基本邏輯名詞

2009/02/25
~ 阿亮 ~

最近看到一些簡單的邏輯 (Logic) 英詞名詞,並不是很清楚,所以,找出來再複習一下。

其實大部份是很簡單的,只是換成英文就不熟了.

Rules of Inference

 Modus Ponens

\displaystyle\begin{array}{l}
p\rightarrow q \\
p \\
\therefore q
\end{array}

 

 Modus Tollens

\begin{array}{l}
p\rightarrow q \\
\neg q \\
\therefore \,\neg p
\end{array}

 

 Hypothetical Syllogism

\begin{array}{l}
p\rightarrow q \\
q\rightarrow r \\
\therefore p\rightarrow r
\end{array}

 

 Disjunctive Syllogism

\begin{array}{l}
p\vee q \\
\neg p \\
\therefore q
\end{array}

 Constructive Dilemma

\begin{array}{l}
(p\rightarrow q) \wedge (r\rightarrow s) \\
p\vee r \\
\therefore q\vee s
\end{array}

 

 Absorption

\begin{array}{l}
\,\\
p\rightarrow q \\
\therefore p\rightarrow (p\wedge q)
\end{array}

 Simplification

\begin{array}{l}
\\
p\wedge q \\
\therefore p
\end{array}

 

 Conjunction

\begin{array}{l}
p \\
q \\
\therefore p\wedge q
\end{array}

 

 Addition

\begin{array}{l}
\\
p \\
\therefore p \vee q
\end{array}

 

 

Rules of Replacement

Double Negation p\leftrightarrow \,\neg\neg p
Commutation \begin{array}{l} \\  (p\vee q)\leftrightarrow  (q\vee p) \\ (p\wedge q)\leftrightarrow  (q\wedge p) \\ \end{array}
Tautology \begin{array}{l} \\  p\leftrightarrow  (p\vee p) \\ p\leftrightarrow  (p\wedge p) \\ \end{array}
Association

\begin{array}{l} \\ \left[p\vee  (q\vee r)\right] \leftrightarrow  \left[(p\vee  q)\vee r\right] \\  \left[p\wedge  (q\wedge r)\right] \leftrightarrow  \left[(p\wedge  q)\wedge r\right] \\ \end{array}

Transposition \begin{array}{l} \\  (p\rightarrow q) \leftrightarrow  (\neg q\rightarrow \neg p) \\ \end{array}
Material Implication \begin{array}{l} \\  (p\rightarrow q) \leftrightarrow  (\neg p\vee q) \\ \end{array}
Exportation  \begin{array}{l} \\  \left[(p\wedge q)\rightarrow r }\right]  \leftrightarrow   \left[p\rightarrow (q\rightarrow r)\right]   \\ \end{array}
Material Equivalence

\begin{array}{l} \\
(p\leftrightarrow q)\leftrightarrow \left[(p\rightarrow q) \wedge (q\rightarrow p)\right] \\
 (p\leftrightarrow q)\leftrightarrow \left[(p\wedge q) \vee (\neg p \,\wedge \neg q)\right] \\
\end{array}

Distribution

\begin{array}{l} \\
\left[p \wedge (q\vee r)\left] \leftrightarrow \left[(p\wedge q)\vee (p\wedge r)\right] \\
\left[p \vee (q\wedge r)\left] \leftrightarrow \left[(p\vee q)\wedge (p\vee r)\right] \\
\end{array}

De Morgan’s Theorems

\begin{array}{l} \\
\neg(p \wedge q)\leftrightarrow (\neg p \,\vee \neg q) \\
\neg(p \vee q)\leftrightarrow (\neg p \,\wedge \neg q) \\
\end{array}

 

 

Bi-conditionals Logical Equivalence

(\forall x)(\psi x\rightarrow \varphi x) \leftrightarrow  \,\neg(\exists x)(\psi x \,\wedge \neg\varphi x)

"Everything in the lake is wet." 

is logically equivalent to

"There isn’t anything in the lake which is not wet."

 

(\exists x)(\psi x\wedge \varphi x) \leftrightarrow  \,\neg(\forall x)(\psi x \rightarrow \,\neg\varphi x)

"There exists at least one individual who is both a native of Boston and of Irish descent."

is logically equivalent to

"It’s not true that no natives of Boston are of Irish descent."

 

(\forall x)(\psi x\rightarrow \neg\varphi x) \leftrightarrow \,\neg(\exists x)(\psi x \wedge \varphi x)

"No residents of Boston are Irish."

is logically equivalent to

"It’s not true that some residents of Boston are Irish."

 

(\exists x)(\psi x \,\wedge \neg\varphi x) \leftrightarrow \,\neg(\forall x)(\psi x \rightarrow \varphi x)

 "Some residents of Boston are not Irish."

is logically equivalent to

"Not all residents of Boston are Irish."

 

 

 



歡迎留言

敝站會審核留言的適宜性,您的留言可能會較晚發佈,而且小弟保留刪除的權利!!

站內搜尋



本站其他服務

本站其他軟體



  • 下一班公車(nextBus)

    這個 app 只要開啟後,就根據定位幫你過濾出附近站牌的時刻表,以及提供相關公車預計到站的時間,方便您在很快時間內確定要坐的哪一班公車


  • 下一班高鐵 (nextTHSR)

    這個 app 只要開啟後,就根據定位幫你過濾出最近高鐵站的時刻表,不用再按任何按鈕了,方便您在很快時間內確定要坐的哪一班高鐵


  • 藝文快訊

    讓你可以輕鬆追蹤含有您想要關注關鍵詞的任何藝文活動訊息,只要有最新的資訊,「藝文快訊」即會推播通知給你.


  • 臉書粉絲專頁搜尋 (FPSearch)

    不用登入臉書即可搜尋臉書粉絲專頁(臉書粉絲團)所公開的文章。您可以指定特定日期範圍之前的文章,也可設定搜尋粉絲專頁內包含特定關鍵詞的文章。


  • 油價快訊App (OilPrices)

    依據油價及匯率,估算台灣下週油價,另外提供三週、一年以及三年的歷史變化,以及週末下午推播通知最新油價預估或公告。


  • 台灣空污警報(AirInfo)

    設定特定站點為推播通知關注點後,當該站點空氣品質變糟時,即時推播通知給您。另外提供站點附近基本天氣預測資料。