{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0001","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「整數 n 可被 4 整除」；命題 Q：「整數 n 是偶數」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260922,"parameters":{"p":"整數 n 可被 4 整除","q":"整數 n 是偶數","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"整數 n 可被 4 整除\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"整數 n 是偶數\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0002","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「整數 n 可被 6 整除」；命題 Q：「整數 n 可被 3 整除」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260923,"parameters":{"p":"整數 n 可被 6 整除","q":"整數 n 可被 3 整除","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"整數 n 可被 6 整除\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"整數 n 可被 3 整除\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0003","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「整數 n 是 9 的倍數」；命題 Q：「整數 n 是 3 的倍數」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260924,"parameters":{"p":"整數 n 是 9 的倍數","q":"整數 n 是 3 的倍數","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"整數 n 是 9 的倍數\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"整數 n 是 3 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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0004","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「整數 n 是質數且大於 2」；命題 Q：「整數 n 是奇數」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260925,"parameters":{"p":"整數 n 是質數且大於 2","q":"整數 n 是奇數","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"整數 n 是質數且大於 2\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"整數 n 是奇數\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0005","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「三角形有三條相等的邊」；命題 Q：「三角形至少有兩條相等的邊」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260926,"parameters":{"p":"三角形有三條相等的邊","q":"三角形至少有兩條相等的邊","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"三角形有三條相等的邊\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"三角形至少有兩條相等的邊\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0006","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「四邊形是正方形」；命題 Q：「四邊形有四個直角」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260927,"parameters":{"p":"四邊形是正方形","q":"四邊形有四個直角","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"四邊形是正方形\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"四邊形有四個直角\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0007","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「整數 n 是 10 的倍數」；命題 Q：「整數 n 是 5 的倍數」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260928,"parameters":{"p":"整數 n 是 10 的倍數","q":"整數 n 是 5 的倍數","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"整數 n 是 10 的倍數\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"整數 n 是 5 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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0008","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","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 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260929,"parameters":{"p":"整數 n 是 12 的倍數","q":"整數 n 是 4 的倍數","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"整數 n 是 12 的倍數\",\"premises\":[\"P→Q\",\"P\"],\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0009","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「整數 n 是完全平方數」；命題 Q：「存在整數 k 使 n=k²」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260930,"parameters":{"p":"整數 n 是完全平方數","q":"存在整數 k 使 n=k²","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"整數 n 是完全平方數\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"存在整數 k 使 n=k²\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0010","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「有理數 r 等於 0」；命題 Q：「r 的絕對值等於 0」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260931,"parameters":{"p":"有理數 r 等於 0","q":"r 的絕對值等於 0","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"有理數 r 等於 0\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"r 的絕對值等於 0\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0011","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「集合 A 是空集合」；命題 Q：「A 沒有元素」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260932,"parameters":{"p":"集合 A 是空集合","q":"A 沒有元素","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"集合 A 是空集合\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"A 沒有元素\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0012","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「函數 f 是常數函數」；命題 Q：「任兩個輸入有相同輸出」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260933,"parameters":{"p":"函數 f 是常數函數","q":"任兩個輸入有相同輸出","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"函數 f 是常數函數\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"任兩個輸入有相同輸出\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0013","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「角的量值是 90°」；命題 Q：「角是直角」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260934,"parameters":{"p":"角的量值是 90°","q":"角是直角","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"角的量值是 90°\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"角是直角\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0014","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「角的量值是 180°」；命題 Q：「角是平角」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260935,"parameters":{"p":"角的量值是 180°","q":"角是平角","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"角的量值是 180°\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"角是平角\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0015","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「整數 n 可被 8 整除」；命題 Q：「整數 n 可被 2 整除」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260936,"parameters":{"p":"整數 n 可被 8 整除","q":"整數 n 可被 2 整除","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"整數 n 可被 8 整除\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"整數 n 可被 2 整除\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0016","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「整數 n 是 15 的倍數」；命題 Q：「整數 n 可被 5 整除」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260937,"parameters":{"p":"整數 n 是 15 的倍數","q":"整數 n 可被 5 整除","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"整數 n 是 15 的倍數\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"整數 n 可被 5 整除\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0017","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「數 x 大於 7」；命題 Q：「x 大於 