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
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive 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, (idemponent) 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 formal logic such as type theory a term zz is an entity/expression of the formal language which is of some type ZZ. One writes z:Zz : Z to express this (a typing judgement). The semantics of terms in Set is: elements of a set, where one writes zZz \in Z. One also says zz is an inhabitant of the type ZZ and that ZZ is an inhabited type if it has a term.

A term z:Zz : Z may depend on free variables xx that are themselves terms x:Xx : X of some other type XX. For instance zx+3z \coloneqq x + 3 may be a term of type ZZ \coloneqq \mathbb{Z} (the integers) which depends on a variable term xx also of type XX \coloneqq \mathbb{Z} the integers. The notation for this in the metalanguage is

x:Xz:Z. x : X \vdash z : Z \,.

Generally here also the type ZZ itself may depend on the variable xx (hence the term zz may be of different type dependending on its free variables) in which case one says that zz is a term of XX-dependent type.


In the metalanguage of type theory called natural deduction, terms are what the term introduction rules produce.

Categorical semantics

Here are comments on the interpretation of types in the categorical semantics of dependent type theory.

In the internal language of any category CC, a morphism

f:BA f : B \to A

is a term f(x)f(x) of type AA where xx is a free variable of type BB, which in type-theoretic symbols is given by

x:Bf(x):A. x\colon B \vdash f(x)\colon A \,.

Last revised on September 26, 2012 at 17:06:01. See the history of this page for a list of all contributions to it.