{
  "id": "2403.01274",
  "version": "v1",
  "published": "2024-03-02T17:32:14.000Z",
  "updated": "2024-03-02T17:32:14.000Z",
  "title": "Modal weak Kleene logics: axiomatizations and relational semantics",
  "authors": [
    "Stefano Bonzio",
    "Nicolò Zamperlin"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "Weak Kleene logics are three-valued logics characterized by the presence of an infectious truth-value. In their external versions, as they were originally introduced by Bochvar and Hallden, these systems are equipped with an additional connective capable of expressing whether a formula is classically true. In this paper we further expand the expressive power of external weak Kleen logics by modalizing them with a unary operator. The addition of an alethic modality gives rise to the two systems $\\B^{\\square}$ and $MPWK$, which have two different readings of the modal operator. We provide these logics with a complete and decidable Hilbert-style axiomatization w.r.t. a three-valued possible worlds semantics. The starting point of these calculi are new axiomatizations for the non-modal bases B and PWKe, which we provide using the recent algebraization results about these two logics. In particular, we prove the algebraizability of $\\PWKe$. Finally some standard extensions of the basic modal systems are provided with their completeness results w.r.t. special classes of frames.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-02T17:32:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03B45"
    ],
    "keywords": [
      "modal weak kleene logics",
      "relational semantics",
      "axiomatization",
      "external weak kleen logics",
      "basic modal systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}