0」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260938,"parameters":{"p":"數 x 大於 7","q":"x 大於 0","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"數 x 大於 7\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"x 大於 0\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0018","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「數 x 小於 -3」；命題 Q：「x 小於 0」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260939,"parameters":{"p":"數 x 小於 -3","q":"x 小於 0","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"數 x 小於 -3\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"x 小於 0\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0019","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「多邊形是五邊形」；命題 Q：「多邊形有五條邊」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260940,"parameters":{"p":"多邊形是五邊形","q":"多邊形有五條邊","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"多邊形是五邊形\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"多邊形有五條邊\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0020","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「矩形的長與寬相等」；命題 Q：「矩形是正方形」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260941,"parameters":{"p":"矩形的長與寬相等","q":"矩形是正方形","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"矩形的長與寬相等\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"矩形是正方形\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0021","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「整數 n 是偶數」；命題 Q：「n² 是偶數」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260942,"parameters":{"p":"整數 n 是偶數","q":"n² 是偶數","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"整數 n 是偶數\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"n² 是偶數\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0022","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「整數 n 可被 3 整除」；命題 Q：「n² 可被 3 整除」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260943,"parameters":{"p":"整數 n 可被 3 整除","q":"n² 可被 3 整除","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"整數 n 可被 3 整除\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"n² 可被 3 整除\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0023","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「分數 a/b 的分母 b 不等於 0」；命題 Q：「a/b 是有定義的分數表示」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260944,"parameters":{"p":"分數 a/b 的分母 b 不等於 0","q":"a/b 是有定義的分數表示","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"分數 a/b 的分母 b 不等於 0\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"a/b 是有定義的分數表示\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0024","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「集合 A 是 B 的子集」；命題 Q：「A 中每個元素也在 B 中」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260945,"parameters":{"p":"集合 A 是 B 的子集","q":"A 中每個元素也在 B 中","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"集合 A 是 B 的子集\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"A 中每個元素也在 B 中\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-ponens-0025","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"肯定前件","stem_zh":"命題 P：「命題 R 與 S 都成立」；命題 Q：「命題 R 成立」。已明示前提為 P→Q 與 P，欲判斷結論為 Q。依 modus ponens，是否可推出 Q？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 P；這正是 modus ponens，故可推出 Q。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus ponens"]},"variant":{"family_id":"qf-proof-modus-ponens","seed":20260946,"parameters":{"p":"命題 R 與 S 都成立","q":"命題 R 成立","premises":["P→Q","P"],"conclusion":"Q","inference_rule":"modus_ponens","semantic_template":"qf-proof-modus-ponens/{\"conclusion\":\"Q\",\"inference_rule\":\"modus_ponens\",\"p\":\"命題 R 與 S 都成立\",\"premises\":[\"P→Q\",\"P\"],\"q\":\"命題 R 成立\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0001","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「整數 n 可被 4 整除」；命題 Q：「整數 n 是偶數」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261022,"parameters":{"p":"整數 n 可被 4 整除","q":"整數 n 是偶數","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"整數 n 可被 4 整除\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"整數 n 是偶數\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0002","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「整數 n 可被 6 整除」；命題 Q：「整數 n 可被 3 整除」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261023,"parameters":{"p":"整數 n 可被 6 整除","q":"整數 n 可被 3 整除","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"整數 n 可被 6 整除\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"整數 n 可被 3 整除\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0003","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「整數 n 是 9 的倍數」；命題 Q：「整數 n 是 3 的倍數」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261024,"parameters":{"p":"整數 n 是 9 的倍數","q":"整數 n 是 3 的倍數","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"整數 n 是 9 的倍數\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"整數 n 是 3 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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0004","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「整數 n 是質數且大於 2」；命題 Q：「整數 n 是奇數」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261025,"parameters":{"p":"整數 n 是質數且大於 2","q":"整數 n 是奇數","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"整數 n 是質數且大於 2\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"整數 n 是奇數\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0005","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「三角形有三條相等的邊」；命題 Q：「三角形至少有兩條相等的邊」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261026,"parameters":{"p":"三角形有三條相等的邊","q":"三角形至少有兩條相等的邊","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"三角形有三條相等的邊\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"三角形至少有兩條相等的邊\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0006","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「四邊形是正方形」；命題 Q：「四邊形有四個直角」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261027,"parameters":{"p":"四邊形是正方形","q":"四邊形有四個直角","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"四邊形是正方形\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"四邊形有四個直角\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0007","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「整數 n 是 10 的倍數」；命題 Q：「整數 n 是 5 的倍數」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261028,"parameters":{"p":"整數 n 是 10 的倍數","q":"整數 n 是 5 的倍數","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"整數 n 是 10 的倍數\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"整數 n 是 5 