Modal Logic with “Most”

In this paper, we axiomatize modal logic extended with the modal operator \(M\varphi \) $M\phi$ saying that “there are strictly more \(\varphi \) $\phi$ -successors than \(\lnot \varphi \) $\neg \phi$ -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.