Negation, in its most basic sense, changes the truth value of a statement to its opposite. It is an operation needed chiefly in logic, mathematics, and grammar.

Logic and mathematics

In logic, logical negation is a unary logical operator that reverses the truth value of its operand.

The negation of the statement p is written in various ways:

• p (which is p with a bar over it)
• ~p
• ¬p
• NOT p
•  !p

It is read as "It is not the case that p", or simply "not p".

~p is true if and only if p is false. For instance, if p denotes the statement "today is Saturday", then its negation ~p is the statement "today is not Saturday".

In classical logic, double negation means affirmation; i.e., the statements p and ~(~p) are logically equivalent. In intuitionistic logic, however, ~~p is a weaker statement than p. Nevertheless, ~~~p and ~p are logically equivalent.

Logical negation can be defined in terms of other logical operations. For example, ~p can be defined as p → F, where → is material implication and F is absolute falsehood. Conversely, one can define F as p & ~p for any proposition p, where & is logical conjunction. The idea here is that any contradiction is false. While these ideas work in both classical and intuitionistic logic, they don't work in Brazilian logic, where contradictions are not necessarily false. But in classical logic, we get a further identity: p → q can be defined as ~p ∨ q, where ∨ is logical disjunction.

Algebraically, logical negation corresponds to the complement in a Boolean algebra (for classical logic) or a Heyting algebra (for intuitionistic logic).

Grammar

In grammar, negation is the process that turns an affirmative statement (I am the walrus) into its opposite denial (I am not the walrus). Nouns as well as verbs can be grammatically negated, by the use of a negative adjective (There is no walrus), a negative pronoun (Nobody is the walrus), or a negative adverb (I never was the walrus).

In English, negation for most verbs other than be and have, or verb phrases in which be, have or do already occur, requires the recasting of the sentence using the dummy auxiliary verb do, which adds little to the meaning of the negative phrase, but serves as a place to attach the negative particles not, or its contracted form -n't, to:

• I have a walrus.
• I haven't a walrus. (rare, but it is still possible to negate have without the auxiliary do.)
• I don't have a walrus. (the most common way in contemporary English.)

In Middle English, the particle not could be attached to any verb:

• I see not the walrus.

In Modern English, these forms fell out of use, and the use of an auxiliary such as do or be is obligatory in most cases:

• I do not see the walrus.
• I am not seeing the walrus.
• I have not seen the walrus.

Curiously, the verb do requires a second instance of itself in order to be marked for negation:

• The walrus doesn't do tricks

not

• The walrus doesn't tricks.

In English, as in most other Germanic languages, the use of double negatives as grammatical intensifiers was formerly in frequent use:

• We don't have no walruses here.

Usage prescriptivists consider this use of double negatives to be a solecism, and condemn it. It makes the rhetorical figure of litotes ambiguous. It remains common in colloquial English.

Other languages have simpler forms of negation; in Latin, simple negation is a matter of adding the negative particles non or ne to the verb. In French, the most basic form of verb negation involves adding the circumflexion ne ... pas to the main verb or its auxiliary; je veux un morse ("I want a walrus"); je ne veux pas un morse ("I do not want a walrus.")