Definition:Logical Not

From ProofWiki
Jump to: navigation, search


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 $p$.

Truth Function

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

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

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:

$v \left({\neg \mathbf A}\right) = \begin{cases} T & : v \left({\mathbf A}\right) = F \\ F & : v \left({\mathbf A}\right) = T \end{cases}$

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 $p$
$\operatorname{N} p$ Łukasiewicz's Polish notation

Also known as

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.