{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-quantifier",
  "semantic_family_id": "sf-quantifier",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "量詞與明示論域"
  },
  "summary": {
    "zh-Hant": "全稱與存在量詞必須綁定論域；古典真值語義下，空論域的全稱為真、存在為假。"
  },
  "statement": {
    "zh-Hant": "在明示論域 U 與述詞 P 下，∀x∈U P(x) 表示每個 U 中元素滿足 P，∃x∈U P(x) 表示至少一個 U 中元素滿足 P。採古典真值語義，¬∀x∈U P(x) 等價於 ∃x∈U ¬P(x)，¬∃x∈U P(x) 等價於 ∀x∈U ¬P(x)。"
  },
  "formula": {
    "tex": "P(x)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>P</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "P",
        "glyph": "P"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "9b33b3c569b4eaaf51f044129ec0732a7e7db11c398becde1408432d6a50381a"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "論域 U 和述詞 P 的語義已明示；本輪採古典真值解讀。"
    },
    {
      "id": "a2",
      "zh-Hant": "題庫限定有限明示整數論域與可判定述詞，完整枚舉只封閉該有限論域。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "predicate",
      "role_zh": "已指定語義的述詞，P(x) 真值依輸入而定",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "predicate_input",
      "role_zh": "述詞輸入；量詞綁定見完整陳述",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-proposition",
      "title_zh": "命題",
      "reason_zh": "量化形成封閉陳述；本輪有限可判定題目可逐項驗證真值。"
    },
    {
      "id": "mko-set",
      "title_zh": "集合",
      "reason_zh": "量詞的限制論域由集合指定。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "全稱要求所有允許元素都通過；存在只需一個允許見證。量詞順序與論域屬於陳述本身，不可任意交換或忽略。",
      "U=∅ 時，沒有反例可違反全稱，因此全稱為真；存在無可用見證，因此為假。否定量詞時同時切換全稱／存在並否定述詞，不能只改符號。",
      "本輪逐一列出有限 U 的整數元素，精確檢查大小、相等或非零整除述詞。結果不外推到全部整數、實數或無限論域，也不宣稱一階邏輯的完備性已形式證明。"
    ],
    "common_misconception_zh": "把幾個通過例子當成無限全稱證明，或把否定全稱誤寫成所有元素皆不滿足。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "有限可判定真值可完整重算；一般量詞推理與古典邏輯形式橋仍未附 Evidence。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "綁定 Forall、Exists、受限量詞、古典否定法則及空論域邊界。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-quantifier.md 整理；2026-10-03 脈衝 R01（依使用者要求提前完成 2026-10-04 的流程），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-03"
  }
}