Two-Sorted Modal Logic for Formal and Rough Concepts

In this paper, we propose two-sorted modal logics for the representation and reasoning of concepts arising from rough set theory (RST) and formal concept analysis (FCA). These logics are interpreted in two-sorted bidirectional frames, which are essentially formal contexts with converse relations. The logic KB contains ordinary necessity and possibility modalities and can represent rough set-based concepts. On the other hand, the logic KF has window modality that can represent formal concepts. We study the relationship between KB and KF by proving a correspondence theorem. It is then shown that, using the formulae with modal operators in KB and KF, we can capture formal concepts based on RST and FCA and their lattice structures.