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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0008","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","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，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261029,"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":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0009","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「整數 n 是完全平方數」；命題 Q：「存在整數 k 使 n=k²」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261030,"parameters":{"p":"整數 n 是完全平方數","q":"存在整數 k 使 n=k²","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"整數 n 是完全平方數\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"存在整數 k 使 n=k²\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0010","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「有理數 r 等於 0」；命題 Q：「r 的絕對值等於 0」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261031,"parameters":{"p":"有理數 r 等於 0","q":"r 的絕對值等於 0","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"有理數 r 等於 0\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"r 的絕對值等於 0\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0011","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「集合 A 是空集合」；命題 Q：「A 沒有元素」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261032,"parameters":{"p":"集合 A 是空集合","q":"A 沒有元素","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"集合 A 是空集合\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"A 沒有元素\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0012","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「函數 f 是常數函數」；命題 Q：「任兩個輸入有相同輸出」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261033,"parameters":{"p":"函數 f 是常數函數","q":"任兩個輸入有相同輸出","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"函數 f 是常數函數\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"任兩個輸入有相同輸出\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0013","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「角的量值是 90°」；命題 Q：「角是直角」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261034,"parameters":{"p":"角的量值是 90°","q":"角是直角","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"角的量值是 90°\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"角是直角\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0014","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「角的量值是 180°」；命題 Q：「角是平角」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261035,"parameters":{"p":"角的量值是 180°","q":"角是平角","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"角的量值是 180°\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"角是平角\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0015","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「整數 n 可被 8 整除」；命題 Q：「整數 n 可被 2 整除」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261036,"parameters":{"p":"整數 n 可被 8 整除","q":"整數 n 可被 2 整除","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"整數 n 可被 8 整除\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"整數 n 可被 2 整除\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0016","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「整數 n 是 15 的倍數」；命題 Q：「整數 n 可被 5 整除」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261037,"parameters":{"p":"整數 n 是 15 的倍數","q":"整數 n 可被 5 整除","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"整數 n 是 15 的倍數\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"整數 n 可被 5 整除\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0017","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「數 x 大於 7」；命題 Q：「x 大於 0」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261038,"parameters":{"p":"數 x 大於 7","q":"x 大於 0","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"數 x 大於 7\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"x 大於 0\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0018","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「數 x 小於 -3」；命題 Q：「x 小於 0」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261039,"parameters":{"p":"數 x 小於 -3","q":"x 小於 0","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"數 x 小於 -3\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"x 小於 0\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0019","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「多邊形是五邊形」；命題 Q：「多邊形有五條邊」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261040,"parameters":{"p":"多邊形是五邊形","q":"多邊形有五條邊","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"多邊形是五邊形\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"多邊形有五條邊\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0020","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「矩形的長與寬相等」；命題 Q：「矩形是正方形」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261041,"parameters":{"p":"矩形的長與寬相等","q":"矩形是正方形","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"矩形的長與寬相等\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"矩形是正方形\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0021","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「整數 n 是偶數」；命題 Q：「n² 是偶數」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261042,"parameters":{"p":"整數 n 是偶數","q":"n² 是偶數","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"整數 n 是偶數\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"n² 是偶數\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0022","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「整數 n 可被 3 整除」；命題 Q：「n² 可被 3 整除」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261043,"parameters":{"p":"整數 n 可被 3 整除","q":"n² 可被 3 整除","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"整數 n 可被 3 整除\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"n² 可被 3 整除\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0023","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「分數 a/b 的分母 b 不等於 0」；命題 Q：「a/b 是有定義的分數表示」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261044,"parameters":{"p":"分數 a/b 的分母 b 不等於 0","q":"a/b 是有定義的分數表示","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"分數 a/b 的分母 b 不等於 0\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"a/b 是有定義的分數表示\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0024","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「集合 A 是 B 的子集」；命題 Q：「A 中每個元素也在 B 中」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261045,"parameters":{"p":"集合 A 是 B 的子集","q":"A 中每個元素也在 B 中","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"集合 A 是 B 的子集\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"A 中每個元素也在 B 中\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-modus-tollens-0025","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"否定後件","stem_zh":"命題 P：「命題 R 與 S 都成立」；命題 Q：「命題 R 成立」。