{
  "abstract": "We study the minimal tense logic $K_t$ from an algebraic and proof-theoretic perspective. We introduce the degree-of-truth-preserving logic associated with the class of tense Boolean algebras. We then introduce a sequent calculus for this logic and establish its soundness and completeness with respect to tense Boolean algebras by means of an adaptation of the Lindenbaum--Tarski construction. Consequently, this calculus provides an additional syntactic presentation of the minimal tense logic $K_t$. We also provide a second, purely syntactic proof of completeness. Furthermore, we establish an algebraic soundness and completeness theorem for Kashima's Gentzen-style calculus. To this end, we develop an adaptation of the Lindenbaum--Tarski construction to Kashima's nested sequent framework, which allows us to construct the algebraic semantics directly from the proof-theoretic system. This yields a direct algebraic completeness proof for Kashima's calculus and connects its nested-sequent formulation with the algebraic semantics of tense Boolean algebras.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01547",
  "author_total": 3,
  "authors": [
    {
      "name": "Martín Figallo"
    },
    {
      "name": "Jonathan Sarmiento"
    },
    {
      "name": "Luis Pezzini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/939zx3p93fcyz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01547",
  "fulltext_url": null,
  "id": "939zx3p93fcyz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01547",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7215891774",
  "pdf_url": "https://arxiv.org/pdf/2610.01547",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/939zx3p93fcyz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01547",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01547"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Tense Logic via Truth Degrees: An Algebraic Completeness Result for Kashima's Calculus",
  "title_text": "Tense Logic via Truth Degrees: An Algebraic Completeness Result for Kashima's Calculus",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/939zx3p93fcyz",
  "venue": "arXiv",
  "versions": []
}