nLab negation

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 $\neg$.

As a logic gate

As a logic gate on bits and as a quantum logic gate on qbits ($X$-Pauli matrix):

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

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

One may 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.

Antithesis interpretation

In the antithesis interpretation of intuitionistic logic, one considers, instead of single propositions, pairs of mutually exclusive propositions, i.e. propositions $p$ and $q$ such that $\neg (p \wedge q)$. These are generalisations of the intuitionistic denial negation since we have by currying the statements $p \to \neg q$ and $q \to \neg p$. The intuitionistic negation is simply a special case of the above since we have the true statements $p \to \neg \neg p$ and $\neg p \to \neg p$ from the law of non-contradiction $\neg (p \wedge \neg p)$.

There is an affine linear logic associated with mutually exclusive propositions where the affine negation of mutually exclusive propositions $p$ and $q$ is just $q$ and $p$ respectively. Thus, we can say that $q$ is the negation of $p$ and vice versa, for mutually exclusive propositions $p$ and $q$. This is very important for constructive mathematics, where the terms “equal/not equal” and related concepts such as “rational number/irrational number” and “algebraic number/transcendental number” are usually defined using equality/tight apartness relation for objects such as the real numbers and the p-adic numbers. These terms only makes sense in this context if one is using the affine negation of the mutually exclusive propositions rather than intuitionistic denial negation, the latter which in general is not an involution on propositions.

In type theory syntax

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

Equivalently, in type theory with equivalence types but without function types, negation is the equivalence type with the empty type: $\not a = a \simeq \varnothing$.

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.

Related entries

• double negation

• complement

• De Morgan duality

• co-Heyting negation

• adjoint cylinder

• strong negation

$\phantom{-}$symbol$\phantom{-}$	$\phantom{-}$in logic$\phantom{-}$
$\phantom{A}$$\in$	$\phantom{A}$element relation
$\phantom{A}$$\,:$	$\phantom{A}$typing relation
$\phantom{A}$$=$	$\phantom{A}$equality
$\phantom{A}$$\vdash$$\phantom{A}$	$\phantom{A}$entailment / sequent$\phantom{A}$
$\phantom{A}$$\top$$\phantom{A}$	$\phantom{A}$true / top$\phantom{A}$
$\phantom{A}$$\bot$$\phantom{A}$	$\phantom{A}$false / bottom$\phantom{A}$
$\phantom{A}$$\Rightarrow$	$\phantom{A}$implication
$\phantom{A}$$\Leftrightarrow$	$\phantom{A}$logical equivalence
$\phantom{A}$$\not$	$\phantom{A}$negation
$\phantom{A}$$\neq$	$\phantom{A}$negation of equality / apartness$\phantom{A}$
$\phantom{A}$$\notin$	$\phantom{A}$negation of element relation $\phantom{A}$
$\phantom{A}$$\not \not$	$\phantom{A}$negation of negation$\phantom{A}$
$\phantom{A}$$\exists$	$\phantom{A}$existential quantification$\phantom{A}$
$\phantom{A}$$\forall$	$\phantom{A}$universal quantification$\phantom{A}$
$\phantom{A}$$\wedge$	$\phantom{A}$logical conjunction
$\phantom{A}$$\vee$	$\phantom{A}$logical disjunction
symbol	in type theory (propositions as types)
$\phantom{A}$$\to$	$\phantom{A}$function type (implication)
$\phantom{A}$$\times$	$\phantom{A}$product type (conjunction)
$\phantom{A}$$+$	$\phantom{A}$sum type (disjunction)
$\phantom{A}$$0$	$\phantom{A}$empty type (false)
$\phantom{A}$$1$	$\phantom{A}$unit type (true)
$\phantom{A}$$=$	$\phantom{A}$identity type (equality)
$\phantom{A}$$\simeq$	$\phantom{A}$equivalence of types (logical equivalence)
$\phantom{A}$$\sum$	$\phantom{A}$dependent sum type (existential quantifier)
$\phantom{A}$$\prod$	$\phantom{A}$dependent product type (universal quantifier)
symbol	in linear logic
$\phantom{A}$$\multimap$$\phantom{A}$	$\phantom{A}$linear implication$\phantom{A}$
$\phantom{A}$$\otimes$$\phantom{A}$	$\phantom{A}$multiplicative conjunction$\phantom{A}$
$\phantom{A}$$\oplus$$\phantom{A}$	$\phantom{A}$additive disjunction$\phantom{A}$
$\phantom{A}$$\&$$\phantom{A}$	$\phantom{A}$additive conjunction$\phantom{A}$
$\phantom{A}$$\invamp$$\phantom{A}$	$\phantom{A}$multiplicative disjunction$\phantom{A}$
$\phantom{A}$$\;!$$\phantom{A}$	$\phantom{A}$exponential conjunction$\phantom{A}$
$\phantom{A}$$\;?$$\phantom{A}$	$\phantom{A}$exponential disjunction$\phantom{A}$

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.

• Michael Shulman, Affine logic for constructive mathematics. Bulletin of Symbolic Logic, Volume 28, Issue 3, September 2022. pp. 327 - 386 (doi:10.1017/bsl.2022.28, arXiv:1805.07518)