2018 台灣大學 邏輯通識課程
107學年度第一學期期中考考題
科目:邏輯(通識課程) 2018/11/05 授課教師:傅皓政臺灣大學107學年度第一學期期中考
請按題號順序作答於答案卷上(Answer all questions on answer sheet provided.)
請建構命題邏輯語言(提示:包括符號與形構規則兩個部分)(10%)
(Construct a suitable language for propositional logic. Hint: two parts
involved, alphabets and formation rules)
二、請判斷下列句式哪些是合宜的句式, ?哪些是不合宜的句式?(10%)
(Please consider the following formulae and distinguish the well-formed formulae from ill-formed ones.)
(a)A ¬ ¬B ∨C (b) ¬ ¬K →H (c) ¬¬P ∧Q ¬ (d)(D →F ↔E) →(D ∧E) (e)(S ∧ ¬W) ∨(W ↔ ¬X ∨S) F
(f)Q； ¬P ↔Q (g)(A →(B ∧¬F)) ∨E (h)(HK →G) →H (i) ¬(F ↔((D ∨G) →(D ∧F))) (j)X ↔Y ∨
合宜的句式:_______________________________________
不合宜的句式:______________________________________
三、判斷下列陳述的真假,並且分別以T與F代表「真」與「假」(10%)
(Please judge the following statcments which are true or false. Notice,please use the symbols＂Ｔ＂and＂Ｆ＂which stand for true and false statements respectivcly)
1·前提與結論不一致的論證可能是無效論證
2·每個前提實際上都為假而且結論實際上為真的論證一定是有效論證。
3·每個前提實際上都為假的論證可能是有效論證
4.每個前提實際上都為真而且結論實際上為假的論證可能是有效論證。
5.前提中出現矛盾句的論證可能是無效論證
6·前提與結論一致的論證必定是有效論證
7.每個前提與結論都是偶真句的論證可能是有效論證
8.有效論證的結論必定是實際上為真
9.前提出現恒真句的論證可能是無效論證。
10.結論為偶真句的論證可能是有效論證
四、請以真質表法判斷下列句式那些是恆真句、矛盾句或者是偶真句。注意：必須列出演算過程。(15%)(Using truth table method shows that each of the following formulae is tautology,contradiction,or indeterminate formula. Note：Computational process is required.)
(a)P ∨((Q ↔P) ∨ ¬P)
(b)(K →L) ∨(L →(K ∧M))
(c)(D →E) ↔( ¬E →¬D)
五、請判斷下列各題中的兩個句式之間式蘊涵或是等值關係。如果是蘊涵關係，以φ ⊧ ψ表示；若為等值關係，則以 ⊧φ↔ψ 表示，必須列出演算過程。(15%)(Use the designate method to determine the semantic relation between the following formulae.If the entsilment relation holds then show them of the form φ ⊧ ψ. On the other hand, show them of the form ⊧φ↔ψ if they are equivalent. Computational process is required.)
(a) ((D∧E)→F)∨(E→¬F) ；D→F
(b)K→(L→M) ；(K∧L)→M
(c)(P↔Q)∨¬(P↔Q) ；(P→Q)→ (¬P→¬Q)
六、請寫出等值於真值表中語句φ的DNF及CNF。(10%)(Find out the DNF and CNF each which is equivalent to the following formulae φ.)
備註：底線只是做區隔 原本底下皆為表格形式
(a)
K ____L____ M_________φ
T____T_____T_________F
T____T_____F_________F
T____F_____T_________T
T____F_____F_________T
F____T_____T_________F
F____T_____F_________T
F____F_____T_________T
F____F_____F_________F
(b)
P____Q____ R_________φ
T ____T____ T_________T
T____T_____F_________T
T____F_____T_________F
T____F_____F_________T
F____T_____T_________F
F____T_____F_________F
F____F_____T_________T
F____F_____F_________F
七、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結構。(20%)(Please use tableaux system to prove whether each of the following argument is valid. And specify a counterexample if it is invalid.)
(a) ├ ((L→(M∧N))∧¬M)→¬L
(b)¬P→Q , Q↔¬R , R ├ P∧Q
八、(a)在說明古典邏輯條件句的真值表時，許多人會覺得某些情況的真假值與直覺判斷似乎有所出入請舉例說明之。(5%)
(Some logicians said that the assignment of conditionals in classical logic seems to be counterintuitive. Please show it in some instances.)
(b)如果你認為古典邏輯對條件句的賦值方式是合理的，請解釋如何消弭上述的問題。反之，如果你認為古典邏輯是不合理的，請顯示你認為能夠反映條件句的真值表，並且嘗試說明你的理由。(5%)
(If you agree with the assignment of conditionals provided by classical logic, then try to explain away the puzzle. However, please contruct one provide your reason for it if you do not agree.)
