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

TL;DR

This paper introduces and studies new algebraic structures with dual negations related to intuitionistic logic, providing semantics and logical frameworks that connect to existing logics like Peirce's and Dunn's negation systems.

Contribution

It defines contrapositionally complemented pseudo-Boolean algebras with two negations, explores their properties, semantics, and logical systems, and relates them to existing algebraic and logical frameworks.

Findings

Two types of negations are characterized and compared.

Relational semantics for the new logics are developed.

Connections with Peirce's logic and Dunn's negation framework are established.

Abstract

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.