|logic||category theory||type theory|
|true||terminal object/(-2)-truncated object||h-level 0-type/unit type|
|false||initial object||empty type|
|proposition||(-1)-truncated object||h-proposition, mere proposition|
|cut rule||composition of classifying morphisms / pullback of display maps||substitution|
|cut elimination for implication||counit for hom-tensor adjunction||beta reduction|
|introduction rule for implication||unit for hom-tensor adjunction||eta conversion|
|disjunction||coproduct ((-1)-truncation of)||sum type (bracket type of)|
|implication||internal hom||function type|
|negation||internal hom into initial object||function type into empty type|
|universal quantification||dependent product||dependent product type|
|existential quantification||dependent sum ((-1)-truncation of)||dependent sum type (bracket type of)|
|equivalence||path space object||identity type|
|equivalence class||quotient||quotient type|
|induction||colimit||inductive type, W-type, M-type|
|higher induction||higher colimit||higher inductive type|
|completely presented set||discrete object/0-truncated object||h-level 2-type/preset/h-set|
|set||internal 0-groupoid||Bishop set/setoid|
|universe||object classifier||type of types|
|modality||closure operator, (idemponent) monad||modal type theory, monad (in computer science)|
|linear logic||(symmetric, closed) monoidal category||linear type theory/quantum computation|
|proof net||string diagram||quantum circuit|
|(absence of) contraction rule||(absence of) diagonal||no-cloning theorem|
|synthetic mathematics||domain specific embedded programming language|
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.
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 , it suffices to specify what should be done for all the (zero) ways that could have been constructed. Thus, we don’t need any hypotheses:
That is, given an element of , we can construct an element of any type . In dependent type theory, we must generalize the eliminator to allow to depend on .
There is no β-reduction rule, since there are no constructors to compose with the eliminator. However, there is an η-conversion rule, which says that for any term in a context including a term of type , we have
As with the -conversion rule for the negative presentation of the unit type, this is ill-defined as a reduction (since we cannot determine from ), but makes sense as an expansion.
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 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” and the negative one is “bottom” .