nLab intrinsic and extrinsic views of typing



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


Deduction and Induction

Constructivism, Realizability, Computability



The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms



The intrinsic and extrinsic views of typing correspond to different interpretations of types, also known as the contrast between “types à la Church” and “types à la Curry”.

Reynolds claims (Rey00),

There are two very different ways of giving denotational semantics to a programming language (or other formal language) with a nontrivial type system. In an intrinsic semantics, only phrases that satisfy typing judgements have meanings. Indeed, meanings are assigned to the typing judgements, rather than to the phrases themselves, so that a phrase that satisfies several judgements will have several meanings.

In contrast, in an extrinsic semantics, the meaning of each phrase is the same as it would be in a untyped language, regardless of its typing properties. In this view, a typing judgement is an assertion that the meaning of a phrase possesses some property.

The terms “intrinsic” and “extrinsic” are recent coinages by the author (Rey98, Chapter 15), but the concepts are much older. The intrinsic view is associated with Alonzo Church, and has been called “ontological” by Leivant (Lev86). The extrinsic view is associated with Haskell Curry, and has been called “semantical” by Leivant.

According to Melliès and Zeilberger (MZ13),

One of the difficulties in giving a clear mathematical definition of the “topic” of type theory is that the word “type” is actually used with two very different intuitive meanings and technical purposes in mind:

(1). Like the syntactician’s parts of speech, as a way of defining the grammar of well-formed expressions. (2). Like the semanticist’s predicates, as a way of identifying subsets of expressions with certain desirable properties.

These two different views of types are often associated respectively with Alonzo Church and Haskell Curry (hence “types à la Church” and “types à la Curry”), while the late John Reynolds referred to these as the intrinsic and the extrinsic interpretations of types (Rey00). In the intrinsic view, all expressions carry a type, and there is no need (or even sense) to consider the meaning of “untyped” expressions; while in the extrinsic view, every expression carries an independent meaning, and typing judgments serve to assert some property of that meaning.


  • John Reynolds, Theories of Programming Languages. Cambridge University Press, Cambridge, England, 1998.
  • Daniel Leivant, Typing and computational properties of lambda expressions, Theoretical Computer Science, 44(1):51–68, 1986.
  • John Reynolds, The Meaning of Types: from Intrinsic to Extrinsic Semantics, BRICS Report RS-00-32, Aarhus University, December 2000. pdf
  • Paul-André Melliès, Noam Zeilberger, Type refinement and monoidal closed bifibrations, arXiv:1310.0263

Attempts to bring both views together in a theory of type refinement:

Last revised on December 31, 2020 at 06:34:22. See the history of this page for a list of all contributions to it.