{
  "id": "2010.15944",
  "version": "v1",
  "published": "2020-10-29T21:05:24.000Z",
  "updated": "2020-10-29T21:05:24.000Z",
  "title": "Contrapositionally Complemented Pseudo-Boolean Algebras and Intuitionistic Logic with Minimal Negation",
  "authors": [
    "Anuj Kumar More",
    "Mohua Banerjee"
  ],
  "comment": "36 pages",
  "categories": [
    "math.LO"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-29T21:05:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03B60",
      "06D20",
      "03B20",
      "03G10",
      "F.4.1"
    ],
    "keywords": [
      "contrapositionally complemented pseudo-boolean algebra",
      "intuitionistic logic",
      "minimal negation",
      "relational semantics",
      "dunns compatibility frames"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 36,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}