Math 2.0

Mathematical logic and foundations Definition

Brouwer–Heyting–Kolmogorov interpretation

Builds on Intuitionistic logicConstructivismConjunctionDisjunctionImplicationNegationUniversal quantifierExistential quantifier

Definition

Let DD be a domain of objects, and let AA, BB and P(x)P(x) be statements about DD. The Brouwer–Heyting–Kolmogorov interpretation says what a proof of a compound statement is, by these clauses.

  1. A proof of A∧BA \wedge B is a pair (a,b)(a, b), where aa is a proof of AA and bb is a proof of BB.
  2. A proof of A∨BA \vee B is a pair (i,c)(i, c) with i∈{0,1}i \in \set{0, 1}. If i=0i = 0, then cc is a proof of AA. If i=1i = 1, then cc is a proof of BB.
  3. A proof of A→BA \to B is a construction that transforms each proof of AA into a proof of BB.
  4. There is no proof of the absurdity ⊥\bot.
  5. A proof of ¬A\neg A is a proof of A→⊥A \to \bot.
  6. A proof of ∀x P(x)\forall x\, P(x) is a construction that transforms each d∈Dd \in D into a proof of P(d)P(d).
  7. A proof of ∃x P(x)\exists x\, P(x) is a pair (d,p)(d, p), where d∈Dd \in D and pp is a proof of P(d)P(d).

These clauses give the meaning of conjunction, disjunction, implication, negation, the universal quantifier and the existential quantifier in intuitionistic logic.

Remarks

Example: A→¬¬AA \to \neg \neg A has a BHK proof. Given a proof aa of AA, the construction f↦f(a)f \mapsto f(a) sends each proof ff of ¬A\neg A to a proof of ⊥\bot, so it is a proof of ¬¬A\neg \neg A. Non-example: there is no known BHK proof of ¬¬A→A\neg \neg A \to A for all AA, because a proof of ¬¬A\neg \neg A gives no construction of a proof of AA. The notion of "construction" is left informal. Kleene's realizability reads it as a computable function, and the Curry–Howard correspondence reads it as a typed lambda term.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

Show the path as a graph
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 quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal 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, definition, no evidence levelBrouwer–Heyting–Kolmo…definition
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. Universal quantifier definition
  14. Propositional logic concept
  15. Mathematical proof concept
  16. Tautology definition
  17. Constructive proof concept
  18. Logical equivalence definition
  19. Double negation law statement
  20. Proof by contradiction concept
  21. Constructivism concept
  22. Brouwer–Heyting–Kolmogorov interpretation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Brouwer–Heyting–Kolmogorov_interpretation
About this page
Tag1VFH9WT1RZ
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Kinddefinition
WritingW0 stub
MSC03F55, 03B20
DefinesBrouwer–Heyting–Kolmogorov interpretation
UsesIntuitionistic logic, Constructivism, Conjunction, Disjunction, Implication, Negation, Universal quantifier, Existential quantifier
Changed2 November 2026

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