已明示前提為 P→Q 與 ¬Q，欲判斷結論為 ¬P。依 modus tollens，是否可推出 ¬P？","answer":{"kind":"choice","value":"可以","choices":["可以","不可以"],"explanation_zh":"前提為 P→Q 與 ¬Q；這正是 modus tollens，故可推出 ¬P。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["材料蘊涵","modus tollens"]},"variant":{"family_id":"qf-proof-modus-tollens","seed":20261046,"parameters":{"p":"命題 R 與 S 都成立","q":"命題 R 成立","premises":["P→Q","¬Q"],"conclusion":"¬P","inference_rule":"modus_tollens","semantic_template":"qf-proof-modus-tollens/{\"conclusion\":\"¬P\",\"inference_rule\":\"modus_tollens\",\"p\":\"命題 R 與 S 都成立\",\"premises\":[\"P→Q\",\"¬Q\"],\"q\":\"命題 R 成立\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0001","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 可被 4 整除」；命題 Q：「整數 n 是偶數」。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":2,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261122,"parameters":{"p":"整數 n 可被 4 整除","q":"整數 n 是偶數","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 可被 4 整除\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"整數 n 是偶數\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0002","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 可被 6 整除」；命題 Q：「整數 n 可被 3 整除」。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":20261123,"parameters":{"p":"整數 n 可被 6 整除","q":"整數 n 可被 3 整除","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 可被 6 整除\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"整數 n 可被 3 整除\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0003","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 9 的倍數」；命題 Q：「整數 n 是 3 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":1,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261124,"parameters":{"p":"整數 n 是 9 的倍數","q":"整數 n 是 3 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 9 的倍數\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"整數 n 是 3 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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0004","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 是質數且大於 2」；命題 Q：「整數 n 是奇數」。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":2,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261125,"parameters":{"p":"整數 n 是質數且大於 2","q":"整數 n 是奇數","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 是質數且大於 2\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"整數 n 是奇數\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0005","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：「三角形有三條相等的邊」；命題 Q：「三角形至少有兩條相等的邊」。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":20261126,"parameters":{"p":"三角形有三條相等的邊","q":"三角形至少有兩條相等的邊","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\":\"三角形有三條相等的邊\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"三角形至少有兩條相等的邊\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0006","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：「四邊形是正方形」；命題 Q：「四邊形有四個直角」。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":1,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261127,"parameters":{"p":"四邊形是正方形","q":"四邊形有四個直角","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\":\"四邊形是正方形\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"四邊形有四個直角\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0007","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 10 的倍數」；命題 Q：「整數 n 是 5 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":2,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261128,"parameters":{"p":"整數 n 是 10 的倍數","q":"整數 n 是 5 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 10 的倍數\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"整數 n 是 5 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"}}
{"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0009","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 是完全平方數」；命題 Q：「存在整數 k 使 n=k²」。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":1,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261130,"parameters":{"p":"整數 n 是完全平方數","q":"存在整數 k 使 n=k²","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 是完全平方數\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"存在整數 k 使 n=k²\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0010","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：「有理數 r 等於 0」；命題 Q：「r 的絕對值等於 0」。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":2,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261131,"parameters":{"p":"有理數 r 等於 0","q":"r 的絕對值等於 0","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\":\"有理數 r 等於 0\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"r 的絕對值等於 0\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0011","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：「集合 A 是空集合」；命題 Q：「A 沒有元素」。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":20261132,"parameters":{"p":"集合 A 是空集合","q":"A 沒有元素","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\":\"集合 A 是空集合\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"A 沒有元素\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0012","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：「函數 f 是常數函數」；命題 Q：「任兩個輸入有相同輸出」。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":1,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261133,"parameters":{"p":"函數 f 是常數函數","q":"任兩個輸入有相同輸出","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\":\"函數 f 是常數函數\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"任兩個輸入有相同輸出\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0013","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：「角的量值是 90°」；命題 Q：「角是直角」。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":2,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261134,"parameters":{"p":"角的量值是 90°","q":"角是直角","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\":\"角的量值是 90°\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"角是直角\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0014","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：「角的量值是 180°」；命題 Q：「角是平角」。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":20261135,"parameters":{"p":"角的量值是 180°","q":"角是平角","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\":\"角的量值是 180°\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"角是平角\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0015","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 可被 8 整除」；命題 Q：「整數 n 可被 2 整除」。