{
  "schema_version": "ocme-reviewed-question-reader-v1",
  "question_id": "q-20260921-proof-invalid-converse-0008",
  "pilot_number": "R0067-34333",
  "number": 34333,
  "family_id": "qf-proof-invalid-converse",
  "title": {
    "zh-Hant": "第 34333 題：無效的肯定後件",
    "en": "Question 34333: Invalid Affirmation of the Consequent"
  },
  "original_question": {
    "schema_version": "ocme-question-v0.1",
    "id": "q-20260921-proof-invalid-converse-0008",
    "batch_id": "batch-20260921-proof-foundations",
    "language": "zh-Hant",
    "source_mko_ids": [
      "mko-proof"
    ],
    "title_zh": "無效的肯定後件",
    "stem_zh": "命題 P：「整數 n 是 12 的倍數」；命題 Q：「整數 n 是 4 的倍數」。已明示前提為 P→Q 與 Q，欲判斷能否一般推出 P。這個 converse 是否有效？",
    "answer": {
      "kind": "choice",
      "value": "不可以",
      "choices": [
        "可以",
        "不可以"
      ],
      "explanation_zh": "令 P=false、Q=true：P→Q 與 Q 都為真，但 P 為假。因此不能由 P→Q 與 Q 一般推出 P。"
    },
    "difficulty": {
      "band": "upper_secondary",
      "level": 3,
      "skills": [
        "材料蘊涵",
        "無效 converse",
        "反例真值指派"
      ]
    },
    "variant": {
      "family_id": "qf-proof-invalid-converse",
      "seed": 20261129,
      "parameters": {
        "p": "整數 n 是 12 的倍數",
        "q": "整數 n 是 4 的倍數",
        "premises": [
          "P→Q",
          "Q"
        ],
        "conclusion": "P",
        "inference_rule": "invalid_converse",
        "counterexample_truth_assignment": {
          "P": false,
          "Q": true
        },
        "semantic_template": "qf-proof-invalid-converse/{\"conclusion\":\"P\",\"counterexample_truth_assignment\":{\"P\":false,\"Q\":true},\"inference_rule\":\"invalid_converse\",\"p\":\"整數 n 是 12 的倍數\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"整數 n 是 4 的倍數\"}"
      }
    },
    "verification": {
      "status": "candidate",
      "checks": [
        "schema_pending",
        "independent_recomputation_pending"
      ],
      "warning_zh": "本題是 candidate 練習；機械重算僅核對明示條件與指定實例，並非形式證明或人工審查。"
    },
    "provenance": {
      "generator_role": "question_producer",
      "generator_model": "GPT-5.6 Terra",
      "generated_at": "2026-09-21",
      "contract_version": "ocme-question-factory-v0.1"
    }
  },
  "original_english_edition": {
    "schema_version": "ocme-question-localization-v0.1",
    "question_id": "q-20260921-proof-invalid-converse-0008",
    "batch_id": "batch-20260921-proof-foundations",
    "locale": "en",
    "translation_state": "ai_candidate",
    "source": {
      "question_sha256": "089c89618a8a88c1a5ba38040e900c7f21f8cc5486ef9cc13a998c155421ff22",
      "family_id": "qf-proof-invalid-converse",
      "seed": 20261129,
      "semantic_template": "qf-proof-invalid-converse/{\"conclusion\":\"P\",\"counterexample_truth_assignment\":{\"P\":false,\"Q\":true},\"inference_rule\":\"invalid_converse\",\"p\":\"整數 n 是 12 的倍數\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"整數 n 是 4 的倍數\"}"
    },
    "fields": {
      "title": "Invalid Affirmation of the Consequent",
      "stem": "Let P mean “the integer n is a multiple of 12” and Q mean “the integer n is a multiple of 4”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?",
      "answer": {
        "kind": "choice",
        "value": "Cannot",
        "choices": [
          "Can",
          "Cannot"
        ],
        "explanation": "Take P=false and Q=true. Then P→Q and Q are both true while P is false, so P does not follow in general."
      },
      "skills": [
        "material implication",
        "invalid converse",
        "counterexample truth assignment"
      ],
      "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."
    }
  },
  "givens": [
    {
      "id": "G001",
      "text": {
        "zh-Hant": "命題 P：「整數 n 是 12 的倍數」；命題 Q：「整數 n 是 4 的倍數」。已明示前提為 P→Q 與 Q，欲判斷能否一般推出 P。這個 converse 是否有效？",
        "en": "Let P mean “the integer n is a multiple of 12” and Q mean “the integer n is a multiple of 4”. The stated premises are P→Q and Q, and the proposed conclusion is P. Under the converse inference, can the conclusion be derived in general?"
      }
    }
  ],
  "steps": [
    {
      "id": "S001",
      "uses": [
        "G001"
      ],
      "before": "source roles = [qf-proof-invalid-converse, [conclusion, counterexample_truth_assignment, inference_rule, p, premises, q]]",
      "after": "derived = [BASE-no-required-operation, no-invented-domain]",
      "relation": "implies",
      "rule_id": "R67_SOURCE_specific_keys_operation_domain_and_numeric_bank",
      "conditions": [],
      "explanation": {
        "zh-Hant": "按本bank核keys與operation/domain：legacy為各表underscore，MOD50用local hyphen；BASE參數較寬不加未知欄限制。Number.isInteger與SAFE兩種來源界分開，原JSON數字表示仍完整保存。",
        "en": "Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。",
          "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S002",
      "uses": [
        "G001",
        "S001"
      ],
      "before": "source roles = [invalid_converse, [P→Q, Q], P]",
      "after": "derived = [[P, Q, P->Q, second premise, both premises, conclusion], [[否, 否, 是, 否, 否, 否], [否, 是, 是, 是, 是, 否], [是, 否, 否, 否, 否, 是], [是, 是, 是, 是, 是, 是]], 否]",
      "relation": "implies",
      "rule_id": "Proof_FULL_truth_table_premise_satisfaction_THEN_conclusion",
      "conditions": [],
      "explanation": {
        "zh-Hant": "把P/Q當一般命題符號，不加入例示文字間額外數學關係。依欄名逐四種truth valuations核兩前提；有效需每個「前提全真」row的結論真。不是由前提形式推它們在現實已真，也不是因果推論。",
        "en": "Treat P/Q as GENERAL proposition symbols,without adding extra arithmetic relations between their illustrative texts.Use the displayed COLUMN LABELS to audit ALL four valuations.Validity requires a true conclusion in EVERY row satisfying BOTH premises.This neither proves the premises factually true nor establishes causation."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。",
          "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof."
