Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels




In computer science and to some extent also in formal logic, sugaring refers to the modification of formal notation (syntax) into a form which is more readable for humans, thereby “sweetening” it for consumption. Syntactic sugar does not add to the functionality or expressivity of the language.

Peter Landin first used the expression ‘syntactic sugaring’ in the context of interpretations of terms in the λ-calculus (Landin64).

The reverse process to expressions in the more formal language is sometimes known as ‘desugaring’.



In a set theory where classes are not “first-class objects” (for example ZFC, but not NBG), classes are only syntactic sugar and have no independent existence. For instance, if one defines a class M{X|φ(X)}M \coloneqq \{ X | \varphi(X) \} by comprehension, then “XMX \in M” is syntactic sugar for “φ(X)\varphi(X)”.


In formal linguistics, natural language versions of formal expressions are described as ‘sugarings’ (Ranta 1994). Often there are many ways to sugar a formal expression into natural language.

Consider the type theoretic expression x:man y:donkey(xownsy)\sum_{x: man} \sum_{y: donkey} (x\;owns\; y). Ranta (1994, p. 68) gives nine natural language sugarings for it by applying three different operations twice:

  • there is a man and there is a donkey and he owns it,
  • there is a man and he owns a donkey,
  • there is a man and there is a donkey that he owns,
  • there is a donkey and a man owns it,
  • a man owns a donkey,
  • there is a donkey that a man owns,
  • there is a man such that there is a donkey and he owns it,
  • there is a man who owns a donkey,
  • there is a man such that there is a donkey that he owns.


  • Peter Landin, The mechanical evaluation of expressions, (pdf)

  • Aarne Ranta, Type-theoretical grammar, Oxford University Press (1994)

  • Aarne Ranta, Type theory and the informal language of mathematics, TYPES ‘93: Proceedings of the international workshop on Types for proofs and programs, pp. 352–365, (pdf)

Last revised on May 7, 2020 at 10:03:15. See the history of this page for a list of all contributions to it.