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":1,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261136,"parameters":{"p":"整數 n 可被 8 整除","q":"整數 n 可被 2 整除","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 可被 8 整除\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"整數 n 可被 2 整除\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0016","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 15 的倍數」；命題 Q：「整數 n 可被 5 整除」。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":2,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261137,"parameters":{"p":"整數 n 是 15 的倍數","q":"整數 n 可被 5 整除","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 15 的倍數\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"整數 n 可被 5 整除\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0017","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：「數 x 大於 7」；命題 Q：「x 大於 0」。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":20261138,"parameters":{"p":"數 x 大於 7","q":"x 大於 0","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\":\"數 x 大於 7\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"x 大於 0\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0018","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：「數 x 小於 -3」；命題 Q：「x 小於 0」。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":1,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261139,"parameters":{"p":"數 x 小於 -3","q":"x 小於 0","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\":\"數 x 小於 -3\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"x 小於 0\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0019","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：「多邊形是五邊形」；命題 Q：「多邊形有五條邊」。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":2,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261140,"parameters":{"p":"多邊形是五邊形","q":"多邊形有五條邊","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\":\"多邊形是五邊形\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"多邊形有五條邊\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0020","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：「矩形的長與寬相等」；命題 Q：「矩形是正方形」。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":20261141,"parameters":{"p":"矩形的長與寬相等","q":"矩形是正方形","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\":\"矩形的長與寬相等\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"矩形是正方形\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0021","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 是偶數」；命題 Q：「n² 是偶數」。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":1,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261142,"parameters":{"p":"整數 n 是偶數","q":"n² 是偶數","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 是偶數\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"n² 是偶數\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0022","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 可被 3 整除」；命題 Q：「n² 可被 3 整除」。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":2,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261143,"parameters":{"p":"整數 n 可被 3 整除","q":"n² 可被 3 整除","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 可被 3 整除\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"n² 可被 3 整除\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0023","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：「分數 a/b 的分母 b 不等於 0」；命題 Q：「a/b 是有定義的分數表示」。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":20261144,"parameters":{"p":"分數 a/b 的分母 b 不等於 0","q":"a/b 是有定義的分數表示","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\":\"分數 a/b 的分母 b 不等於 0\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"a/b 是有定義的分數表示\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0024","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：「集合 A 是 B 的子集」；命題 Q：「A 中每個元素也在 B 中」。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":1,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261145,"parameters":{"p":"集合 A 是 B 的子集","q":"A 中每個元素也在 B 中","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\":\"集合 A 是 B 的子集\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"A 中每個元素也在 B 中\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-invalid-converse-0025","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：「命題 R 與 S 都成立」；命題 Q：「命題 R 成立」。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":2,"skills":["材料蘊涵","無效 converse","反例真值指派"]},"variant":{"family_id":"qf-proof-invalid-converse","seed":20261146,"parameters":{"p":"命題 R 與 S 都成立","q":"命題 R 成立","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\":\"命題 R 與 S 都成立\",\"premises\":[\"P→Q\",\"Q\"],\"q\":\"命題 R 成立\"}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0001","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都是偶數」。明示 witness 為 n=1。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=1 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261222,"parameters":{"domain":"integers","universal_statement":"每個整數都是偶數","predicate_id":"even","bound":null,"witness":1,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":null,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"even\",\"universal_statement\":\"每個整數都是偶數\",\"witness\":1}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0002","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都大於 0」。明示 witness 為 n=0。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=0 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261223,"parameters":{"domain":"integers","universal_statement":"每個整數都大於 0","predicate_id":"positive","bound":null,"witness":0,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":null,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"positive\",\"universal_statement\":\"每個整數都大於 0\",\"witness\":0}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0003","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都大於或等於 0」。明示 witness 為 n=-1。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=-1 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261224,"parameters":{"domain":"integers","universal_statement":"每個整數都大於或等於 0","predicate_id":"nonnegative","bound":null,"witness":-1,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":null,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"nonnegative\",\"universal_statement\":\"每個整數都大於或等於 0\",\"witness\":-1}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0004","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都是 3 的倍數」。