theorem 2.2 this means that a Heyting algebra validates ρ 1 if and only if it validates ρ 2 .
Derivability should be contrasted with admissibility; we will call a rule Γ/Δ admissible in IPC (or admissible tout court) iff it is valid in the free Heyting algebra with countably many generators. Taking into consideration the disjunction property of IPC, it is known (see e.g. [16,24]) that this is equivalent to either one of the following conditions: (1) every substitution making all members of Γ a theorem in IPC makes also some member of Δ a theorem of IPC, and (2) adding Γ/Δ to IPC does not lead to the derivability of new theorems.
A set of rules R is said to form an admissible basis for a logic L if every rule admissible in L is derivable from R.

Closed Domain Condition and Stable Canonical Rules for Heyting Algebras
We recall some definitions and results from [1].
Definition 2.3. Let X = (X, ≤) and Y = (Y, ≤) be Esakia spaces and let f : X → Y be a map. We call f stable if it is continuous and orderpreserving.
It can be shown that Definition 2.3 can be dualized in the following way. Let A and B be Heyting algebras; then h : A → B is a bounded lattice morphism iff the dual Esakia morphism h * : B * → A * is stable.
Definition 2.4. Let X = (X, ≤) and Y = (Y, ≤) be Esakia spaces, f : X → Y be a map, and U be a clopen subset of Y . We say that f satisfies the closed domain condition holds for all x ∈ X. Let D be a collection of clopen subsets of Y . We say that f : X → Y satisfies the closed domain condition (CDC) for D if f satisfies CDC for each U ∈ D.
Stable canonical rules are introduced in the following definition: ) Let A be a finite Heyting algebra, D ⊆ A 2 , and B be an arbitrary Heyting algebra. Then the following are equivalent: The interesting point about stable rules is the following completeness theorem:

Dichotomy Property and Admissible Basis for IPC
Let V n be the rule: Theorem 3.1. The rule V n is admissible for each n ∈ ω.
Proof. We have to show that if σ is a substitution such that none of σq → σp 1 ,..., σq → σp n is a theorem of IPC, then ( Suppose that a stable canonical rule γ(F, D) has the following property. Given an Esakia space W and a clopen up-set Y ⊆ W and a stable surjective map f : We will show that under the assumption of the lemma this rule is equivalent to (2) ⇒ (1) is clear. Now assume that (the Heyting algebra dual to the Esakia space) W does not validate (2). We show that then it does not validate (1).
. This means that there is a stable surjective f : Y → F . By the condition of the lemma f is extended to stable surjectivef : W → F , implying W |= (1).
The following definition will be our main ingredient for a semantic characterization of admissibility of a stable canonical rule 1 : We will see below that the triviality condition plays the same role for stable canonical rules as the existence of tight predecessors in the context of [18, Theorem 4.9 (iv)].
Theorem 3.4. The following are equivalent: is not equivalent to an assumption-free rule.
Proof. (2) ⇒ (1). We know that all V n are admissible, i.e. valid in the free Heyting algebra on infinitely many generators. Since moreover γ(A, D) is derivable from {V n : n ∈ ω}, we conclude that γ(A, D) is also valid on this algebra, i.e. is admissible.
(3) ⇒ (2). Let F = A * and suppose γ(F, D) is not derivable from {V n : n ∈ ω}. Then, by Theorem 2.2, there is an Esakia space W validating all V n 's and refuting γ(F, D). The latter means that there is a stable surjective f : W → F satisfying CDC for D. We will now show that γ(F, D) is trivial. In what follows, we will employ the Heyting algebra W * ; in particular, implication will be understood in the sense of this algebra.
Since W validates the rules V n for each n ∈ ω, and none of q → p s are the whole of W , it follows that neither ( s∈S p s ) → q → s∈S p s is the whole of W ; in particular, ( s∈S p s ) → q \ s∈S p s is not empty. As the topology on F is discrete, p s and q are clopen sets. Thus both ( s∈S p s ) → q and s∈S p s are clopen too, and we may actually pick a maximal element y of ( s∈S p s ) → q \ s∈S p s .
