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Hybrid logics with Sahlqvist axioms

Hybrid logics with Sahlqvist axioms

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6 Hybrid logics with Sahlqvist axioms property states that whenever this obtains, there exists a formula ϑ in which p does not occur, such that every Λ-model M globally satisfying Γ globally satisfies p ↔ ϑ. The relevant formula ϑ is called an explicit definition of p relative to Γ and Λ. Usually, the Beth property is proved as a corollary of the interpolation property. Unfortunately, it is known that interpolation fails for hybrid logic [1]. In what follows, we will show that the basic hybrid logic satisfies a restricted version of the interpolation, that is strong enough to entail the Beth property. In fact, this restricted version of interpolation holds not only for the basic hybrid logic, but for many other hybrid logics as well. Let us say that a modal or hybrid logic Λ has interpolation over proposition letters if whenever φ → ψ ∈ Λ, there exists a formula ϑ, such that φ → ϑ ∈ Λ and ϑ → ψ ∈ Λ, and all proposition letters (but not necessarily the nominals) occurring in ϑ occur both in φ and in ψ. Theorem 4.1 (Interpolation for hybrid logic) Let Σ be a set of modal Sahlqvist formulas. If all axioms in Σ have universal Horn frame correspondents, then K H(@) Σ has interpolation over proposition letters. Proof. We again apply the non-standard modal semantics of the hybrid language, and use the fact that every modal logic axiomatized by Sahlqvist formulas with universal Horn correspondents has interpolation over proposition letters [9, Corollary B.4.1]. By assumption, all axioms in Σ are modal Sahlqvist formulas with universal Horn correspondents. Furthermore, with the exception of [self-dual], all first-order correspondents of axioms in ∆ are universal Horn sentences. The [self-dual] itself is equivalent to the conjunction of ¬@ i ¬p → @ i p and ¬@ i ¬⊤. The former is a modal Sahlqvist formula with a universal Horn correspondent and the latter is a formula without proposition letters. Rautenberg [10] proved that extending a modal logic that has interpolation over proposition letters with formulas without proposition letters yields again a logic with interpolation over proposition letters. Corollary 4.2 (The Beth property for hybrid logic) Let Σ be a set of modal Sahlqvist formulas not containing nominals or satisfaction operators. If all axioms in Σ have universal Horn frame correspondents, then K H(@) Σ has Beth’s definability property. Proof. Let Γ be any set of hybrid formulas, and p a proposition letter, such that Γ implicitly defines p in K H(@) Σ. Let p ′ be a new proposition letter, and let Γ[p/p ′ ] be the result of replacing all occurrences of p in Γ by p ′ . Let K Σ be the frame class defined by Σ. Then by definition, every model based on a frame in K Σ that globally satisfies Γ ∪ Γ[p/p ′ ] globally satisfies p ↔ p ′ . Now, let Γ ✷ be the set of formulas {✷ n φ, @ i ✷ n φ | φ ∈ Γ, n ∈ ω, i ∈ nom} and let Γ ✷ [p/p ′ ] be obtained from Γ ✷ by replacing all occurrences of p by p ′ . Claim: For each model M based on a frame in K Σ , and for each world w of M, if M, w |= Γ ✷ ∪ Γ ✷ [p/p ′ ] then M, w |= p ↔ p ′ . Proof of claim: Suppose M, w |= Γ ✷ ∪ Γ ✷ [p/p ′ ]. Let M w be the submodel of M generated by w. By closure under generated subframes, the underlying frame of M w

Hybrid logics with Sahlqvist axioms 7 is in K Σ . Moreover, by construction of Γ ✷ and Γ ✷ [p/p ′ ], M w globally satisfies Γ and Γ[p/p ′ ]). It follows that M w , w |= p ↔ p ′ and hence M, w |= p ↔ p ′ . ⊣ Since Σ consists of modal Sahlqvist formulas, K Σ is a ∆-elementary frame class (i.e., it is defined by a set of first-order formulas). We may therefore apply compactness and conclude from the claim that there are φ 1 , . . . , φ n ∈ Γ ✷ such that K Σ |= ( ∧ ) ( ∧ φ k ∧ φ k [p/p ′ ] ) → (p ↔ p ′ ) and hence k≤n K Σ |= ( p ∧ ∧ ) φ k k≤n k≤n → (( ∧ φ k [p/p ′ ] ) → p ′) By Theorem 3.4 K Σ |= φ is equivalent to saying that φ ∈ K H(@) Σ. Thus by the interpolation theorem there is a formula ϑ such that k≤n 1. The proposition letters p and p ′ do not occur in ϑ. 2. K Σ |= (p ∧ ∧ k≤n φ k) → ϑ. 3. K Σ |= ϑ → (( ∧ k≤n φ k[p/p ′ ]) → p ′ ) It follows from 2. and 3. by uniform substitution that K Σ |= ( ∧ k≤n φ k) → (p ↔ ϑ). Hence, whenever a model based on a frame in K Σ globally satisfies Γ, it globally satisfies p ↔ ϑ. 5 Combining pure and Sahlqvist axioms As we mentioned in the introduction, not every Sahlqvist axiom corresponds to a pure axiom. It is natural to ask if completeness obtains when we extend the basic hybrid logic K + H(@) with a combination of pure and Sahlqvist axioms. The answer is negative. Theorem 5.1 There is a pure formula π and a modal Sahlqvist formula σ such that the hybrid logic K + H(@) {π, σ} is not complete for the frame class defined by π ∧ σ. Proof. Consider the following axioms (the first-order frame conditions they define are given as well): [cr] ✸✷p → ✷✸p ∀xyz(Rxy ∧ Rxz → ∃u(Ryu ∧ Rzu)) [nogrid] ✸(i ∧ ✸j) → ✷(✸j → i) ∀xyzu(Rxy ∧ Rxz ∧ Ryu ∧ Rzu → y = z) [func] ✸p → ✷p ∀xyz(Rxy ∧ Rxz → y = z) [cr] and [func] are Sahlqvist formulas and [nogrid] is pure. As can be easily seen from the first-order correspondents, every frame validating [cr] and [nogrid] validates [func]. However, we will show that [func] is not derivable in K + H(@) {[cr], [nogrid]}. Consider ω ω , i.e., the countably branching tree of infinite depth. Let F be the general frame whose domain is ω ω , whose accessibility relation is the child relation, and in which the admissible sets are exactly the finite and co-finite sets. It is not hard to see that this is indeed a general frame. We will claim that every axiom of

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