Math 2.0

Mathematical logic and foundations Concept

Simply typed lambda calculus

Builds on Lambda calculusType theoryCurry–Howard correspondence

Summary

The simply typed lambda calculus is the lambda calculus in which every term carries a type built from base types with the function arrow →\to. It was introduced by Alonzo Church in 1940 as a typed version of his calculus. Typing rules out self-application, so every typed term has a normal form and computation always ends. It is the smallest serious type theory, and through the Curry–Howard correspondence it matches the implication fragment of intuitionistic logic.

Reading path

To understand this page from the ground up, read these 23 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…conceptSimply typed lambda calculus, concept, no evidence levelSimply typed lambda c…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
  23. Simply typed lambda calculus concept

Where this leads

16 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Simply_typed_lambda_calculus
About this page
Tag2DP940895E
URIm2://2DP940895E
Kindconcept
WritingW0 stub
MSC03B40, 68N18, 03B38
UsesLambda calculus, Type theory, Curry–Howard correspondence
Same as
Changed3 November 2026

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