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The article is a study of two algebraic structures, the `contrapositionallycomplemented pseudo-Boolean algebra' (ccpBa) and `contrapositionally $\vee$complemented pseudo-Boolean algebra' (c$\vee$cpBa). The algebras have recentlybeen obtained from a topos-theoretic study of categories of rough sets. Thesalient feature of these algebras is that there are two negations, oneintuitionistic and another minimal in nature, along with a condition connectingthe two operators. We study properties of these algebras, give examples, andcompare them with relevant existing algebras. `Intuitionistic Logic withMinimal Negation (ILM)' corresponding to ccpBas and its extension ILM-${\vee}$for c$\vee$cpBas, are then investigated. Besides its relations withintuitionistic and minimal logics, ILM is observed to be related to Peirce'slogic. With a focus on properties of the two negations, two kinds of relationalsemantics for ILM and ILM-${\vee}$ are obtained, and an inter-translationbetween the two semantics is provided. Extracting features of the two negationsin the algebras, a further investigation is made, following logical studies ofnegations that define the operators independently of the binary operator ofimplication. Using Dunn's logical framework for the purpose, two logics$K_{im}$ and $K_{im-{\vee}}$ are presented, where the language does not includeimplication. $K_{im}$-algebras are reducts of ccpBas. The negations in thealgebras are shown to occupy distinct positions in an enhanced form of Dunn'sKite of negations. Relational semantics for $K_{im}$ and $K_{im-{\vee}}$ aregiven, based on Dunn's compatibility frames. Finally, relationships areestablished between the different algebraic and relational semantics for thelogics defined in the work.

Contrapositionally Complemented Pseudo-Boolean Algebras and Intuitionistic Logic with Minimal Negation