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. (shrink)