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Mathematical logic and foundations Definition

Type theory

Builds on Russell's paradoxSetFunction

Definition

A type theory is a formal system with these parts.

  1. A collection of types, built from base types by type constructors.
  2. A collection of terms, built from variables and constants by term constructors.
  3. Judgments, in particular the typing judgment Γ⊢t:A\Gamma \vdash t : A, which says that in the context Γ=(x1:A1,…,xn:An)\Gamma = (x_1 : A_1, \dots, x_n : A_n) the term tt has type AA.
  4. Inference rules, which say which judgments can be derived from which, and a notion of computation or equality on terms.

Every well-formed term has a type, and a term can only be used where its type allows.

Remarks

Example: Church's simply typed lambda calculus is a type theory. Its types are built from base types by the arrow A→BA \to B. Its rules include: if Γ,x:A⊢t:B\Gamma, x : A \vdash t : B, then Γ⊢λx. t:A→B\Gamma \vdash \lambda x.\, t : A \to B; and if Γ⊢f:A→B\Gamma \vdash f : A \to B and Γ⊢a:A\Gamma \vdash a : A, then Γ⊢f a:B\Gamma \vdash f\, a : B. Non-example: the untyped lambda calculus is not a type theory, because its terms carry no types. In it, the self-application λx. x x\lambda x.\, x\, x is a legal term, while in the simply typed calculus it has no type. In dependent type theories, types may depend on terms, as in the type Vec(n)\mathrm{Vec}(n) of lists of length nn. Unlike a set, a term has its type as part of its form, and "tt has type AA" is a judgment, not a proposition that can be true or false. This is how type theory avoids Russell's paradox.

Reading path

To understand this page from the ground up, read these 28 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionProposition, 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 levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptModus ponens, concept, no evidence levelModus ponensconceptExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionPropositional logic, concept, no evidence levelPropositional logicconceptSubset, definition, no evidence levelSubsetdefinitionMathematical proof, concept, no evidence levelMathematical proofconceptUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTautology, definition, no evidence levelTautologydefinitionLogical equivalence, definition, no evidence levelLogical equivalencedefinitionFunction, definition, no evidence levelFunctiondefinitionDouble 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, definition, no evidence levelType theorydefinition
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. Proposition concept
  3. Set concept
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Negation definition
  9. Predicate definition
  10. Ordered pair definition
  11. Biconditional definition
  12. Truth table concept
  13. Modus ponens concept
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Propositional logic concept
  18. Subset definition
  19. Mathematical proof concept
  20. Uniqueness quantifier definition
  21. Binary relation definition
  22. Tautology definition
  23. Logical equivalence definition
  24. Function definition
  25. Double negation law statement
  26. Proof by contradiction concept
  27. Russell's paradox statement
  28. Type theory definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Type_theory
About this page
Tag42A2M9S2C1
URIm2://42A2M9S2C1
Kinddefinition
WritingW0 stub
MSC03B38, 03B16
DefinesType theory
UsesRussell's paradox, Set, Function
Changed2 November 2026

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