Negation

Negation

NOT
Definition	${\displaystyle \lnot {x}}$
Truth table	${\displaystyle (01)}$
Logic gate
Normal forms
Disjunctive	${\displaystyle \lnot {x}}$
Conjunctive	${\displaystyle \lnot {x}}$
Zhegalkin polynomial	${\displaystyle 1\oplus x}$
Post's lattices
0-preserving	no
1-preserving	no
Monotone	no
Affine	yes
Self-dual	yes
v t e

Logical connectives
NOT ${\displaystyle \neg A}$, ${\displaystyle -A}$, ${\displaystyle {\overline {A}}}$, ${\displaystyle \sim A}$ AND ${\displaystyle A\land B}$, ${\displaystyle A\cdot B}$, ${\displaystyle AB}$, ${\displaystyle A\&B}$, ${\displaystyle A\&\&B}$ NAND ${\displaystyle A{\overline {\land }}B}$, ${\displaystyle A\uparrow B}$, ${\displaystyle A\mid B}$, ${\displaystyle {\overline {A\cdot B}}}$ OR ${\displaystyle A\lor B}$, ${\displaystyle A+B}$, ${\displaystyle A\mid B}$, ${\displaystyle A\parallel B}$ NOR ${\displaystyle A{\overline {\lor }}B}$, ${\displaystyle A\downarrow B}$, ${\displaystyle {\overline {A+B}}}$ XNOR ${\displaystyle A\ {\text{XNOR}}\ B}$ └ equivalent ${\displaystyle A\equiv B}$, ${\displaystyle A\Leftrightarrow B}$, ${\displaystyle A\leftrightharpoons B}$ XOR ${\displaystyle A{\underline {\lor }}B}$, ${\displaystyle A\oplus B}$ └nonequivalent ${\displaystyle A\not \equiv B}$, ${\displaystyle A\not \Leftrightarrow B}$, ${\displaystyle A\nleftrightarrow B}$ implies ${\displaystyle A\Rightarrow B}$, ${\displaystyle A\supset B}$, ${\displaystyle A\rightarrow B}$ converse ${\displaystyle A\Leftarrow B}$, ${\displaystyle A\subset B}$, ${\displaystyle A\leftarrow B}$
Related concepts
Propositional calculus Predicate logic Boolean algebra Truth table Truth function Boolean function Functional completeness Scope (logic)
Applications
Digital logic Programming languages Mathematical logic Philosophy of logic
Category
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In logic, negation, also called the logical not or logical complement, is an operation that takes a proposition ${\displaystyle P}$ to another proposition "not ${\displaystyle P}$", written ${\displaystyle \neg P}$, ${\displaystyle {\mathord {\sim }}P}$ or ${\displaystyle {\overline {P}}}$. It is interpreted intuitively as being true when ${\displaystyle P}$ is false, and false when ${\displaystyle P}$ is true.^[1][2] For example, if ${\displaystyle P}$ is "Spot runs", then "not ${\displaystyle P}$" is "Spot does not run".

Negation is a unary logical connective. It may furthermore be applied not only to propositions, but also to notions, 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 ${\displaystyle P}$ is the proposition whose proofs are the refutations of ${\displaystyle P}$.

An operand of a negation is a negand,^[3] or negatum.^[3]