{
  "abstract": "Zadeh fuzzy logic is normally introduced by assigning scalar membership degrees and postulating standard complement, minimum conjunction, and maximum disjunction. We derive this calculus from a strictly positive atomless normalized Boolean measure algebra. Equality of measure sends Boolean complement uniquely to $1-x$, but it cannot send meet to a single-valued operation: the exact scalar image of conjunction is the overlap interval $[(x+y-1)^+,\\min\\{x,y\\}]$. We quantify the information lost under scalarization, prove that no nontrivial measure-preserving compression can retain meet, and show that the overlap operation is associative as a hyperoperation. Zadeh conjunction $\\min\\{x,y\\}$ is then forced as the unique nondecreasing idempotent selection and is realized globally by nested event chains; its De Morgan dual is Zadeh disjunction $\\max\\{x,y\\}$. Thus the min--max--standard-negation calculus follows from measure theory once the nesting commitment is made explicit. The \\Luk{} and product conjunctions arise as minimum-overlap and independent-factor selections, clarifying alternative scalarizations. Exact minimax results, an ordered-valued extension, and a three-dimensional realization obstruction delimit what scalar fuzzy logic retains from Boolean structure.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28510",
  "author_total": 1,
  "authors": [
    {
      "name": "Angshul Majumdar"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7n6b5ma620hnx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28510",
  "fulltext_url": null,
  "id": "7n6b5ma620hnx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28510",
  "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.GM"
    ],
    [
      "openalex",
      "1703|Advanced Algebra and Logic"
    ]
  ],
  "openalex": "W7214310398",
  "pdf_url": "https://arxiv.org/pdf/2609.28510",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7n6b5ma620hnx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28510",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28510"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "A Measure-Theoretic Derivation of Zadeh Fuzzy Logic",
  "title_text": "A Measure-Theoretic Derivation of Zadeh Fuzzy Logic",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/7n6b5ma620hnx",
  "venue": "arXiv",
  "versions": []
}