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wikipedia
In logic, negation, also called logical complement, is an operation that essentially takes a proposition p to another proposition "not p", written ¬p, which is interpreted intuitively as being true when p is false and false when p is true. Negation is thus a unary (single-argument) logical connective. It may be applied as an operation on propositions, truth values, or semantic values more generally. In classical logic negation is normally identified with the truth function that takes truth to falsity and vice versa. In intuitionistic logic, according to the Brouwer–Heyting–Kolmogorov interpretation, the negation of a proposition p is the proposition whose proofs are the refutations of p.

Classical 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. So, if statement A is true, then ¬A (pronounced "not A") would therefore be false; and conversely, if ¬A is true, then A would be false.

Classical negation can be defined in terms of other logical operations. For example, ¬p can be defined as p → F, where "→" is logical consequence 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 do not 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: "not p, or q".
Algebraically, classical negation corresponds to complementation in a Boolean algebra, and intuitionistic negation to pseudocomplementation in a Heyting algebra. These algebras provide a semantics for classical and intuitionistic logic respectively.

Double negation
Within a system of classical logic, double negation, that is, the negation of the negation of a proposition p, is logically equivalent to p. Expressed in symbolic terms, ¬¬p ⇔ p. In intuitionistic logic, a proposition implies its double negation but not conversely. This marks one important difference between classical and intuitionistic negation. Algebraically, classical negation is called an involution of period two.
However, in intuitionistic logic we do have the equivalence of ¬¬¬p and ¬p. Moreover, in the propositional case, a sentence is classically provable if its double negation is intuitionistically provable. This result is known as Glivenko's theorem.
Distributivity
De Morgan's laws provide a way of distributing negation over conjunction and disjunction:
, and
.
Linearity
In Boolean algebra, a linear function is one such that:
If there exists a0, a1, ..., an {0,1} such that f(b1, ..., bn) = a0 ⊕ (a1 b1) ⊕ ... ⊕ (an bn), for all b1, ..., bn {0,1}.
Another way to express this is that each variable always makes a difference in the truth-value of the operation or it never makes a difference. Negation is a linear logical operator.
[edit]Self dual
In Boolean algebra a self dual function is one such that:
f(a1, ..., an) = ~f(~a1, ..., ~an) for all a1, ..., an {0,1}. Negation is a self dual logical operator.

I have no idea what I just read