We claim that then for each y > y we have y ∈ q. Indeed since ( s∈S p s ) → q is an upset and y belongs to it, also y will belong to it. But then y / ∈ s∈S p s is impossible by maximality of y, so y ∈ s∈S p s , hence y ∈ q.
Let us now check that f (y) fulfils the triviality conditions for S. For the first condition just note that y / For the second condition, suppose d ∩↑f (y) = ∅ for d ∈ D, then by the CDC of f we have that there is y ≥ y such that f (y ) ∈ d. Thus, either y = y and then f (y ) = f (y) ∈ d ∩ {f (y)} or y > y and then, as we have seen, is trivial. We show that then it is equivalent to an assumption-free rule. We use Lemma 3.2. Let W be an Esakia space, Y ⊆ W a clopen up-set and f : Y → F a stable surjective map satisfying CDC for D. We extend f to some f l : W → F with the same properties.
Proof. It follows from the minimality of S that Y ∪Y S is an up-set. Indeed, if x ∈ Y ∪Y S and x y, then either y ∈ Y and then we are done, or, provided This finishes the proof of the claim.
We now extend f tof with dom(f ) = Y ∪ Y S . We put where s is such that S ⊆ ↑s and for all d It is easy to see thatf is order-preserving. Now we also show thatf is continuous. Indeed, for every Since Y S is a clopen set the continuity follows.
Finally, we show thatf satisfies CDC. The relevant case is So we extended f tof on Y ∪ Y S . We need to show that by repeating this procedure we cover the whole of W . This holds since the following is true: if some S ⊆ F has been used for further extension of the map according to the above procedure, then this same S can never occur again during any subsequent extensions.
Indeed let f k , resp. f n be any further extensions of f to Y k , resp. Y n , k < n < ω. Suppose we have used some S for f k ; then it cannot happen that S can be also used for f n .
Suppose, to the contrary, that S occurs as one of the candidates to build f n . Then in particular It thus follows that after each next extension at least one subset of F is excluded from all subsequent extension steps. Thus after some step n there will be no w / ∈ Y n and no S left with the property f n w = S. Which just means that there is no w outside Y n , i.e. Y n = W .
(1) ⇒ (4) Suppose γ(A, D) is admissible and equivalent to an assumptionfree rule /Δ. Then by the definition of admissibility any substitution makes one of the formulas in Δ a theorem of IPC. Hence /Δ is valid on any Heyting algebra. However, A |= γ(A, D), which is a contradiction.
Corollary 3.6. A stable canonical rule γ(A, D) has the following dichotomy property: it is either admissible or equivalent to an assumption-free rule.
Let Ξ be the set of all subformulas of formulas in Γ ∪ Δ. Then Ξ is finite. Let m be the cardinality of Ξ. Since the bounded lattice reduct of Heyting algebras is locally finite, up to isomorphism, there are only finitely many pairs (A, D) satisfying the following two conditions: (i) A is a finite Heyting algebra that is at most m-generated as a bounded distributive lattice and A |= Γ/Δ. Proof. By Theorem 2.7 and the above.

Duality for Modal Algebras
We use [5,6,19,28] as our main references for the basic theory of normal modal logics, including their algebraic and relational semantics, and the dual equivalence between modal algebras and modal spaces (descriptive Kripke frames).
A modal algebra is a pair A = (A, ♦), where A is a Boolean algebra and ♦ is a unary operator on A that commutes with finite joins. As usual, the dual operator is defined as ¬♦¬. A modal homomorphism between two modal algebras is a Boolean homomorphism h satisfying h(♦a) = ♦h(a). Let MA be the category of modal algebras and modal homomorphisms.
A modal space (or descriptive Kripke frame) is a pair X = (X, R), where X is a Stone space (zero-dimensional compact Hausdorff space) and R is a binary relation on X satisfying the conditions: Let A = (A, ♦) be a modal algebra and let X = (X, R) be its dual space. Then it is well known that R is reflexive iff a ♦a for all a ∈ A, and R is transitive iff ♦♦a ♦a for all a ∈ A. A modal algebra A is a K4-algebra if ♦♦a ♦a holds in A, and it is an S4-algebra if in addition a ♦a holds in A. S4-algebras are also known as closure algebras, interior algebras, or topological Boolean algebras. Let K4 be the full subcategory of MA consisting of K4-algebras, and let S4 be the full subcategory of K4 consisting of S4-algebras. A modal space X = (X, R) is a transitive space if R is transitive, and it is a quasi-ordered space if R is reflexive and transitive.
