Definition:Logical Not

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Definition

The logical not or negation operator is a unary connective whose action is to reverse the truth value of the statement on which it operates.

$\neg p$ is defined as: $p$ is not true, or It is not the case that $p$ is true.

Thus the statement $\neg p$ is called the negation of $p$.

$\neg p$ is voiced not $p$.

In the statement $\neg p$, the scope of $\neg$ is $p$.

From the above, we see that the boolean interpretations for $\mathbf A$ under the model $\mathcal M$ are:

$\left({\neg \mathbf A}\right)_\mathcal M = \begin{cases} T & : \mathbf A_\mathcal M = F \\ F & : \mathbf A_\mathcal M = T \end{cases}$

The logical not connective defines the truth function $f^\neg$ as follows:

	$\displaystyle $	$\displaystyle f^\neg \left({F}\right)$	$=$	$\displaystyle T$	$\displaystyle $
	$\displaystyle $	$\displaystyle f^\neg \left({T}\right)$	$=$	$\displaystyle F$	$\displaystyle $

The truth table of $\neg p$ is as follows:

$\begin{array}{|c||c|} \hline p & \neg p \\ \hline F & T \\ T & F \\ \hline \end{array}$

Various symbols are encountered that denote the concept of the logical not:

Symbol	Origin	Known as
$\neg p$
$\mathsf{NOT}\ p$
$\sim p$ or $\tilde p$	Bertrand Russell and Alfred North Whitehead: Principia Mathematica (1910)	tilde
$- p$
$\bar p$		Bar $p$
$p'$		$p$ prime or $p$ complement
$! p$		Bang $p$
$\operatorname{N} p$	Łukasiewicz's Polish notation

	\(\displaystyle \)	\(\displaystyle f^\neg \left({F}\right)\)	\(=\)	\(\displaystyle T\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle f^\neg \left({T}\right)\)	\(=\)	\(\displaystyle F\)	\(\displaystyle \)