        }
      },
      "arithmetic": null
    },
    {
      "id": "S003",
      "uses": [
        "G001",
        "S002"
      ],
      "before": "source roles = [否, 是, [P→Q, Q]]",
      "after": "derived = [[否, 是, 是, 是, 是, 否], 否]",
      "relation": "implies",
      "rule_id": "Proof_CONVERSE_explicit_Boolean_countervaluation_BOTH_premises_true",
      "conditions": [],
      "explanation": {
        "zh-Hant": "P假Q真時P→Q真且Q真，但結論P假，故一般不能affirm consequent。這是一般規則反例；若另有特定P/Q等價定理，須額外證明，不能偷加到本推論。",
        "en": "With P false/Q true,the implication and Q are BOTH true while conclusion P is false.The converse inference is therefore invalid IN GENERAL.A special equivalence theorem for particular P/Q would require an EXTRA proof,not a silently added premise."
      },
      "reverse_audit": {
        "kind": "premise_and_result_audit",
        "explanation": {
          "zh-Hant": "以本題原始角色、完整前提及欄序反查本步；區分單向推論與等價，並保留原題與英文、MKO、AST及審查狀態。不把有限核對冒充獨立審查或一般形式證明。",
          "en": "Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof."
        }
      },
      "arithmetic": null
    }
  ],
  "math_model": {
    "result": "不可以"
  },
  "conclusion": {
    "canonical_value": "不可以",
    "explanation": {
      "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。",
      "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged."
    }
  },
  "definition_note": {
    "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。",
    "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate."
  },
  "source": {
    "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db",
    "question_binding": {
      "batch_id": "batch-20260921-proof-foundations",
      "batch_path": "public/data/questions/batches/batch-20260921-proof-foundations/questions.jsonl",
      "batch_sha256": "16755575d0aefd29143eb27fb780816bbf67cc3ca2da5ecfa277cea483cf877f",
      "original_line": 58,
      "original_line_sha256": "c66ae62e553f46f1fd35df13a05d262ddca6a6a1a55400661bd9c0b4f4736db4",
      "question_compact_sha256": "089c89618a8a88c1a5ba38040e900c7f21f8cc5486ef9cc13a998c155421ff22",
      "english_row_compact_sha256": "cc5dd3e65daf2d61522695fb48efeda7171134d788eb12acd662d26519b84cb5",
      "english_manifest_path": "public/data/i18n/editions/batch-20260921-proof-foundations.en.json",
      "english_rows_path": "public/data/i18n/questions/batch-20260921-proof-foundations.en.jsonl"
    },
    "mko_bindings": {
      "mko-proof": {
        "sha256": "72f6009b912caa3c05a2948a97ca422367574a19ad5fc665964a540875417b91",
        "english_sha256": "1c8ec335b48f7d913d3057b2d632c371b092ca869f799b056b72464d948504fe",
        "original_path": "public/data/mko/proof.json",
        "english_path": "public/data/i18n/editions/mko-proof.en.json"
      }
    },
    "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff",
    "selected_part_sha256": "6bb156cec7bdb6baf2497762702fc01f94fd1db55a5926bc575b12ff9237a4de",
    "original_candidate_sha256": "ae334e851b6667c96e2281bb90b194cd1c07fd4c93120dc918be4a8d2c0789df",
    "original_candidate_line_sha256": "ae334e851b6667c96e2281bb90b194cd1c07fd4c93120dc918be4a8d2c0789df"
  },
  "historical_source_state": {
    "review_state": {
      "derivation": "candidate_authored_and_locally_checked",
      "source_question_state": "candidate",
      "independent_AI_or_human_review": "NOT_PERFORMED",
      "formalization": "not_formalized",
      "Lean": "NOT_RUN"
    },
    "scope_note": {
      "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。",
      "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged."
    },
    "label": "Frozen producer metadata; current per-question review is separate below."
  },
  "revision": {
    "id": "review-20261026-012-p68",
    "review_date": "2026-10-26",
    "owner_authorized_reader_publication": true,
    "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.",
    "review_receipts": {
      "semantic_review_sha256": "c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c",
      "question_first_witness_sha256": "12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57"
    },
    "status": "second_context_reviewed",
    "mathematical_semantics": true,
    "bilingual_semantics": true,
    "steps_reviewed": 3,
    "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction",
    "exact_question_first_check": {
      "model_sha256": "a15eafefa7f1b1837022abda8c85e5e9f0e46e80b377c80dce1d3bde832c7145",
      "arithmetic_steps": 0
    },
    "uses_answer_as_premise": false,
    "human_review": "NOT_PERFORMED",
    "formalization": "not_formalized",
    "Lean": "NOT_RUN",
    "Canonical_truth_promotion": false,
    "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged."
  }
}