Sahlqvist Completeness Theory for Hybrid Logic with Downarrow Binder

Abstract

Abstract In the present paper, we continue the research in Zhao (2021, Logic J. IGPL) to develop the Sahlqvist completeness theory for hybrid logic with satisfaction operators and downarrow binders $\mathcal {L}( @, {\downarrow })$. We define the class of restricted Sahlqvist formulas for $\mathcal {L}( @, {\downarrow })$ following the ideas in Conradie and Robinson (2017, J. Logic Comput., 27, 867–900), but we follow a different proof strategy which is purely proof-theoretic, namely showing that for every restricted Sahlqvist formula $\varphi $ and its hybrid pure correspondence $\pi $, $\textbf {K}_{\mathcal {H}( @, {\downarrow })}+\varphi $ proves $\pi $; therefore, $\textbf {K}_{\mathcal {H}( @, {\downarrow })}+\varphi $ is complete with respect to the class of frames defined by $\pi $, using a modified version $\textsf {ALBA}^{{\downarrow }}_{\textsf {Modified}}$ of the algorithm $\textsf {ALBA}^{{\downarrow }}$ defined in Zhao (2021, Logic J. IGPL).