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谓词演算 - Wikipedia

谓词演算

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数理逻辑谓词演算谓词逻辑命题函数演算是用来描述数学理论的形式系统。

谓词演算是命题演算的扩展,命题演算对于描述更复杂的数学结构是不充分的。从文法上说谓词演算在现存的命题演算上增加了“谓词-主词结构”和量词。主词是给定的个体群组(集合)的一个成员的名字,而谓词是在这个群组上的关系,一元谓词在哲学中称为性质,在数学中称为指示函数,在数理逻辑中称为布尔值函数

[编辑] 恒等式

\neg \forall x P(x) \Leftrightarrow \exists x \neg P(x)
\neg \exists x P(x) \Leftrightarrow \forall x \neg P(x)
\forall x \forall y P(x,y) \Leftrightarrow \forall y \forall x P(x,y)
\exists x \exists y P(x,y) \Leftrightarrow \exists y \exists x P(x,y)
\forall x P(x) \land \forall x Q(x) \Leftrightarrow \forall x (P(x) \land Q(x))
\exists x P(x) \lor \exists x Q(x) \Leftrightarrow \exists x (P(x) \lor Q(x))

[编辑] 推理规则

\exists x \forall y P(x,y) \Rightarrow \forall y \exists x P(x,y)
\forall x P(x) \lor \forall x Q(x) \Rightarrow \forall x (P(x) \lor Q(x))
\exists x (P(x) \land Q(x)) \Rightarrow \exists x P(x) \land \exists x Q(x)
\exists x P(x) \land \forall x Q(x) \Rightarrow \exists x (P(x) \land Q(x))
\forall x P(x) \Rightarrow P(c) (c 如果为变量,则在 P(x) 中一定不能被其他量词约束)
P(c) \Rightarrow \exists x P(x) (x 在 P(c) 中一定不能自由出现)

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