A multi-modal logic for disagreement and exhaustiveness

The paper explores two basic types of relations betwen objects of a Pawlak-style information system generated by the values of some attribute of those objects: disagreement (disjoint sets of values) and exhaustiveness (sets of values adding up to the whole universe of the attribute). Out of these two fundamental types of relations, most other types of relations on objects of an information system considered in the literature can be derived - as, for example, indiscernibility, similarity and complementarity. The algebraic properties of disagreement and indiscernibility relations are explored, and a representation theorem for each of these two types of relations is proved. The notions of disagreement and exhaustiveness relations for a single attribute are extended to relations generated by arbitrary sets of attributes, yielding two families of relations parametrized by sets of attributes. They are used as accessibility relations to define a multi-modal logic with modalities corresponding to the lower and upper approximation of a set in Pawlak's rough set theory. Finally, a complete Rasiowa-Sikorski deduction system for that logic is developed.