﻿ 命题逻辑中等价公式的证明方法探讨

# 命题逻辑中等价公式的证明方法探讨On the Proof Methods of Propositional Equivalences in Propositional Logic

Abstract: Propositional formulas are the basic contents in propositional logic. It is an important problem to determine whether two propositional formulas are logically equivalent. The paper summarizes six methods about how to prove the equivalence of two propositional formulas by specific examples, and discusses the relations of these methods.

1. 引言

2. 命题公式等价的相关概念

3. 命题公式等价的证明方法

$\begin{array}{l}\left(P\to \left(Q\vee R\right)\right)\wedge \left(¬P\vee \left(Q↔R\right)\right)\\ ⇔\left(¬P\vee Q\vee R\right)\wedge \left(¬P\vee \left(\left(Q\to R\right)\wedge \left(R\to Q\right)\right)\right)\\ ⇔¬P\vee \left(\left(Q\vee R\right)\wedge \left(¬Q\vee R\right)\wedge \left(Q\vee ¬R\right)\right)\\ ⇔¬P\vee \left(R\wedge \left(Q\vee ¬R\right)\right)\\ ⇔P\to \left(R\wedge \left(Q\vee ¬R\right)\right)\end{array}$

1) $\left(P\vee Q\right)\wedge \left(¬P\wedge Q\right)⇔¬P\wedge Q$

2) $¬\left(P\wedge Q\right)\to \left(¬P\wedge Q\right)⇔¬P\vee Q$

$\left(P\vee Q\right)\wedge \left(¬P\wedge Q\right)⇔\left(P\wedge \left(¬P\wedge Q\right)\right)\vee \left(Q\wedge \left(¬P\wedge Q\right)\right)⇔¬P\wedge Q$

2) 注意到2)式的右边 $¬P\vee Q$ 恰好是1)式右边 $¬P\wedge Q$ 的对偶式。而2)式左边

$¬\left(P\wedge Q\right)\to \left(¬P\wedge Q\right)⇔\left(P\wedge Q\right)\vee \left(¬P\vee Q\right)$

4. 结论

[1] Rosen, K.H. (1999) Discrete Mathematics and Its Applications. China Machine Press, Beijing.

[2] Kolman, B., Busby, R.C. and Ross, C.S. (2006) Discrete Mathematics Structures. 5th Edition, Higher Education Press, Beijing.

[3] 左孝凌, 李为鉴, 刘永才. 离散数学[M]. 上海: 上海科学技术文献出版社, 1982.

[4] 杜君花, 梁红梅, 马艳萍. 数理逻辑的若干应用[J]. 高师理科学刊, 2018(9): 56-60.

[5] 王礼萍, 张树功. 重言式和矛盾式的代数化证明[J]. 计算机与数字工程, 2009(37): 17-21.

[6] 徐小萍. 命题公式类型的判定方法[J]. 廊坊师范学院学报(自然科学版), 2010(10): 7-10.

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