For a clopen subset Y ⊆ X of a transitive space (X, R), a point y ∈ Y is called quasi-maximal if for any x ∈ Y with yRx we have xRy. It is known that any point of any clopen subset sees a quasi-maximal point of this subset (see e.g. [6,Theorem 10.36]).
Let TS be the full subcategory of MS consisting of transitive spaces, and let QS be the full subcategory of TS consisting of quasi-ordered spaces. Then the dual equivalence of MA and MS restricts to the dual equivalence of K4 and TS, which restricts further to the dual equivalence of S4 and QS.

Multi-conclusion Modal Rules
We use greek letters γ, δ, . . . , ϕ, ψ, . . . to denote formulas built up from propositional variables using the connectives ¬, ∧, ∨, →, ⊥, , ♦. A valuation on a modal algebra A = (A, ♦) is a map associating an element of A with every propositional variable. It is then extended to all modal formulas in a standard way. A Kripke frame is a pair (X, R) where X is a set and R is a binary relation on X. A Kripke model is a triple (X, R, V ), where (X, R) is a Kripke frame and V is a valuation on the powerset Boolean algebra of X with ♦ := R −1 . We use letters M, N, . . . for Kripke models and the notation M, x |= ϕ to mean that x belongs to V (ϕ), where V is the valuation of the Kripke model M. The notation M |= ϕ ('ϕ is valid in M') means that M, x |= ϕ holds for all x from the underlying frame of M. We let K, K4, S4 stand for the set of formulas which are valid in all modal algebras, K4-modal algebras, S4-modal algebras, respectively (as it is well-known, we can equivalently use validity in the corresponding classes of Kripke models).
A transitive normal modal multi-conclusion consequence relation is a set S of modal rules such that (1) ϕ/ϕ ∈ S.

(7) If Γ/Δ ∈ S and σ is a substitution, then σ(Γ)/σ(Δ) ∈ S.
We denote the least transitive normal modal multi-conclusion consequence relation by S K4 . For a set R of multi-conclusion modal rules, let S K4 + R be the least transitive normal modal multi-conclusion consequence relation containing R. If S = S K4 + R, then we say that S is axiomatized by R or that R is a basis for S. Whenever Γ/Δ belongs to S K4 + R we say that Γ/Δ is derivable from R.
A K4 algebra A validates a multi-conclusion rule Γ/Δ provided for every valuation v on A, if v(γ) = 1 for all γ ∈ Γ, then v(δ) = 1 for some δ ∈ Δ. If A validates Γ/Δ, we write A |= Γ/Δ. The following result is proved in [4,18]: Theorem 4.1. Γ/Δ is derivable from R iff every K4-algebra validating all rules in R also validates Γ/Δ. Admissibility of rules in modal calculi is defined similarly to the intuitionistic case (described in 2.2) and has similar properties.

Closed Domain Conditions and Stable Canonical Rules for Modal Algebras
We now introduce the key concepts of stable homomorphisms and the closed domain condition, and show how the two relate to each other. For the proofs of the results stated in this subsection, the reader is referred to [1]. It is easy to see that h : A → B is stable iff h( a) ≤ h(a) for each a ∈ A. Stable homomorphisms were considered in [3] under the name of semi-homomorphisms and in [13] under the name of continuous morphisms.
Let D be a collection of clopen subsets of Y . We say that f : X → Y satisfies the closed domain condition (CDC) for D if f satisfies CDC for each U ∈ D.    (2) We say that h satisfies the closed domain condition (CDC) for D ⊆ A if h satisfies CDC for each a ∈ D.
We now come to stable canonical rules:  It was proved in [1] that every multi-conclusion consequence relation above K is axiomatizable by stable canonical rules (relative to arbitrary finite modal algebras -not only to those validating K4-axiom). The same proof can easily be extended to our multi-conclusion consequence relations above K4. Thus, we have the following theorem.

