= Negation
{title2=$\neg P$}
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= Logical negation
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= NOT
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<Logical negation> exchanges zero and one: $\neg0=1$ and $\neg1=0$. For a numerical <bit> $x$, this is $1-x$; applying <logical negation> twice restores $x$. It acts on the entire <propositional formula> in its scope. Parentheses therefore matter: $\neg(P\vee Q)$ says that neither condition holds, while $(\neg P)\vee Q$ also accepts every case where $Q$ holds. At $(P,Q)=(1,1)$ their values are zero and one.
On a <set>, <logical negation> becomes taking a <complement> in a fixed <universal set>. In <Boolean operations on events>, “not even” selects $\{1,3,5\}$. “Not both even and at least four” selects $\{1,2,3,5\}$; it does not require both qualifications to fail. <De Morgan's laws> make the scope of these compound negations precise.
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