Grammar
In grammar, negation is the process that turns an affirmative statement (I am Australian) into its opposite denial (I am not Australian). The linguist D. Biber refers to two types of negation, synthetic ('no', 'neither' or 'nor' negation) and analytic ('not' negation). For example, "He is neither here nor there" (synthetic) or "He is not here" (analytic). Nouns as well as verbs can be grammatically negated, by the use of a negative adjective (There is no chicken), a negative pronoun (Nobody is American here), or a negative adverb (I never was American).
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 chicken.
I haven't a chicken. (Rare, but it is still possible to negate have without the auxiliary do.)
I don't have a chicken. (The most common way in contemporary English.)
In Middle English, the particle not could be attached to any verb:
I see not the chicken.
In Modern English, these forms fell out of use, and the use of an auxiliary verb such as do or be is obligatory in most cases:
I do not see the chicken.
I have not seen the chicken.
The verb do also follows this rule, and therefore requires a second instance of itself in order to be marked for negation:
The chicken doesn't do tricks
not The chicken doesn't tricks.
In English, as in most other Germanic languages (and many non-Germanic languages), the use of double negatives as grammatical intensifiers was formerly in frequent use:
We don't have no chickens 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. In Ancient Greek, a simple negative (οὐ or μὴ) following another simple or compound negative (e.g., οὐδείς, no one) results in an affirmation, whereas a compound negative following a simple or compound negative strengthens the negation.