  A formal context consists of objects, properties, and the incidence relation
between them. Various notions of concepts defined with respect to formal
contexts and their associated algebraic structures have been studied
extensively, including formal concepts in formal concept analysis (FCA), rough
concepts arising from rough set theory (RST), and semiconcepts and
protoconcepts for dealing with negation. While all these kinds of concepts are
associated with lattices, semiconcepts and protoconcepts additionally yield an
ordered algebraic structure, called double Boolean algebras. As the name
suggests, a double Boolean algebra contains two underlying Boolean algebras.
  In this paper, we investigate logical and algebraic aspects of the
representation and reasoning about different concepts with respect to formal
contexts. We present two-sorted modal logic systems \textbf{KB} and \textbf{KF}
for the representation and reasoning of rough concepts and formal concepts
respectively. Then, in order to represent and reason about both formal and
rough concepts in a single framework, these two logics are unified into a
two-sorted Boolean modal logic \textbf{BM}, in which semiconcepts and
protoconcepts are also expressible. Based on the logical representation of
semiconcepts and protoconcepts, we prove the characterization of double Boolean
algebras in terms of their underlying Boolean algebras. Finally, we also
discuss the possibilities of extending our logical systems for the
representation and reasoning of more fine-grained information in formal
contexts.
