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 formal logic one speaks of a “fragment” of a theory when referring to just part of it. Specifically, when rules of type formation or some of the inference rules are omitted, one says that one retains a “fragment” of the original theory.

This terminology is jargon and has not been formalized. What exactly counts as a fragment of a theory and what not differs a bit from author to author.


First-order logic in type theory

In type theory first order logic of propositions is contained via “propositions as types” essentially as the sub-theory dealing with types of type Prop, hence with bracket types or what in homotopy type theory are called (-1)-types.

Some authors speak of the “first-order fragment” of type theory when referring to this restriction (e.g. Bezem-Hendriks-Nivelle 02, Hendriks 13).

Multiplicative conjunction in linear logic

By default linear logic has a wealth of conjunctions. Discarding or ignoring all of them except the multiplicative conjunction yield the fragment of multiplicative linear logic.


  • Marc Bezem, Dimitri Hendriks, Hans de Nivelle, Automated Proof Construction in Type Theory using Resolution, 2002 (pdf)

  • Dimitri Hendriks, Metamathematics in Coq, 2013 (pdf)

Last revised on November 2, 2014 at 20:08:57. See the history of this page for a list of all contributions to it.