Bi-superintuitionistic logics for rough sets

Bi-intuitionistic logic, also called Heyting-Brouwer logic, is a logic based on Heyting and Brouwerian algebras. A rough set logic based on regular double Stone algebra is regarded as the extension of bi-intuitionistic logic without intuitionistic and dual intuitionistic implication. In this paper, we discuss the aspects of bi-superintuitionistic logics which are stronger than bi-intuitionistic logic as a foundation for rough set logics. We propose some bi-superintuitionistic logics with a Kripke semantics and natural deduction. These logics can serve as foundations for reasoning about rough and vague information.