nLab
negation

In logic, the negation of a statement $p$ is a statement $\neg p$ which is true if and only if $p$ is false.

In classical logic, we have the double negation law:

$$\neg \neg p\equiv p.$$\neg\neg{p} \equiv p.

In intuitionistic logic, we only have

$$\neg \neg p⊣p,$$\neg\neg{p} \dashv p,

while in paraconsistent logic, we instead have

$$\neg \neg p⊢p.$$\neg\neg{p} \vdash p.

You can interpret intuitionistic negation as ‘denial’ and paraconsistent negation as ‘doubt’. So when one says that one doesn't deny $p$, that's weaker than actually asserting $p$; while when one says that one doesn't doubt $p$, that's stronger than merely asserting $p$. Paraconsistent logic has even been applied to the theory of law: if $p$ is a judgment that normally requires only the preponderance of evidence, then $\neg \neg p$ is a judgment of $p$ beyond reasonable doubt.

Linear logic features (at least) three different forms of negation, one for each of the above. (The default meaning of the term ‘negation’ in linear logic, $p^{\perp }$, is the one that satisfies the classical double-negation law.)

Accordingly, negation mediates de Morgan duality in classical and linear logic but not in intuitionistic or paraconsistent logic.

In a topos, the negation of an object $A$ is the object $0^A$; this matches the intuitionistic notion of negation in that there is a natural morphism $A\to 0^{0^A}$ but not the other way around.