# Contents

## Idea

In (classical) logic, the negation of a statement $p$ is a statement $\neg{p}$ which is true if and only if $p$ is false. Hence, viewed algebraically, the negation corresponds to the complement operator of the corresponding Boolean algebra which satisfies $a\wedge\neg a=\bot$ as well as $a\vee \neg a=\top$.

More generally, as different logics correspond to different types of lattices, one calls negation antitone, or polarity reversing, lattice operators that mimic or approximate the algebraic and proof-theoretic behavior of $\neg$.

## Negation in different logics

In classical logic, we have the double negation law:

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

In intuitionistic logic, we only have

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

while in paraconsistent logic, we instead have

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

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, $p^\bot$, 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 type theory syntax

In usual type theory syntax negation is obtained as the function type into the empty type: $\not a = a \to \emptyset$.

## In categorical semantics

The categorical semantics of negation is the internal hom into the initial object: $\not = [-, \emptyset]$.

In a topos, the negation of an object $A$ (a proposition under the propositions as types-interpretation) is the internal hom object $0^A$, where $0 = \emptyset$ denotes the initial object.

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.

## References

• Y. Gauthier, A Theory of Local Negation: The Model and some Applications , Arch. Math. Logik 25 (1985) pp.127-143. (gdz)

• H. Wansing, Negation , pp.415-436 in Goble (ed.), The Blackwell Guide to Philosophical Logic , Blackwell Oxford 2001.

Revised on February 4, 2016 18:52:33 by Toby Bartels (64.89.58.198)