Dichotomy Property and Admissible Basis for K4
From now on, all Kripke frames and modal spaces are assumed to be transitive. Below + ϕ abbreviates ϕ ∧ ϕ; in a transitive Kripke frame/modal space (X, R), R + abbreviates R ∪ id and → S stands for {w ∈ X | ∃s ∈ S sR + w}. We may also use the notation ↑S for {w ∈ X | ∃s ∈ S sRw}. When we say that S is an up-set we mean S = → S. If S is a singleton set {y}, then we use ↑y and → y instead of ↑{y} and → {y}, respectively. Notations → S, ↓S, ↓{y} and → {y} are defined dually (notice that R −1 (S) is the same as ↓S).
Let F = (W, R) be a frame dual to a finite K4-algebra A = (A, ♦). We denote the set {β(a) : a ∈ D} by D. We will also denote (abusing notation) the stable canonical rule ρ (A, D) by ρ(F, D).
Let S ,m n be the rule: and T m n be the rule: Proof. The foregoing proof is essentially an adjustment of the proof of Theorem 3.1.
(1) We have to show that if σ is a substitution such that none of  ∨ + σq)). This means that for any k and any w in M with M, w |= σr k one has M, w |= σr k ∨ + σq for all w with wRw . But any such w is either in some M i and then M, w |= + σq, or w = , and then because of wRw also w = , so M, w |= σr k . In both cases M, w |= σr k ∨ + σq.
(2) The rule (T m n ) is proved to be admissible in a similar way (this time, an irreflexive extra root is needed).  ρ(F, D) is equivalent to an assumption-free rule.
Proof. Let ρ(A, D) be the rule ϕ ϕ 1 | · · · | ϕ n (1) We will show that under the assumption of the lemma this rule is equivalent to (2) ⇒ (1) is clear. Now assume that a transitive modal space (W, R) does not validate (2). We show that then it does not validate (1).
. This means that there is a stable surjective f : Y → F satisfying CDC for D. By the condition of the lemma f can be extended to a stable surjective mapf : W → F satisfying CDC for D, implying W |= (1).
The following is a modal analogue of Definition 3.3.     ρ(F, D). The latter means that there is a stable surjective f : We will first show that there exist x • and x • satisfying the conditions of Definition 5.3(1)- (2). In what follows we are working in the modal algebra (W, R) * ; all connectives and modal operators are taken in this algebra. For = {v 1 , . . . , v } be a finite set of clopens of W . Since f is continuous and F is discrete, p s and r k are clopens, while q is a clopen up-set in W since f is also stable. In particular, q and + q have the same underlying set. Moreover, for all s ∈ S we have + q p s . Indeed, for any w s ∈ f −1 (s) we have that w s ∈ q but w s / ∈ p s = W \ f −1 (s). This means that the conclusion of the rule S ,m n is falsified on W . It follows that W falsifies the premise of that rule as well. Hence there exists w C ∈ W such that w C ∈ l=1 ( v l → v l ), w C ∈ k∈F (r k → (r k ∪ + q)) and w C / ∈ s∈S p s . The latter can be equivalently written as w C ∈ s∈S ♦f −1 (s). We thus obtain that of clopens of W has finite intersection property. Since W is compact, the intersection of all these clopens is nonempty, i.e. there is w ∈ W that belongs to all of these clopens. That is, w belongs to all clopens of the form v → v (which means that w is reflexive), and also w ∈ k∈F (r k → (r k ∪ + q)) and w ∈ s∈S ♦f −1 (s). By the latter, we have that for every s ∈ S there is a w such that wRw and f (w ) = s. In other words, f being stable, ↑f (w) ⊇ S. Let x := f (w). Then condition (1) of Definition 5.3 is met (notice that x is reflexive because w is reflexive and f is stable). We now show that condition (2) is met as well.
