  The article is a study of two algebraic structures, the `contrapositionally
complemented pseudo-Boolean algebra' (ccpBa) and `contrapositionally $\vee$
complemented pseudo-Boolean algebra' (c$\vee$cpBa). The algebras have recently
been obtained from a topos-theoretic study of categories of rough sets. The
salient feature of these algebras is that there are two negations, one
intuitionistic and another minimal in nature, along with a condition connecting
the two operators. We study properties of these algebras, give examples, and
compare them with relevant existing algebras. `Intuitionistic Logic with
Minimal Negation (ILM)' corresponding to ccpBas and its extension ILM-${\vee}$
for c$\vee$cpBas, are then investigated. Besides its relations with
intuitionistic and minimal logics, ILM is observed to be related to Peirce's
logic. With a focus on properties of the two negations, two kinds of relational
semantics for ILM and ILM-${\vee}$ are obtained, and an inter-translation
between the two semantics is provided. Extracting features of the two negations
in the algebras, a further investigation is made, following logical studies of
negations that define the operators independently of the binary operator of
implication. Using Dunn's logical framework for the purpose, two logics
$K_{im}$ and $K_{im-{\vee}}$ are presented, where the language does not include
implication. $K_{im}$-algebras are reducts of ccpBas. The negations in the
algebras are shown to occupy distinct positions in an enhanced form of Dunn's
Kite of negations. Relational semantics for $K_{im}$ and $K_{im-{\vee}}$ are
given, based on Dunn's compatibility frames. Finally, relationships are
established between the different algebraic and relational semantics for the
logics defined in the work.
