nLab negation



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels




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

Negation in different logics

In classical logic, we have the double negation law:

¬¬pp. \neg\neg{p} \;\equiv\; p \,.

In intuitionistic logic, we only have

¬¬pp, \neg\neg{p} \;\dashv\; p \,,

while in paraconsistent logic, we instead have

¬¬pp. \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 pp, that's weaker than actually asserting pp; while when one says that one doesn't doubt pp, that's stronger than merely asserting pp. Paraconsistent logic has even been applied to the theory of law: if pp is a judgment that normally requires only the preponderance of evidence, then ¬¬p\neg\neg{p} is a judgment of pp 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 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: ¬a=a\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: ¬a=a\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 AA (a proposition under the propositions as types-interpretation) is the internal hom object 0 A0^A, where 0=0 = \emptyset denotes the initial object.

This matches the intuitionistic notion of negation in that there is a natural morphism A0 0 AA \to 0^{0^A} but not the other way around.

\phantom{-}symbol\phantom{-}\phantom{-}in logic\phantom{-}
A\phantom{A}\inA\phantom{A}element relation
A\phantom{A}:\,:A\phantom{A}typing relation
A\phantom{A}\vdashA\phantom{A}A\phantom{A}entailment / sequentA\phantom{A}
A\phantom{A}\topA\phantom{A}A\phantom{A}true / topA\phantom{A}
A\phantom{A}\botA\phantom{A}A\phantom{A}false / bottomA\phantom{A}
A\phantom{A}\LeftrightarrowA\phantom{A}logical equivalence
A\phantom{A}\neqA\phantom{A}negation of equality / apartnessA\phantom{A}
A\phantom{A}\notinA\phantom{A}negation of element relation A\phantom{A}
A\phantom{A}¬¬\not \notA\phantom{A}negation of negationA\phantom{A}
A\phantom{A}\existsA\phantom{A}existential quantificationA\phantom{A}
A\phantom{A}\forallA\phantom{A}universal quantificationA\phantom{A}
A\phantom{A}\wedgeA\phantom{A}logical conjunction
A\phantom{A}\veeA\phantom{A}logical disjunction
symbolin type theory (propositions as types)
A\phantom{A}\toA\phantom{A}function type (implication)
A\phantom{A}×\timesA\phantom{A}product type (conjunction)
A\phantom{A}++A\phantom{A}sum type (disjunction)
A\phantom{A}00A\phantom{A}empty type (false)
A\phantom{A}11A\phantom{A}unit type (true)
A\phantom{A}==A\phantom{A}identity type (equality)
A\phantom{A}\simeqA\phantom{A}equivalence of types (logical equivalence)
A\phantom{A}\sumA\phantom{A}dependent sum type (existential quantifier)
A\phantom{A}\prodA\phantom{A}dependent product type (universal quantifier)
symbolin linear logic
A\phantom{A}\multimapA\phantom{A}A\phantom{A}linear implicationA\phantom{A}
A\phantom{A}\otimesA\phantom{A}A\phantom{A}multiplicative conjunctionA\phantom{A}
A\phantom{A}\oplusA\phantom{A}A\phantom{A}additive disjunctionA\phantom{A}
A\phantom{A}&\&A\phantom{A}A\phantom{A}additive conjunctionA\phantom{A}
A\phantom{A}\invampA\phantom{A}A\phantom{A}multiplicative disjunctionA\phantom{A}
A\phantom{A}!\;!A\phantom{A}A\phantom{A}exponential conjunctionA\phantom{A}


  • 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.