Zhiguang Zhao

In this paper, we axiomatize modal logic extended with the modal operator $$M\varphi $$ saying that “there are strictly more $$\varphi $$ -successors than $$\lnot \varphi $$ -successors”, both in the class of image-finite Kripke frames and in the class of all Kripke frames. We follow the proof strategy of van der Hoek (Int J Uncertain Fuzziness Knowl Based Syst 4(1):45–60, 1996.), and prove a characterization result of finite majority structures which are capable of representing finite cardinality measures and a characterization result of finite extended majority structures which are capable of representing cardinality measures.

Modal logic with counting is obtained from basic modal logic by adding cardinality comparison formulas of the form $ \#\varphi \succsim \#\psi $, stating that the cardinality of successors satisfying $ \varphi $ is larger than or equal to the cardinality of successors satisfying $ \psi $. It is different from graded modal logic where basic modal logic is extended with formulas of the form $ \Diamond _{k}\varphi $ stating that there are at least $ k$-many different successors satisfying $ \varphi $. In this paper, we investigate the axiomatization of ML(#) with respect to different frame classes, such as image-finite frames and arbitrary frames. Drawing inspiration from existing works, we employ a similar proof strategy that uses the characterization of binary relations on finite Boolean algebras capable of representing generalized probability measures or finite (respectively arbitrary) cardinality measures. Our main result shows that any formula not provable in the Hilbert system can be refuted within a finite (respectively arbitrary) cardinality measure Kripke frame with a finite domain. We then transform this finite (respectively arbitrary) cardinality measure Kripke frame into a Kripke frame in the corresponding class, refuting the unprovable formula.

In the present paper, we study the correspondence and canonicity theory of modal subordination algebras and their dual Stone space with two relations, generalizing correspondence results for subordination algebras in [13–15, 25]. Due to the fact that the language of modal subordination algebras involves a binary subordination relation, we will find it convenient to use the so-called quasi-inequalities and $\varPi _{2}$-statements. We use an algorithm to transform (restricted) inductive quasi-inequalities and (restricted) inductive $\varPi _{2}$-statements to equivalent first-order correspondents on the dual Stone spaces with two relations with respect to arbitrary (resp. admissible) valuations.

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).