{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-proposition",
  "semantic_family_id": "sf-proposition",
  "type": "concept",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "命題",
    "en": "Proposition"
  },
  "summary": {
    "zh-Hant": "命題是在指定邏輯脈絡中可被斷言或否定，並可談論真假的有意義陳述。"
  },
  "statement": {
    "zh-Hant": "本條目採入門古典邏輯語境：命題 P 可以為真或為假；P 是命題不表示 P 已被證明為真。"
  },
  "formula": {
    "tex": "P",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mi>P</mi></math>",
    "semantic_ast": {
      "type": "symbol",
      "name": "P",
      "glyph": "P"
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "5c62e091b8c0565f1bafad0dad5934276143ae2ccef7a5381e8ada5b1a8d26d2"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "P 是指定邏輯脈絡中良構且有意義的命題。"
    },
    {
      "id": "a2",
      "zh-Hant": "真與假的二分敘述限於本條目採用的入門古典語境。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "proposition_variable",
      "role_zh": "命題變項",
      "scope": "concept"
    }
  ],
  "dependencies": [],
  "explanation": {
    "paragraphs_zh": [
      "命題是可以被斷言或否定的陳述。真的命題與假的命題都仍是命題；真假是命題的語義狀態，而不是判定某段文字是否屬於命題的唯一先決條件。",
      "證明用來支持某命題為真，但『P 是命題』不等於『P 已經有證明』。在 Lean 中可先有 P : Prop，再另外詢問是否存在 P 的證明項。",
      "像 x>0 這類含未指定自由變數的開放句，需經代入、量化或明確指定脈絡，才成為可在該封閉語境中評估的陳述。"
    ],
    "common_misconception_zh": "常見誤解是把疑問句、命令句或未封閉的開放表達式直接當作命題，或把『是命題』誤解成『已證明為真』。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "本條目是概念候選稿；研究來源不是 OCME Evidence，尚未建立計算檢查或 Lean 形式證明。",
    "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": "以 P : Prop 建立命題與證明項的最小 Lean 語義橋，並清楚區分古典真假語義所需的額外原理。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-proposition.md 的來源與語義決定整理；內容為 OCME 原創改寫，待人工審查。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-08-07"
  }
}