Sahlqvist correspondence theory for second-order propositional modal logic

Abstract

Abstract Modal logic with propositional quantifiers (i.e. second-order propositional modal logic ($\textsf {SOPML}$)) has been considered since the early time of modal logic. Its expressive power and complexity are high, and its van Benthem–Rosen theorem and Goldblatt–Thomason theorem have been proved by ten Cate (2006, J. Philos. Logic, 35, 209–223). However, the Sahlqvist theory of $\textsf {SOPML}$ has not been considered in the literature. In the present paper, we fill in this gap. We develop the Sahlqvist correspondence theory for $\textsf {SOPML}$, which covers and properly extends existing Sahlqvist formulas in basic modal logic. We define the class of Sahlqvist formulas for $\textsf {SOMPL}$ step by step in a hierarchical way, each formula of which is shown to have a first-order correspondent over Kripke frames effectively computable by an algorithm $\textsf {ALBA}^{\textsf {SOMPL}}$. In addition, we show that certain $\varPi _2$-rules correspond to $\varPi _2$-Sahlqvist formulas in $\textsf {SOMPL}$, which further correspond to first-order conditions, and that even for very simple $\textsf {SOMPL}$ Sahlqvist formulas, they could already be non-canonical.