  The notion of a context in formal concept analysis and that of an
approximation space in rough set theory are unified in this study to define a
Kripke context. For any context (G,M,I), a relation on the set G of objects and
a relation on the set M of properties are included, giving a structure of the
form ((G,R), (M,S), I). A Kripke context gives rise to complex algebras based
on the collections of protoconcepts and semiconcepts of the underlying context.
On abstraction, double Boolean algebras (dBas) with operators and topological
dBas are defined. Representation results for these algebras are established in
terms of the complex algebras of an appropriate Kripke context. As a natural
next step, logics corresponding to classes of these algebras are formulated. A
sequent calculus is proposed for contextual dBas, modal extensions of which
give logics for contextual dBas with operators and topological contextual dBas.
The representation theorems for the algebras result in a protoconcept-based
semantics for these logics.
