type formation


Deduction and Induction

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

equivalence classquotientquotient type

inductioncolimitinductive type, W-type, M-type

higher inductionhigher colimithigher inductive 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, (idemponent) 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 type theory a type formation rule is a natural deduction step roughly of the form

A:TypeB:TypeF(A,B,):Type \frac{ \vdash\; A \colon Type \;\;\; \vdash \;B \colon Type \;\;\; \cdots }{ \vdash \; F(A,B,\cdots) \colon Type }

which says that given types A,B,A, B, \cdots , there is a new type F(A,B,)F(A,B, \cdots).

For instance for product types it says that

A:TypeB:TypeA×B:Type. \frac{\vdash \; A \colon Type \;\;\; \vdash \; B \colon Type}{\vdash \; A \times B \colon Type} \,.

In natural deduction the type formation rule for a kind of type in the first in a quadruple of rules that come with every kind of type:

  1. type formation rule

  2. term introduction rule

  3. term elimination rule

  4. computation rule


Relation to category theory

Under the relation between type theory and category theory, the categorical semantics of type formation essentially corresponds to certain universal constructions in category theory.

Last revised on September 29, 2012 at 19:05:21. See the history of this page for a list of all contributions to it.