{
  "$schema":"../../../schemas/mko-v0.4.schema.json","schema_version":"mko-v0.4","id":"mko-logical-connectives","semantic_family_id":"sf-logical-connectives","type":"definition","version":"0.10.0","language":"zh-Hant",
  "titles":{"zh-Hant":"邏輯連接詞","en":"Logical connectives"},
  "summary":{"zh-Hant":"否定、合取、析取、蘊涵與雙條件用來由既有命題形成複合命題。"},
  "statement":{"zh-Hant":"對命題 P、Q，¬P 表示否定，P∧Q 表示兩者皆成立，P∨Q 表示至少一者成立，P→Q 表示若 P 則 Q，P↔Q 表示兩方向皆成立。"},
  "formula":{"tex":"P\\land Q","mathml":"<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>P</mi><mo>∧</mo><mi>Q</mi></mrow></math>","semantic_ast":{"type":"logical_and","left":{"type":"symbol","name":"P","glyph":"P"},"right":{"type":"symbol","name":"Q","glyph":"Q"}},"compiler":{"id":"ocme-formula-core","version":"0.4.0","source_sha256":"b285e5915858c4dcba1f721a295d10bf16cc2fe2376b19433365e7fecdb03593"}},
  "assumptions":[{"id":"a1","zh-Hant":"P 與 Q 是具有明確命題語境的命題。"}],
  "symbols":[{"token":"P","role":"left_proposition","role_zh":"左命題","scope":"definition"},{"token":"Q","role":"right_proposition","role_zh":"右命題","scope":"definition"},{"token":"∧","role":"conjunction","role_zh":"合取連接詞","scope":"representative_formula"}],
  "dependencies":[{"id":"mko-proposition","title_zh":"命題","reason_zh":"邏輯連接詞以一個或兩個命題形成複合命題。"}],
  "explanation":{"paragraphs_zh":["合取 P∧Q 只有在 P、Q 都成立時成立；析取 P∨Q 採含括或，允許兩者同時成立。","材料蘊涵 P→Q 是命題關係，不自動表示時間先後或因果；雙條件 P↔Q 則要求兩方向的蘊涵。","符號的作用域與括號會改變複合命題的結構，閱讀時必須先辨識最外層連接詞。"],"common_misconception_zh":"常見誤解是把析取當成排他或、把蘊涵當成因果，或只證一個方向便宣稱雙條件。"},
  "proofs":[],"computational_companions":[],
  "verification":{"computational_status":"not_evaluated_no_computational_claim","warning_zh":"真值表可檢查有限命題指派，但不構成全部推理規則的形式證明。","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":"分別綁定 Not、And、Or、箭頭與 Iff 的精確形式聲明及證據。"},
  "provenance":{"origin":"ocme_local_ai_materialization_candidate","note_zh":"依 research/atlas-logical-connectives.md 整理；待人工審查。","content_license":"CC-BY-SA-4.0","code_license":"MIT","created_at":"2026-09-16"}
}