Rough and Near: Modal History of Two Theories

Abstract

Near sets were introduced by J.F. Peters in 2007 in the context and within the conceptual framework of rough sets, which were initiated by Z. Pawlak in the early 1980s. However, due to further evolution and development, near set theory has become an independent field of study. For this reason, nowadays, the relationships between near set theory and rough set theory are not easy to spot. In this short paper we would like to re-define near sets and to re-think their foundations and relationships to/bearing on rough sets. To this end we translate the basic concepts of near set theory into the framework of modal logic, which has already been successfully applied to rough sets. The concept of nearness of sets, however, was originally defined globally (that is, with respect to the whole underlying space), but modal logic is intrinsically local: the logical value of a formula is computed with respect to a single point and its neighbourhood. Our approach to near sets is local in the very same sense: we are concerned with nearness of sets seen from the perspective of a single point. Interestingly, this local perspective brings together rough set theory and near set theory, revealing their deep theoretical connections. Therefore, what we offer is a modal and algebraic “shared history” of the two theories at issue.