Bimodal Logic with the Irreflxive Modality

This paper proposes a bimodal logic with an additional modality (called the irreflexive modality), which corresponds semantically to the intersection of the accessibility relation and the inequality. First, we show that we can define, within this framework, several properties that are undefinable in the unimodal language; irreflexivity is one of such properties. Second, with respect to the frame expressivity, we compare our language with the unimodal language and another bimodal language with the difference operator that is studied by de Rijke. Finally, we give a Hilbert-style axiomatization of our logic and prove that certain familiar modal systems, such as S4 and S5, enjoy Kripke completeness in our language.