  A non-distributive two-sorted hypersequent calculus \textbf{PDBL} and its
modal extension \textbf{MPDBL} are proposed for the classes of pure double
Boolean algebras and pure double Boolean algebras with operators respectively.
A relational semantics for \textbf{PDBL} is next proposed, where any formula is
interpreted as a semiconcept of a context. For \textbf{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 \textbf{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
\textbf{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 \textbf{MPDBL}, these
approximation operators and their properties can be captured.
