Math 2.0

Mathematical logic and foundations Constructive mathematics Concept

Brouwer–Heyting–Kolmogorov interpretation

Builds on Intuitionistic logicConstructivism

Summary

The Brouwer–Heyting–Kolmogorov interpretation, or BHK interpretation, explains the meaning of the logical connectives in intuitionistic logic. It says what counts as a proof of a compound statement in terms of proofs of its parts. It is an informal explanation, not a formal semantics, and it expresses the view of constructivism. Realizability and the Curry–Howard correspondence are precise versions of it.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

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Intuitionistic logic, concept, no evidence levelIntuitionistic logicconceptProposition, concept, no evidence levelPropositionconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptLaw of excluded middle, statement, no evidence levelLaw of excluded middlestatementModus ponens, concept, no evidence levelModus ponensconceptExistential quantifier, definition, no evidence levelExistential quantifierdefinitionPropositional logic, concept, no evidence levelPropositional logicconceptMathematical proof, concept, no evidence levelMathematical proofconceptTautology, definition, no evidence levelTautologydefinitionConstructive proof, concept, no evidence levelConstructive proofconceptLogical equivalence, definition, no evidence levelLogical equivalencedefinitionDouble negation law, statement, no evidence levelDouble negation lawstatementProof by contradiction, concept, no evidence levelProof by contradictionconceptConstructivism, concept, no evidence levelConstructivismconceptBrouwer–Heyting–Kolmogorov interpretation, concept, no evidence levelBrouwer–Heyting–Kolmo…concept
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Intuitionistic logic concept
  2. Proposition concept
  3. Conjunction definition
  4. Disjunction definition
  5. Implication definition
  6. Negation definition
  7. Predicate definition
  8. Biconditional definition
  9. Truth table concept
  10. Law of excluded middle statement
  11. Modus ponens concept
  12. Existential quantifier definition
  13. Propositional logic concept
  14. Mathematical proof concept
  15. Tautology definition
  16. Constructive proof concept
  17. Logical equivalence definition
  18. Double negation law statement
  19. Proof by contradiction concept
  20. Constructivism concept
  21. Brouwer–Heyting–Kolmogorov interpretation concept

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Brouwer–Heyting–Kolmogorov_interpretation
About this page
Tag7B53ZGXXEQ
URIm2://7B53ZGXXEQ
Kindconcept
WritingW0 stub
MSC03F55, 03B20
UsesIntuitionistic logic, Constructivism
Same as
Changed2 November 2026

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