Contents

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 $\neg$.

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.

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.

$\phantom{-}$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
symbolin 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)
symbolin 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}$

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.

Last revised on January 4, 2023 at 01:52:39. See the history of this page for a list of all contributions to it.