This entry is about the notion in type theory. For the unrelated notion of the same name in model theory see at type (in model theory).


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 modern logic, we understand that every variable should have a type, or domain of discourse or be of some sort. For instance we say that if a variable nn is constrained to be an integer then “nn is of integer type” or “of type \mathbb{Z}”. The usual formal expression from set theory for this – nn \in \mathbb{Z} – is then often written n:n \colon \mathbb{Z}

We speak of typed logic if this typing of variables is enforced by the metalanguage. In formulations of a theory the types are often called sorts. More generally, type theory formalizes reasoning with such typed variables. See there for more

(Untyped logic may be seen as simply a special case, in which there is only a single unique type. Thus, untyped logic has one type, not no type.)


Reasoning with types is formalized in natural deduction (which in turn is formalized in a logical framework such as Elf).

Behaviour of types is specified by a 4-step set of rules

  1. type formation

  2. term introduction

  3. term elimination

  4. computation rules


Deep relations between type theory, category theory and computer science relate types to other notions, such as objects in a category. See at computational trinitarianism for more on this.

type, type theory

dependent type, dependent type theory, Martin-Löf dependent type theory

homotopy type, homotopy type theory