明示 witness 為 n=1。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=1 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261225,"parameters":{"domain":"integers","universal_statement":"每個整數都是 3 的倍數","predicate_id":"multiple","bound":3,"witness":1,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":3,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"multiple\",\"universal_statement\":\"每個整數都是 3 的倍數\",\"witness\":1}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0005","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都是 5 的倍數」。明示 witness 為 n=2。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=2 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261226,"parameters":{"domain":"integers","universal_statement":"每個整數都是 5 的倍數","predicate_id":"multiple","bound":5,"witness":2,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":5,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"multiple\",\"universal_statement\":\"每個整數都是 5 的倍數\",\"witness\":2}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0006","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都等於 0」。明示 witness 為 n=1。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=1 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261227,"parameters":{"domain":"integers","universal_statement":"每個整數都等於 0","predicate_id":"zero","bound":null,"witness":1,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":null,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"zero\",\"universal_statement\":\"每個整數都等於 0\",\"witness\":1}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0007","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都大於 5」。明示 witness 為 n=5。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=5 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261228,"parameters":{"domain":"integers","universal_statement":"每個整數都大於 5","predicate_id":"greater","bound":5,"witness":5,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":5,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"greater\",\"universal_statement\":\"每個整數都大於 5\",\"witness\":5}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0008","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都小於 10」。明示 witness 為 n=10。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=10 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261229,"parameters":{"domain":"integers","universal_statement":"每個整數都小於 10","predicate_id":"less","bound":10,"witness":10,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":10,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"less\",\"universal_statement\":\"每個整數都小於 10\",\"witness\":10}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0009","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都是偶數」。明示 witness 為 n=-3。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=-3 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261230,"parameters":{"domain":"integers","universal_statement":"每個整數都是偶數","predicate_id":"even","bound":null,"witness":-3,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":null,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"even\",\"universal_statement\":\"每個整數都是偶數\",\"witness\":-3}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0010","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都大於 0」。明示 witness 為 n=-1。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=-1 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261231,"parameters":{"domain":"integers","universal_statement":"每個整數都大於 0","predicate_id":"positive","bound":null,"witness":-1,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":null,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"positive\",\"universal_statement\":\"每個整數都大於 0\",\"witness\":-1}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0011","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都是 7 的倍數」。明示 witness 為 n=3。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=3 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261232,"parameters":{"domain":"integers","universal_statement":"每個整數都是 7 的倍數","predicate_id":"multiple","bound":7,"witness":3,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":7,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"multiple\",\"universal_statement\":\"每個整數都是 7 的倍數\",\"witness\":3}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0012","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都等於 0」。明示 witness 為 n=-2。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=-2 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261233,"parameters":{"domain":"integers","universal_statement":"每個整數都等於 0","predicate_id":"zero","bound":null,"witness":-2,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":null,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"zero\",\"universal_statement\":\"每個整數都等於 0\",\"witness\":-2}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0013","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都大於 -4」。明示 witness 為 n=-4。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=-4 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261234,"parameters":{"domain":"integers","universal_statement":"每個整數都大於 -4","predicate_id":"greater","bound":-4,"witness":-4,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":-4,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"greater\",\"universal_statement\":\"每個整數都大於 -4\",\"witness\":-4}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0014","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都小於 0」。明示 witness 為 n=0。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=0 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261235,"parameters":{"domain":"integers","universal_statement":"每個整數都小於 0","predicate_id":"less","bound":0,"witness":0,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":0,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"less\",\"universal_statement\":\"每個整數都小於 0\",\"witness\":0}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0015","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都是 2 的倍數」。明示 witness 為 n=5。