nLab Brouwer-Heyting-Kolmogorov interpretation



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
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/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


Constructivism, Realizability, Computability



The Brouwer-Heyting-Kolmogorov interpretation of intuitionistic logic is a description of proofs of propositions in intuitionistic logic as functions, often computable functions, where it is also called the realizability interpretation.

This is otherwise known as the paradigm of propositions as types and proofs as programs, and in a precise form as the Curry-Howard correspondence. See there for more.

The name “Brouwer-Heyting-Kolmogorov” is due to Troelstra, and it is a matter of some dispute whether Brouwer’s name should be included. Brouwer never explicitly formulated any interpretation of this sort, and remained against all formalism his entire life. Moreover, Escardo-Xu have shown that Brouwer’s famous intuitionistic theorem “all functions \mathbb{N}^{\mathbb{N}} \to \mathbb{N} are continuous” is actually inconsistent under a literal version of this interpretation (i.e. without including propositional truncation). Thus, perhaps it should only be called the “Heyting-Kolmogorov” interpretation.


  • Wikipedia, BHK interpretation

  • L. E. J. Brouwer, Points and Spaces , CJM 6 (1954) pp.1-17. (pdf)

  • H. Freudenthal , Zur intuitionistischen Deutung logischer Formeln , Comp. Math. 4 (1937) pp.112-116. (pdf)

  • A. Heyting , Die intuitionistische Grundlegung der Mathematik , Erkenntnis 2 (1931) pp.106-115.

  • A. Heyting , Bemerkungen zu dem Aufsatz von Herrn Freudenthal “Zur intuitionistischen Deutung logischer Formeln” , Comp. Math. 4 (1937) pp.117-118. (pdf)

  • A. Kolmogoroff, Zur Deutung der intuitionistischen Logik , Math. Z. 35 (1932) pp.58-65. (gdz)

  • G. Kreisel, Mathematical Logic , pp.95-195 in Saaty (ed.), Lectures on Modern Mathematics III , Wiley New York 1965.

  • E. G. F. Díez, Five observations concerning the intended meaning of the intuitionistic logical constants , J. Phil. Logic 29 no. 4 (2000) pp.409–424 . (preprint)

  • Jean-Yves Girard et al., Proofs and Types , CUP 1989.

  • Anne Sjerp Troelstra, Principles of Intuitionism , Springer Heidelberg 1969. (§2)

  • Anne Sjerp Troelstra, Aspects of Constructive Mathematics , pp.973-1052 in Barwise (ed.), Handbook of Mathematical Logic , Elsevier Amsterdam 1977.

  • Anne Sjerp Troelstra, History of Constructivism in the Twentieth Century (1991). (preprint)

  • Wouter Pieter Stekelenburg, Realizability Categories , (arXiv:1301.2134).

  • Martin Escardo and Chuangjie Xu, The inconsistency of a Brouwerian continuity principle with the Curry–Howard interpretation . (pdf)

Links to many papers on realizability and related topics may be found here.

For a comment see also

Last revised on June 16, 2015 at 08:02:45. See the history of this page for a list of all contributions to it.