A non-distributive logic for semiconcepts of a context and its modal extension with semantics based on Kripke contexts

A non-distributive two-sorted hypersequent calculus $\mathbf{\text{PDBL}}$ and its modal extension $\mathbf{\text{MPDBL}}$ are proposed for the classes of pure double Boolean algebras and pure double Boolean algebras with operators respectively. A relational semantics for $\mathbf{\text{PDBL}}$ is next proposed, where any formula is interpreted as a semiconcept of a context. For $\mathbf{\text{MPDBL}}$, the relational semantics is based on Kripke contexts, and a formula is interpreted as a semiconcept of the underlying context. The systems are shown to be sound and complete with respect to the relational semantics. Adding appropriate sequents to $\mathbf{\text{MPDBL}}$ results in logics with semantics based on reflexive, symmetric or transitive Kripke contexts. One of these systems is a logic for topological pure double Boolean algebras. It is demonstrated that, using $\mathbf{\text{PDBL}}$, the basic notions and relations of conceptual knowledge can be expressed and inferences involving negations can be obtained. Further, drawing a connection with rough set theory, lower and upper approximations of semiconcepts of a context are defined. It is then shown that, using the formulae and sequents involving modal operators in $\mathbf{\text{MPDBL}}$, these approximation operators and their properties can be captured.