Logic and mathematics
The negation of the statement p is written in various ways:
- p (which is p with a bar over it)
- NOT p
It is read as "It is not the case that p", or simply "not p".
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.
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.) or a negative pronoun (Nobody is 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
- The walrus doesn't tricks.
- We don't have no walruses here.
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.")