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 $\Box\cdots\Box p$ (often abbreviated as $\Box^i p$ for $0 \leq i < \omega$).
• A Sahlqvist antecedent is a formula constructed using ∧, ∨, and $\Diamond$ 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 $\Box$ (unrestricted), and using ∨ on formulas with no common variables.

Examples of Sahlqvist formulas

$p \rightarrow \Diamond p$

Its first-order corresponding formula is $\forall x \; Rxx$, and it defines all reflexive frames

$p \rightarrow \Box\Diamond p$

Its first-order corresponding formula is $\forall x \forall y [Rxy \rightarrow Ryx]$, and it defines all symmetric frames

$\Diamond \Diamond p \rightarrow \Diamond p$ or $\Box p \rightarrow \Box \Box p$

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

$\Diamond p \rightarrow \Diamond \Diamond p$ or $\Box \Box p \rightarrow \Box p$

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

$\Box p \rightarrow \Diamond p$

Its first-order corresponding formula is $\forall x \exists y \; Rxy$, and it defines all right-unbounded frames (also called serial)

$\Diamond\Box p \rightarrow \Box\Diamond p$

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

Examples of non-Sahlqvist formulas

$\Box\Diamond p \rightarrow \Diamond \Box p$

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

$\Box(\Box p \rightarrow p) \rightarrow \Box p$

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

$(\Box\Diamond p \rightarrow \Diamond \Box p) \land (\Diamond\Diamond q \rightarrow \Diamond q)$

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 $\forall x[\forall y(Rxy \rightarrow \exists z[Ryz]) \rightarrow \exists y(Rxy \wedge \forall z[Ryz \rightarrow z = y])]$) 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.