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A non-distributive two-sorted hypersequent calculus \textbf{PDBL} and itsmodal extension \textbf{MPDBL} are proposed for the classes of pure doubleBoolean algebras and pure double Boolean algebras with operators respectively.A relational semantics for \textbf{PDBL} is next proposed, where any formula isinterpreted as a semiconcept of a context. For \textbf{MPDBL}, the relationalsemantics is based on Kripke contexts, and a formula is interpreted as asemiconcept of the underlying context. The systems are shown to be sound andcomplete with respect to the relational semantics. Adding appropriate sequentsto \textbf{MPDBL} results in logics with semantics based on reflexive,symmetric or transitive Kripke contexts. One of these systems is a logic fortopological pure double Boolean algebras. It is demonstrated that, using\textbf{PDBL}, the basic notions and relations of conceptual knowledge can beexpressed and inferences involving negations can be obtained. Further, drawinga connection with rough set theory, lower and upper approximations ofsemiconcepts of a context are defined. It is then shown that, using theformulae and sequents involving modal operators in \textbf{MPDBL}, theseapproximation operators and their properties can be captured.

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