nLab empty type



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
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/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 type theory the empty type is the type with no term.

In a model by categorical semantics, this is an initial object. In set theory, it is an empty set. In logic, especially the propositions as types interpretation of type theory, it represents falsehood, and constructing a term of an empty type represents a contradiction; thus functions into the empty type are regarded as the negation of a proposition.


Like all type constructors in type theory, to characterize the empty type we must specify how to build it, how to construct elements of it, how to use such elements, and the computation rules.

The way to build the empty type is trivial: it exists.

:Type\frac{ }{\emptyset\colon Type}

As a positive type

The empty type is most naturally presented as a positive type, so that the constructor rules are primary. However, since the empty type is supposed to contain no elements, there are no constructor rules.

The eliminator rules are derived from the constructor rules in the usual way: to use a term e:e\colon \emptyset, it suffices to specify what should be done for all the (zero) ways that ee could have been constructed. Thus, we don’t need any hypotheses:

e:abort C(e):C\frac{ }{e\colon \emptyset \vdash abort_C(e)\colon C}

That is, given an element of \emptyset, we can construct an element of any type CC. In dependent type theory, we must generalize the eliminator to allow CC to depend on \emptyset.

There is no beta-reduction rule, since there are no constructors to compose with the eliminator. However, there is an eta-conversion rule, which says that for any term e:c:Ce\colon \emptyset\vdash c\colon C in a context including a term of type \emptyset, we have

abort C(e) ηc. abort_C(e) \;\leftrightarrow_\eta\; c.

As with the η\eta-conversion rule for the negative presentation of the unit type, this is ill-defined as a reduction (since we cannot determine cc from abort C(e)abort_C(e)), but makes sense as an expansion.

The positive presentation of the empty type can be regarded as a particular sort of inductive type. In Coq syntax:

Inductive empty : Type :=

Coq implements the beta reduction rule, but not the eta (although eta equivalence is provable for the inductively defined identity types, using the dependent eliminator mentioned above).

As a negative type

As for binary sum types, it is possible to present the empty type as a negative type as well, but only if we allow sequents with multiple conclusions. This is common in sequent calculus presentations of classical logic, but not as common in type theory and almost unheard of in dependent type theory.

The two definitions are provably equivalent, but only using the contraction rule and the weakening rule. Thus, in linear logic they become distinct; the positive empty type is “zero” 0\mathbf{0} and the negative one is “bottom” \bot.

Last revised on June 16, 2022 at 07:45:53. See the history of this page for a list of all contributions to it.