{
  "id": "1907.04477",
  "version": "v1",
  "published": "2019-07-10T01:16:21.000Z",
  "updated": "2019-07-10T01:16:21.000Z",
  "title": "The First Epsilon Theorem in Pure Intuitionistic and Intermediate Logics",
  "authors": [
    "Matthias Baaz",
    "Richard Zach"
  ],
  "categories": [
    "math.LO"
  ],
  "abstract": "By adding a $\\tau$ operator, versions of Hilbert's $\\varepsilon$-calculus can be obtained for intermediate propositional logics including intuitionistic logic. It is well known that such calculi, in contrast to the $\\varepsilon$-calculus for classical logic, are not conservative. In particular, any $\\varepsilon$-$\\tau$ calculus for an intermediate logic proves all classically valid quantifier shift principles. The resulting calculi are however conservative over their propositional fragments. One important result pertaining to the $\\varepsilon$-calculus is the first epsilon theorem, which is closely related to Herbrand's theorem for existential formulas. It is shown that finite-valued G\\\"odel logics have the first epsilon theorem, and no other intermediate logic does. However, there are partial results for other intermediate logics including G\\\"odel-Dummett logic, the logic of weak excluded middle, and intuitionistic logic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-10T01:16:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03F05",
      "03B20",
      "03B55"
    ],
    "keywords": [
      "first epsilon theorem",
      "intermediate logic",
      "pure intuitionistic",
      "classically valid quantifier shift principles",
      "intuitionistic logic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}