Negating

Negating is the act of denying, contradicting, or reversing the truth value of a statement or quantity. In logic and mathematics, negation is typically an operation that maps a proposition p to its negation ¬p; in arithmetic, negating a real number x yields -x, its additive inverse.

In formal logic, ¬ is a unary operator. The truth value of ¬p is the opposite of p. In classical logic, double negation holds: ¬¬p is equivalent to p. De Morgan's laws describe how negation distributes over conjunction and disjunction: ¬(p ∧ q) ≡ ¬p ∨ ¬q, and ¬(p ∨ q) ≡ ¬p ∧ ¬q. Some logical systems, such as intuitionistic logic, do not accept all these equivalences in the same way.

In mathematics, negation extends to signs, such as the additive inverse of a number, and to set-theoretic complements relative to a universal set. The negation of a statement is not the same as its contradictory in all contexts, but both express forms of opposition to the original claim.

In computer science, boolean negation is the NOT operation. It is often denoted by NOT or the exclamation mark ! and flips a true value to false and vice versa; bitwise negation flips all bits in a binary representation. Negation underpins control flow, conditionals, and the evaluation of predicates, and interacts with other logical operators through laws akin to De Morgan's.

In linguistics and everyday language, negation is expressed with adverbs and particles such as not, never, no, as well as with prefixes like un-, in-, non-. Negation can affect scope, emphasis, and polarity, and is analyzed across syntax, semantics, and pragmatics in different languages.