基本邏輯名詞 -阿亮遇見熊

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