Negation

Negation flips a statement’s truth value: if a proposition P is true, its negation “not P” is false, and if P is false, “not P” is true. This operation, also called the logical not or logical complement, works on propositions, notions, truth values, or semantic values. The negated proposition is written as ¬P, ∼P, P′, or P̅. For instance, if P is “The dog runs,” then ¬P is “The dog does not run.” The thing being negated is called the negand or negatum.

In classical logic, negation is a truth function that swaps truth and falsity. Intuitionistic logic treats the negation of P as the proposition whose proofs are refutations of P, following the Brouwer–Heyting–Kolmogorov interpretation. Negation can be defined using other logical operations: ¬P can be expressed as P → ⊥ (where → is logical consequence and ⊥ is absolute falsehood), or ⊥ can be defined as P ∧ ¬P for any P (where ∧ is logical conjunction).

These definitions work in classical and intuitionistic logic but not in paraconsistent logic, where contradictions aren’t automatically false. Negation can also be built from NAND or NOR. Algebraically, classical negation matches complementation in a Boolean algebra, while intuitionistic negation matches pseudocomplementation in a Heyting algebra.

Notation

Notation varies by context. Besides ¬, ∼, P′, and P̅, Polish notation uses Np. In set theory, ∉ indicates “not in the set of”: U ∉ A means U is not a member of A. Regardless of symbol, ¬P can be read as “it is not the case that P,” “not that P,” or simply “not P.”

Precedence

Precedence rules reduce parentheses: ¬ has higher precedence than ∧, ∧ higher than ∨, and ∨ higher than →. So ¬P ∧ Q means (¬P) ∧ Q.

Double negation

Key properties include double negation, distributivity, linearity, and self-duality. In classical logic, ¬¬P is equivalent to P (an involution of period two).

In intuitionistic logic, P implies ¬¬P, but not vice versa; however, ¬¬¬P implies ¬P holds. Glivenko’s theorem states that a proposition is classically provable if its double negation is intuitionistically provable. De Morgan’s laws distribute negation over disjunction and conjunction: ¬(P ∨ Q) is equivalent to ¬P ∧ ¬Q, and ¬(P ∧ Q) is equivalent to ¬P ∨ ¬Q. Negation is a linear logical operator in Boolean algebra—each variable always or never changes the truth value—and it is self-dual, meaning ¬(¬a) = a for all a.

Negations of quantifiers

For quantifiers in first-order logic, negating a universal quantifier (∀) gives an existential quantifier (∃), and vice versa. For example, “all humans are mortal” (∀x P(x)) negates to “there exists a human who is not mortal” (∃x ¬P(x)).

Rules of inference for classical negation in natural deduction include negation introduction (reductio ad absurdum: from a derivation of both P and ¬P from Q, infer ¬Q), negation elimination (ex falso quodlibet: from P and ¬P, infer any proposition), and double negation elimination (from ¬¬P, infer P). Intuitionistic negation uses the same rules except double negation elimination.

Negation introduction says that if an absurdity follows from P, then P is false (classically) or refutable (intuitionistically). Negation elimination says anything follows from an absurdity; sometimes it’s formulated with a primitive absurdity sign ⊥, where P and ¬P yield ⊥. In intuitionistic logic, ¬P is defined as P → ⊥, making negation introduction and elimination special cases of implication rules.