課程名稱︰ 邏輯 課程性質︰ 通識 A4 課程教師︰ 傅皓政 開課學院： 開課系所︰ 考試日期（年月日）︰2019/06/17 考試時限（分鐘）：90分鐘 試題 : 一、建構初階邏輯語言 (提示：包括符號與形構規則兩個部分)。 (10%) Construct a suitable language for first-order (or predicate) logic. Hint: two parts should be involved, alphabets and formation rules. 二、請以亞里斯多德的方法證明下列三段論的論證是否為有效論證，如為無效論證請寫出 具有說服力的反例。 (15%) Please use the Aristotle's method to prove whether the arguments of the following syllogisms are valid arguments. For the invalid arguments, please write a persuasive counterexample. (a) M A P (b) S E M (c) S A M M A S P E M M A P ------- ------- ------- S A P S E P S I P 三、請將下列日常語言的語句翻譯為初階邏輯語言的表達式。 (15%) Please translate the following ordinary language statements into expressions of the first-order logic language. Cx: x是市民(x is a citizen) ； Hxy: x幫助y (x will help y) (a) 沒有市民會幫助任何一個市民。 No citizen will help any citizen. (b) 沒有市民會幫助每個市民。 No citizen will help every citizen. (c) 每個市民都會幫助某些市民。 Every citizen will help some citizens 四、請以真值樹法證明下列語法序列是否為有效論證，若為無效論證請顯示其反例節構。 (20%) Please use the tableaux method to prove whether the following sequence is a valid arguments. If it is an invalid argument, please show its counter-example structure. (a) (∀x)(GxＶHx), (∀x)(Hx→Kx) ├ (∀x)GxＶ(∀x)Kx (b) (∃x)Lx→(∀x)(Kx→(∃y)Mxy), (∀x)(∀y)(Mxy→Nx) ├ (∀x)((LxΛKx)→(∃y)Ny) 五、請以實例說明下列謬誤，並說明為什麼它是這類謬誤。 (10%) Please exemplify the following fallacies and explain why it is such a fallacies. (a) 人身攻擊的謬誤 Argument against the person (b) 滑坡謬誤 Slippery Slope Fallacy 六、請完成下列演算，作答時需連同題目寫在答案卷上。 (20%) Please complete the following calculations, and write the answer sheet together with the diagram. (a) ├ ﹁(P↔﹁P) P↔﹁P P↔﹁P ------- ------- P P→﹁P ﹁P ﹁P→P -------------- ---------------- P ﹁P P ﹁P ----------------- ------------------- ⊥ ⊥ --------- ------- ﹁P P ----------------------------------------- ⊥ ------------- ﹁(P↔﹁P) (b) (∀x)(Lx→(∃y)(LyΛMxy)), (∃x)(LxΛ(∀y)((LyΛMxy)→Nxy)) ├ (∃x)(∃y)((LxΛLy)→Nxy) 1. (∀x)(Lx→(∃y)(LyΛMxy)) __Pr__ 2. (∃x)(LxΛ(∀y)((LyΛMxy)→Nxy)) __Pr__ 3. LaΛ(∀y)((LyΛMay)→Nay) ______ 4. La ______ 5. La→(∃y)(LyΛMay) ______ 6. (∃y)(LyΛMay) ______ 7. LbΛMab ______ 8. (∀y)((LyΛMay)→Nay) ______ 9. (LbΛMab)→Nab ______ 10. Nab ______ 11. NabＶ﹁La ______ 12. (NabＶ﹁La)Ｖ﹁Lb ______ 13. NabＶ(﹁LaＶ﹁Lb) ______ 14. NabＶ﹁(LaΛLb) ______ 15. ﹁(LaΛLb)ＶNab ______ 16. LaΛLb→Nab ______ 17. (∃y)((LaΛLy→Nay) ______ 18. (∃x)(∃y)((LxΛLy)→Nxy) ______ 七、請說明羅素確定描述詞理論的內容及其重要性為何。 (10%) Please briefly explain the content and significance of Russell's theory of description. *題目後附推論規則 1. 樹狀自然演繹法推論規則(rules of inferences: tree-likely natural deduction) 2. 線性自然演繹法推論規則(rules of inferences: linearly natural deduction) --- --

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