---
aliases: [negation, negations]
---
#logic
## Definition
> [!tldr] Definition
> The **negation** of a [[Propositions|proposition]] is another proposition with the opposite truth values. In mathematical notation, the negation of the proposition $P$ is $\neg P$ which is pronounced "not $P
quot;. In this way the negation is a [[connective]] that modifies only one [[Propositions|proposition]].
Notes:
- In Python, the logical negation symbol is the exclamation point `!`. In Excel, it's the word `NOT`.
- The English phrasing of a negation is often just putting the word "not" in the original [[Propositions|proposition]], but this sometimes leads to unintelligible statements. Often, some care is needed to phrase a negation in a way that makes sense to a listener or reader.
- The [[Truth tables|truth table]] for the negation of a statement is built simply by reversing the truth values of the original. The basic truth table for $\neg P$ is:
| $P$ | $\neg P$ |
| ----- | -------- |
| True | False |
| False | True |
- Related: [[Negation of quantified predicates]]
## Examples
The negation of the statement "Bob had pizza for lunch" is "Bob did not have pizza for lunch".
The negation of the statement "Bob had pizza for lunch and went for a walk" is more complicated. If we let $P$ represent the statement "Bob had pizza for lunch" and $W$ "Bob went for a walk" then the original statement is the [[Conjunction]] $P \wedge W$. Its truth table is
| $P$ | $W$ | $P \wedge W$ |
| ----- | ----- | ------------ |
| True | True | True |
| True | False | False |
| False | True | False |
| False | False | False |
The negation of this statement, $\neg (P \wedge W)$, has the truth table:
| $P$ | $W$ | $P \wedge W$ | $\neg (P \wedge W)$ |
| ----- | ----- | ------------ | ------ |
| True | True | True | False |
| True | False | False | True |
| False | True | False | True |
| False | False | False | True |
This statement in English would read, "It is not the case that Bob had pizza for lunch and went for a walk". This is awkward, so we might want to rephrase it in some way. An *incorrect* way to phrase the negation is: "Bob *did not* have pizza for lunch and *did not* go for a walk". This would be the statement $(\neg P) \wedge (\neg W)$ whose truth table is
| $P$ | $W$ | $(\neg P)$ | $(\neg W)$ | $(\neg P) \wedge (\neg W)$ |
| ----- | ----- | ---------- | ---------- | -------------------------- |
| True | True | False | False | False |
| True | False | False | True | False |
| False | True | True | False | False |
| False | False | True | True | True |
And this truth table does not have the exact opposite values as $P \wedge W$: The values differ in the second and third rows. Finding a simply-phrased statement that is [[logically equivalent]] to the original is a task to be done in the course. It turns out that [[DeMorgan's Laws]] state that $\neg (P \wedge W)$ is equivalent to $(\neg P) \vee (\neg W)$, that is, "Either Bob didn't have pizza for lunch or he did not go for a walk".
## Resources
<iframe src="https://player.vimeo.com/video/585958504?h=0b7769f78c#t=5m28s" width="640" height="360" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen></iframe>
<p><a href="https://vimeo.com/585958504#t=5m28s">Screencast 2.2: Logical connectives</a> from <a href="https://vimeo.com/user132700952">Robert Talbert</a> on <a href="https://vimeo.com">Vimeo</a>.</p>