Logical negation

Logical negation is an operation on one logical value, typically the value of a proposition, that produces a value of true when its operand is false and a value of false when its operand is true.

A truth table for $\mathrm {NOT} ~p,$ also written $\lnot p,$ appears below:

${\text{Logical Negation}}$

$p$	$\lnot p$
$\mathrm {F}$	$\mathrm {T}$
$\mathrm {T}$	$\mathrm {F}$

The negation of a proposition $p$ may be found notated in various ways in various contexts of application, often merely for typographical convenience.  Among these variants are the following:

${\text{Variant Notations}}$

${\text{Notation}}$	${\text{Vocalization}}$
${\bar {p}}$	$p$ bar
${\tilde {p}}$	$p$ tilde
$p'$	$p$ prime $p$ complement
$!p$	bang $p$

A logical graph for $\lnot p$ is shown below:

The traversal string of this graph is ${\texttt {(}}p{\texttt {)}}.$  The proposition $\lnot p$ may be taken as a Boolean function $f(p)$ having the abstract type $f:\mathbb {B} \to \mathbb {B} ,$ where $\mathbb {B} =\{0,1\}$ is interpreted in such a way that $0$ means $\mathrm {false}$ and $1$ means $\mathrm {true} .$

A Venn diagram for $\lnot p$ indicates the region where $\lnot p$ is true by means of a distinctive color or shading.  In this case the region is a single cell, as shown below:

Resources

• Logic Syllabus

Document history

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.

• Logical Negation, InterSciWiki
• Logical Negation, Wikiversity
• Logical Negation, Wikipedia