Since w ∈ k∈F (r k → (r k ∪ + q)), in particular we have w ∈ (r x → (r x ∪ + q)). Since w ∈ r x , we obtain that w ∈ (r x ∪ + q) = (¬r x → + q). This means that any w such that wRw and f (w ) = x will be necessarily in + q. Now if d ∩ ↑x = ∅ for some d ∈ D, then as x = f (w), by the CDC of f there is w such that wRw and f (w ) ∈ d. Then, either f (w ) = x and then f (w ) ∈ d ∩ {x}, or f (w ) = x and then as we have seen Next we show that there exists an x • satisfying the conditions of Definition 5.3(3)-(4). As above, for s ∈ S let p s = W \f −1 (s) ⊆ W , let q = f −1 ( → S) and let r k = f −1 (k) for k ∈ F . Again, the conclusion of the rule T m n is falsified on W and consequently W falsifies the premise of that rule as well. Thus there is w ∈ k∈F (♦r k → ♦(r k ∧ + q)) and w / ∈ s∈S p s . By the latter, we have that for every s ∈ S there is a w such that wRw and f (w ) = s. In other words, f being stable, ↑f (w) ⊇ S. Let x := f (w). Then condition (3) of Definition 5.3 is met. For condition (4), consider d ∈ D such that d ∩ ↑f (w) = ∅. Then, since f satisfies CDC for D, there is an u ∈ W with wRu and f (u) ∈ d. Thus w ∈ ♦r k for k = f (u), as r k = f −1 (k); since w ∈ ♦r k → ♦(r k ∧ + q), there is w such that wRw and w ∈ r k ∩ + q, which means in particular that f (w ) = k ∈ d ∩ → S, as wanted. Putting x • = x we deduce that ρ(F, D) is trivial • and hence, trivial.
(4) ⇒ (3). Suppose ρ(A, D) is trivial. We show that then it is equivalent to an assumption-free rule. Using Lemma 5.2, it suffices to extend any stable surjective map f : Y → F from a clopen up-set Y ⊆ W of a transitive modal space (W, R) to F satisfying CDC for D to anf : W → F with the same properties. For Proof. That Y ∪ Y S is an up-set follows from minimality of S. Indeed, if x ∈ Y ∪ Y S and xRy, then either y ∈ Y and then we are done, or, provided y / ∈ Y , then, since Y is an up-set, also so by minimality of S necessarily f y = S. The latter means y ∈ Y S , so \Y is clopen. This finishes the proof of the claim.
We now extend f tof with Y domf = Y ∪ Y S . Recall that, by the triviality of (F, D), there exist two (not necessarily distinct) points s • , s • such that (i) S ⊆ ↑s • and d ∩ ↑s We distinguish two cases, depending whether S has a reflexive root or not. Case (I): S has a reflexive root s ∈ S. We put: It is easy to see thatf is stable (s is reflexive). Now we also show thatf is continuous. Indeed, for every Since the latter is a clopen set, continuity follows. Also,f satisfies CDC: the relevant case is when Thus, there is w ∈ Y with wR + w and f (w ) = s. Since w / ∈ Y and w ∈ Y , we have wRw . We can use the fact that f satisfies the CDC: since w ∈ Y = dom(f ) and ↑f (w) = ↑s = ↑f (w ), we get ↑f (w ) ∩ d = ∅ and also f (↑w ) ∩ d = ∅; as a consequencef (↑w) ∩ d is also not empty.
Case (II): S does not have a reflexive root. We further distinguish two subcases, depending whether there are irreflexive R + -quasi-maximal points in Y S or not. Notice that such points form the clopen antichain there are no irreflexive quasi-maximal points in Y S . Then, as noted above, every point in Y S can see a quasimaximal reflexive point in it. We put: It is easy to see thatf is stable (s • is reflexive). Now we also show thatf is continuous. Indeed, for every Since the latter is a clopen set, continuity follows. Also,f satisfies CDC: the relevant case is when d ∩ ↑f (w) = ∅ for d ∈ D, w ∈ Y S . We havef (w) = s • by construction and d ∩ ( In case (i), we pick a quasi-maximal reflexive w in Y S such that wR + w : sincef (w ) = s • , we have thatf (↑w) ∩ d contains s • and is not empty. In case (ii), recall that It is easy to see that f 0 is stable (points in Y • S are irreflexive). Now we also show that f 0 is continuous. Indeed, for every Since the latter is a clopen set, continuity follows. Also, f 0 satisfies CDC: the relevant case is It follows that quasi-maximal points in Y S are all reflexive. We then can continue as in Subcase (II.1) above and get an extensionf .
