Philosophy:Sahlqvist formula

In modal logic, Sahlqvist formulas are a certain kind of modal formula with remarkable properties. The Sahlqvist correspondence theorem states that every Sahlqvist formula is canonical, and corresponds to a first-order definable class of Kripke frames. Sahlqvist's definition characterizes a decidable set of modal formulas with first-order correspondents. Since it is undecidable, by Chagrova's theorem, whether an arbitrary modal formula has a first-order correspondent, there are formulas with first-order frame conditions that are not Sahlqvist [Chagrova 1991] (see the examples below). Hence Sahlqvist formulas define only a (decidable) subset of modal formulas with first-order correspondents.

Definition

Sahlqvist formulas are built up from implications, where the consequent is positive and the antecedent is of a restricted form.

• A boxed atom is a propositional atom preceded by a number (possibly 0) of boxes, i.e. a formula of the form [math]\displaystyle{ \Box\cdots\Box p }[/math] (often abbreviated as [math]\displaystyle{ \Box^i p }[/math] for [math]\displaystyle{ 0 \leq i \lt \omega }[/math]).
• A Sahlqvist antecedent is a formula constructed using ∧, ∨, and [math]\displaystyle{ \Diamond }[/math] from boxed atoms, and negative formulas (including the constants ⊥, ⊤).
• A Sahlqvist implication is a formula A → B, where A is a Sahlqvist antecedent, and B is a positive formula.
• A Sahlqvist formula is constructed from Sahlqvist implications using ∧ and [math]\displaystyle{ \Box }[/math] (unrestricted), and using ∨ on formulas with no common variables.

Examples of Sahlqvist formulas

[math]\displaystyle{ p \rightarrow \Diamond p }[/math]

Its first-order corresponding formula is [math]\displaystyle{ \forall x \; Rxx }[/math], and it defines all reflexive frames

[math]\displaystyle{ p \rightarrow \Box\Diamond p }[/math]

Its first-order corresponding formula is [math]\displaystyle{ \forall x \forall y [Rxy \rightarrow Ryx] }[/math], and it defines all symmetric frames

[math]\displaystyle{ \Diamond \Diamond p \rightarrow \Diamond p }[/math] or [math]\displaystyle{ \Box p \rightarrow \Box \Box p }[/math]

Its first-order corresponding formula is [math]\displaystyle{ \forall x \forall y \forall z [(Rxy \land Ryz) \rightarrow Rxz] }[/math], and it defines all transitive frames

[math]\displaystyle{ \Diamond p \rightarrow \Diamond \Diamond p }[/math] or [math]\displaystyle{ \Box \Box p \rightarrow \Box p }[/math]

Its first-order corresponding formula is [math]\displaystyle{ \forall x \forall y [Rxy \rightarrow \exists z (Rxz \land Rzy)] }[/math], and it defines all dense frames

[math]\displaystyle{ \Box p \rightarrow \Diamond p }[/math]

Its first-order corresponding formula is [math]\displaystyle{ \forall x \exists y \; Rxy }[/math], and it defines all right-unbounded frames (also called serial)

[math]\displaystyle{ \Diamond\Box p \rightarrow \Box\Diamond p }[/math]

Its first-order corresponding formula is [math]\displaystyle{ \forall x \forall x_1 \forall z_0 [Rxx_1 \land Rxz_0 \rightarrow \exists z_1 (Rx_1z_1 \land Rz_0z_1)] }[/math], and it is the Church-Rosser property.

Examples of non-Sahlqvist formulas

[math]\displaystyle{ \Box\Diamond p \rightarrow \Diamond \Box p }[/math]

This is the McKinsey formula; it does not have a first-order frame condition.

[math]\displaystyle{ \Box(\Box p \rightarrow p) \rightarrow \Box p }[/math]

The Löb axiom is not Sahlqvist; again, it does not have a first-order frame condition.

[math]\displaystyle{ (\Box\Diamond p \rightarrow \Diamond \Box p) \land (\Diamond\Diamond q \rightarrow \Diamond q) }[/math]

The conjunction of the McKinsey formula and the (4) axiom has a first-order frame condition (the conjunction of the transitivity property with the property [math]\displaystyle{ \forall x[\forall y(Rxy \rightarrow \exists z[Ryz]) \rightarrow \exists y(Rxy \wedge \forall z[Ryz \rightarrow z = y])] }[/math]) but is not equivalent to any Sahlqvist formula.

Kracht's theorem

When a Sahlqvist formula is used as an axiom in a normal modal logic, the logic is guaranteed to be complete with respect to the elementary class of frames the axiom defines. This result comes from the Sahlqvist completeness theorem [Modal Logic, Blackburn et al., Theorem 4.42]. But there is also a converse theorem, namely a theorem that states which first-order conditions are the correspondents of Sahlqvist formulas. Kracht's theorem states that any Sahlqvist formula locally corresponds to a Kracht formula; and conversely, every Kracht formula is a local first-order correspondent of some Sahlqvist formula which can be effectively obtained from the Kracht formula [Modal Logic, Blackburn et al., Theorem 3.59].

References

• L. A. Chagrova, 1991. An undecidable problem in correspondence theory. Journal of Symbolic Logic 56:1261–1272.
• Marcus Kracht, 1993. How completeness and correspondence theory got married. In de Rijke, editor, Diamonds and Defaults, pages 175–214. Kluwer.
• Henrik Sahlqvist, 1975. Correspondence and completeness in the first- and second-order semantics for modal logic. In Proceedings of the Third Scandinavian Logic Symposium. North-Holland, Amsterdam.