零阶逻辑

零阶逻辑是在与布尔函数、一元谓词演算、命题逻辑或句子逻辑有关主题的从业人员中流行的术语。使用这个术语的好处是它确立了更高的抽象层次，在其中上述这些主题之间的很无关紧要的区别可以在这个中肯的同构下被包容。

向着最初的方向，表1列出了具体类型X × Y → B和抽象类型 B × B → B的十六个函数在零阶逻辑的不同语言中的等价表达。

**表1. 在两个变量上的命题形式**
L₁	L₂	L₃	L₄	L₅	L₆
	x :	1 1 0 0
	y :	1 0 1 0
f₀	f₀₀₀₀	0 0 0 0	( )	假	0
f₁	f₀₀₀₁	0 0 0 1	(x)(y)	非x且非y	¬x ∧ ¬y
f₂	f₀₀₁₀	0 0 1 0	(x) y	y且非x	¬x ∧ y
f₃	f₀₀₁₁	0 0 1 1	(x)	非x	¬x
f₄	f₀₁₀₀	0 1 0 0	x (y)	x且非y	x ∧ ¬y
f₅	f₀₁₀₁	0 1 0 1	(y)	非y	¬y
f₆	f₀₁₁₀	0 1 1 0	(x, y)	x不等于y	x ≠ y
f₇	f₀₁₁₁	0 1 1 1	(x y)	不x且y	¬x ∨ ¬y
f₈	f₁₀₀₀	1 0 0 0	x y	x且y	x ∧ y
f₉	f₁₀₀₁	1 0 0 1	((x, y))	x等于y	x = y
f₁₀	f₁₀₁₀	1 0 1 0	y	y	y
f₁₁	f₁₀₁₁	1 0 1 1	(x (y))	非x若非y	x → y
f₁₂	f₁₁₀₀	1 1 0 0	x	x	x
f₁₃	f₁₁₀₁	1 1 0 1	((x) y)	非y若非x	x ← y
f₁₄	f₁₁₁₀	1 1 1 0	((x)(y))	x或y	x ∨ y
f₁₅	f₁₁₁₁	1 1 1 1	(( ))	真	1

六种语言

对十六个布尔函数的六种语言按如下次序方便的描述：

语言L₃描述每个布尔函数f : B² → B，通过四个布尔值的序列 (f(1,1), f(1,0), f(0,1), f(0,0))的方式。这样的一个序列，可能在另一个次序下，并可能带有逻辑值F和T分别替代布尔值0和1，并通常显示为真值表中的一列。

语言L₂以f_i的形式列出十六个函数，这里的索引i是从L₃中的布尔值序列形成的位串。

语言L₁符号表示布尔函数f_i带有同L₂中的二进制索引等价的十进制索引i。

语言L₄使用逻辑连结词表达了十六个函数，通过以乘积的方式连接函数名字或命题表达式的方式，加上极小否定算子家族来表示，用下列各种符号给出几个极小否定算子：

{\begin{matrix}(\ )&=&0&=&{\mbox{false}}\\(x)&=&{\tilde {x}}&=&x'\\(x,y)&=&{\tilde {x}}y\lor x{\tilde {y}}&=&x'y\lor xy'\\(x,y,z)&=&{\tilde {x}}yz\lor x{\tilde {y}}z\lor xy{\tilde {z}}&=&x'yz\lor xy'z\lor xyz'\end{matrix}}

还要注意 $(x,y)\!$ 是同 $x+y\!$ 与 $x\neq y$ 相同的函数，并且 $(x,y)\!$ 和 $(x,y,z)\!$ 的包含析取表示可以替代为排斥析取而不影响意义，因为参与析取的项已经是分离了的。但是，函数 $(x,y,z)\!$ 和函数 $x+y+z\!$ 不是相同的东西。

语言L₅列出了这十六个函数的日常语言表达，它们是多种同义的表达中最简单的。

语言L₆以在形式逻辑中常用的一些符号表达了十六个函数。

参见

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