Definition:Logical Not

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Definition

The logical not or (logical) 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
It is not the case that $p$ is true
It is false that $p$
$p$ is false.


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


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


Truth Function

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

\(\ds \map {f^\neg} \F\) \(=\) \(\ds \T\)
\(\ds \map {f^\neg} \T\) \(=\) \(\ds \F\)


Truth Table

The characteristic truth table of the negation operator $\neg p$ is as follows:

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


Boolean Interpretation

The truth value of $\neg \mathbf A$ under a boolean interpretation $v$ is given by:

$\map v {\neg \mathbf A} = \begin {cases} \T & : \map v {\mathbf A} = \F \\ \F & : \map v {\mathbf A} = \T \end {cases}$


Notational Variants

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

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


Also known as

The (logical) negation of a statement is also referred to as the denial of that statement.

Treatments which consider logical connectives as functions may refer to this operator as the contradictory function.


Also see

  • Results about logical not can be found here.


Sources

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