Sahlqvist formula explained

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 class of Kripke frames definable by a first-order formula.

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 … \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 → \Diamond p

Its first-order corresponding formula is

\forall x Rxx , and it defines all reflexive framesp → \Box\Diamond p

Its first-order corresponding formula is

\forall x \forall y [Rxy → Ryx] , and it defines all symmetric frames\Diamond \Diamond p → \Diamond p or \Box p → \Box \Box p

Its first-order corresponding formula is

\forall x \forall y \forall z [(Rxy \land Ryz) → Rxz] , and it defines all transitive frames\Diamond p → \Diamond \Diamond p or \Box \Box p → \Box p

Its first-order corresponding formula is

\forall x \forall y [Rxy → \exists z (Rxz \land Rzy)] , and it defines all dense frames\Box p → \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 → \Box\Diamond p

Its first-order corresponding formula is

\forall x \forall x _{1} \forall z _{0} [Rxx _{1} \land Rxz _{0} → \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 → \Diamond \Box p

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

\Box(\Box p → p) → \Box p

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

(\Box\Diamond p → \Diamond \Box p) \land (\Diamond\Diamond q → \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 → \exists z[Ryz]) → \exists y(Rxy \wedge \forall z[Ryz → 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 basic 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.