Sahlqvist formula

In modal logic, Sahlqvist formulae 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.

• A boxed atom is a propositional atom preceded by a number (possibly 0) of boxes, i.e. a formula of the form $\Box \cdots \Box p$.
• A Sahlqvist antecedent is a formula constructed using ∧, ∨, and $\Diamond$ from boxed atoms, and negative formulae (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 antecedents using ∧ and $\Box$ (unlimited), and using ∨ on formulae with no common variables.

Examples of non-Sahlqvist formulae

• The McKinsey formula $\Box \Diamond p\rightarrow \Diamond \Box p$ does not have a first-order frame condition, hence is not Sahlqvist.
• The Löb axiom $\Box (\Box p\rightarrow p)\rightarrow \Box p$ is not Sahlqvist, again because it does not have a first-order frame condition.

Sahlqvist's definition characterises a decidable set of formulae. Since it is undecidable, by Chagrova's theorem, whether an arbitrary modal formula has a first-order frame condition, there are formulae with first-order frame conditions that are not Sahlqvist (Chagrova 1991).

• $(\Box \Diamond p\rightarrow \Diamond \Box p)\land (\Diamond \Diamond p\rightarrow \Diamond q)$ (the conjunction of the McKinsey formula and the 4 axiom) has a first-order frame condition but is not equivalent to any Sahlqvist formula.

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.