Math 2.0

Mathematical logic and foundations Proof theory Concept

Curry–Howard correspondence

Builds on Lambda calculusIntuitionistic logicType theory

Summary

The Curry–Howard correspondence is the match between proofs in intuitionistic logic and programs in typed lambda calculus. A proposition corresponds to a type, and a proof of the proposition corresponds to a program of that type. Simplifying a proof corresponds to running the program. It is the basis of proof assistants built on type theory, where checking a proof means checking the type of a program.

Reading path

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

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Element, definition, no evidence levelElementdefinitionIntuitionistic logic, concept, no evidence levelIntuitionistic logicconceptLambda calculus, concept, no evidence levelLambda calculusconceptProposition, concept, no evidence levelPropositionconceptSet, concept, no evidence levelSetconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptModus ponens, concept, no evidence levelModus ponensconceptMathematical proof, concept, no evidence levelMathematical proofconceptPropositional logic, concept, no evidence levelPropositional logicconceptTautology, definition, no evidence levelTautologydefinitionLogical equivalence, definition, no evidence levelLogical equivalencedefinitionDouble negation law, statement, no evidence levelDouble negation lawstatementProof by contradiction, concept, no evidence levelProof by contradictionconceptRussell's paradox, statement, no evidence levelRussell's paradoxstatementType theory, concept, no evidence levelType theoryconceptCurry–Howard correspondence, concept, no evidence levelCurry–Howard correspo…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. Element definition
  2. Intuitionistic logic concept
  3. Lambda calculus concept
  4. Proposition concept
  5. Set concept
  6. Set definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Negation definition
  11. Biconditional definition
  12. Truth table concept
  13. Modus ponens concept
  14. Mathematical proof concept
  15. Propositional logic concept
  16. Tautology definition
  17. Logical equivalence definition
  18. Double negation law statement
  19. Proof by contradiction concept
  20. Russell's paradox statement
  21. Type theory concept
  22. Curry–Howard correspondence concept

Where this leads

3 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Curry–Howard_correspondence
About this page
TagA77FER1Q5H
URIm2://A77FER1Q5H
Kindconcept
WritingW0 stub
MSC03B40, 03F03, 68N18
UsesLambda calculus, Intuitionistic logic, Type theory
Same as
Changed2 November 2026

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