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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
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.

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.

You can 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 \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 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.

basic symbols used in logic

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
A\phantom{A}\otimesA\phantom{A}A\phantom{A}multiplicative conjunctionA\phantom{A}
A\phantom{A}\oplusA\phantom{A}A\phantom{A}multiplicative disjunctionA\phantom{A}