So we extended f tof . We need to show that by repeating this procedure we cover the whole of W . This holds since the following is true: if some S ⊆ F has been used for further extension of the map according to the above procedure, then this same S can never occur again during any subsequent extensions.
Indeed let f k , resp. f n be any further extensions of f to Y k , resp. Y n , k < n. Suppose we have used some S for f k ; then it cannot happen that S can be also used for f n .
Suppose, to the contrary, that S occurs as one of the candidates to build f n . Then in particular S = f n−1 In fact by minimality of S, f k−1 w cannot be included in S, so f k−1 w \ S is nonempty. Now note that since f n−1 is an extension of f k−1 , one has f n−1 w ⊇ f k−1 w , hence also f n−1 w \ S is nonempty, which contradicts the equality S = f n−1 w above. It thus follows that after each next extension at least one subset of F is excluded from all subsequent extension steps. Thus after some step n there will be no w / ∈ Y n and no S left with the property f n w = S. Which just means that there is no w outside Y n , i.e. Y n = W .
Corollary 5.6. A canonical rule ρ(F, D) has the following dichotomy property: it is either admissible or equivalent to an assumption-free rule.
Proof. The proof is similar to the proof of Corollary 3.7. Proof. The proof is similar to the proof of Corollary 3.8.
To conclude, we mention that the above results also hold for S4, with the following modifications: (i) rules (T m n ) should be removed from the admissible basis; (ii) rules (S ,m n ) are kept, but can be simplified (we do not need the parameter either, because the conjuncts l=1 ( v l → v l ) are now valid formulas); (iii) in Definition 5.3, conditions (3)-(4) are removed (thus a stable canonical rule is trivial in the new S4 sense iff it was just trivial • in the old sense).
Remark 5.9. It is an open question whether the techniques developed in this paper would adapt well to fragments of IPC (or modal logics) and subreducts of Heyting algebras (or modal algebras). The implication and implicationconjunction-negation fragments of IPC are structurally complete, but not the implication-negation fragment (admissibility for the latter fragment is axiomatized in [7], see [20] for the positive fragment). Explicit axiomatizations for the admissible rules of the implication-disjunction fragment of IPC and pseudo-complemented distributive lattices are still lacking, however. there exists a lift, i.e. a homomorphism a : P → A with b = pa. It is well known that free algebras are projective, that a retract of a projective algebra is projective, and that an algebra is projective if and only if it is a retract of a free algebra.
For modal and Heyting algebras we can generalise the notion of projectivity to D-projectivity. We will discuss only the modal K4-case here. Let (P, D) be a pair where P is a K4-algebra and D ⊆ P . For brevity, let us call a map h : P → A a D-morphism if h is a stable homomorphism satisfying CDC for D. We will denote D-morphisms by h : P A. For a subset D ⊆ P of a K4-algebra P we will call the algebra P Dprojective if any diagram A P B p b a of K4-algebras has a D-lift, that is, for any surjective modal homomorphism p and any D-morphism b there is a D-morphism a with pa = b. It can be shown that P is D-projective if and only if it is a D-retract of a free K4algebra. The latter means that there exists a modal homomorphism p : F → P from a free K4-algebra to P and a D-morphism f : P F with pf = id P . Then our main theorem 5.4 is nothing but a characterisation of finite D-projective K4-algebras. Namely it follows from the main theorem that for a finite K4-algebra P and D ⊆ P , TFAE: (1) P is D-projective, (2) ρ(P, D) is not admissible, (3) The dual of P satisfies the triviality conditions of Definition 5.3. Thus, in terms of D-projectivity we have the following dichotomy property: for any finite K4-algebra P and any subset D ⊆ P , the stable canonical rule ρ(P, D) is not admissible if and only if P is Dprojective.
Remark 5.11. Admissibility and unification over the basic (non-transitive) modal logic K are long-standing open problems. Although the proofs of this paper do not apply to K directly, we note that unlike the canonical rules of [18], stable canonical rules axiomatize consequence relations over K. It remains open whether stable canonical rules could be applicable in analysing admissibility for non-transitive logics: in particular, whether they could be used in obtaining some dichotomy property for K. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.