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

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


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.