Difference between revisions of "Logical negation"

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'''Logical negation''' is an [[logical operation|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.
 
'''Logical negation''' is an [[logical operation|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.
  
The [[truth table]] of '''NOT p''' (also written as '''~p''' or '''¬p''') is as follows:
+
The [[truth table]] of <math>\operatorname{NOT}~ p,</math> also written <math>\lnot p,\!</math> appears below:
  
 
<br>
 
<br>
  
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; font-weight:bold; text-align:center; width:40%"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:45%"
|+ '''Logical Negation'''
+
|+ style="height:30px" | <math>\text{Logical Negation}\!</math>
|- style="background:#e6e6ff"
+
|- style="height:40px; background:#f0f0ff"
! style="width:20%" | p
+
| style="width:50%" | <math>p\!</math>
! style="width:20%" | &not;p
+
| style="width:50%" | <math>\lnot p\!</math>
 
|-
 
|-
| F || T
+
| <math>\operatorname{F}</math> || <math>\operatorname{T}</math>
 
|-
 
|-
| T || F
+
| <math>\operatorname{T}</math> || <math>\operatorname{F}</math>
 
|}
 
|}
  
 
<br>
 
<br>
  
The logical negation of a proposition '''p''' is notated in different ways in various contexts of discussion and fields of application.  Among these variants are the following:
+
The negation of a proposition <math>p\!</math> may be found notated in various ways in various contexts of application, often merely for typographical convenience.  Among these variants are the following:
  
 
<br>
 
<br>
  
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:#f8f8ff; width:40%"
+
{| align="center" border="1" cellpadding="8" cellspacing="0" width="45%"
|+ '''Variant Notations'''
+
|+ style="height:30px" | <math>\text{Variant Notations}\!</math>
|- style="background:#e6e6ff"
+
|- style="height:40px; background:#f0f0ff"
! style="text-align:center" | Notation
+
| width="50%" align="center" | <math>\text{Notation}\!</math>
! Vocalization
+
| width="50%" | <math>\text{Vocalization}\!</math>
 
|-
 
|-
| style="text-align:center" | <math>\bar{p}</math>
+
| align="center" | <math>\bar{p}\!</math>
| bar ''p''
+
| <math>p\!</math> bar
 
|-
 
|-
| style="text-align:center" | <math>p'\!</math>
+
| align="center" | <math>\tilde{p}\!</math>
| ''p'' prime,<p> ''p'' complement
+
| <math>p\!</math> tilde
 
|-
 
|-
| style="text-align:center" | <math>!p\!</math>
+
| align="center" | <math>p'\!</math>
| bang ''p''
+
| <math>p\!</math> prime<br> <math>p\!</math> complement
 +
|-
 +
| align="center" | <math>!p\!</math>
 +
| bang <math>p\!</math>
 
|}
 
|}
  

Revision as of 02:06, 16 May 2012

This page belongs to resource collections on Logic and Inquiry.

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.

The truth table of \(\operatorname{NOT}~ p,\) also written \(\lnot p,\!\) appears below:


\(\text{Logical Negation}\!\)
\(p\!\) \(\lnot p\!\)
\(\operatorname{F}\) \(\operatorname{T}\)
\(\operatorname{T}\) \(\operatorname{F}\)


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 \(p\!\)


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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.

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