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=5 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261236,"parameters":{"domain":"integers","universal_statement":"每個整數都是 2 的倍數","predicate_id":"multiple","bound":2,"witness":5,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":2,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"multiple\",\"universal_statement\":\"每個整數都是 2 的倍數\",\"witness\":5}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0016","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都大於或等於 0」。明示 witness 為 n=-9。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=-9 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261237,"parameters":{"domain":"integers","universal_statement":"每個整數都大於或等於 0","predicate_id":"nonnegative","bound":null,"witness":-9,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":null,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"nonnegative\",\"universal_statement\":\"每個整數都大於或等於 0\",\"witness\":-9}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0017","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都是 11 的倍數」。明示 witness 為 n=4。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=4 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261238,"parameters":{"domain":"integers","universal_statement":"每個整數都是 11 的倍數","predicate_id":"multiple","bound":11,"witness":4,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":11,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"multiple\",\"universal_statement\":\"每個整數都是 11 的倍數\",\"witness\":4}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0018","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都是偶數」。明示 witness 為 n=9。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=9 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261239,"parameters":{"domain":"integers","universal_statement":"每個整數都是偶數","predicate_id":"even","bound":null,"witness":9,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":null,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"even\",\"universal_statement\":\"每個整數都是偶數\",\"witness\":9}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0019","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都大於 0」。明示 witness 為 n=-8。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=-8 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261240,"parameters":{"domain":"integers","universal_statement":"每個整數都大於 0","predicate_id":"positive","bound":null,"witness":-8,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":null,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"positive\",\"universal_statement\":\"每個整數都大於 0\",\"witness\":-8}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0020","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都大於 100」。明示 witness 為 n=100。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=100 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261241,"parameters":{"domain":"integers","universal_statement":"每個整數都大於 100","predicate_id":"greater","bound":100,"witness":100,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":100,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"greater\",\"universal_statement\":\"每個整數都大於 100\",\"witness\":100}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0021","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都小於 -2」。明示 witness 為 n=-2。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=-2 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261242,"parameters":{"domain":"integers","universal_statement":"每個整數都小於 -2","predicate_id":"less","bound":-2,"witness":-2,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":-2,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"less\",\"universal_statement\":\"每個整數都小於 -2\",\"witness\":-2}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0022","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都是 4 的倍數」。明示 witness 為 n=6。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=6 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261243,"parameters":{"domain":"integers","universal_statement":"每個整數都是 4 的倍數","predicate_id":"multiple","bound":4,"witness":6,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":4,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"multiple\",\"universal_statement\":\"每個整數都是 4 的倍數\",\"witness\":6}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0023","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都等於 0」。明示 witness 為 n=12。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=12 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":3,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261244,"parameters":{"domain":"integers","universal_statement":"每個整數都等於 0","predicate_id":"zero","bound":null,"witness":12,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":null,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"zero\",\"universal_statement\":\"每個整數都等於 0\",\"witness\":12}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0024","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都大於或等於 0」。明示 witness 為 n=-12。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=-12 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":1,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261245,"parameters":{"domain":"integers","universal_statement":"每個整數都大於或等於 0","predicate_id":"nonnegative","bound":null,"witness":-12,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":null,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"nonnegative\",\"universal_statement\":\"每個整數都大於或等於 0\",\"witness\":-12}"}},"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"}}
{"schema_version":"ocme-question-v0.1","id":"q-20260921-proof-counterexample-universal-0025","batch_id":"batch-20260921-proof-foundations","language":"zh-Hant","source_mko_ids":["mko-proof"],"title_zh":"全稱命題的反例","stem_zh":"論域明示為整數。全稱命題是「每個整數都是 9 的倍數」。明示 witness 為 n=10。這個 witness 是否反駁該全稱命題？","answer":{"kind":"choice","value":"是","choices":["是","否"],"explanation_zh":"n=10 在整數論域內，且不具有該命題要求的性質；所以它是這一全稱命題的有效反例，僅反駁此命題。"},"difficulty":{"band":"upper_secondary","level":2,"skills":["全稱命題","反例 witness"]},"variant":{"family_id":"qf-proof-counterexample-universal","seed":20261246,"parameters":{"domain":"integers","universal_statement":"每個整數都是 9 的倍數","predicate_id":"multiple","bound":9,"witness":10,"counterexample_role":"falsifies_the_stated_universal_only","semantic_template":"qf-proof-counterexample-universal/{\"bound\":9,\"counterexample_role\":\"falsifies_the_stated_universal_only\",\"domain\":\"integers\",\"predicate_id\":\"multiple\",\"universal_statement\":\"每個整數都是 9 的倍數\",\"witness\":10}"}},"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"}}