natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
In logic, the existential quantifier (or particular quantifier) “” is a quantifier used to express, given a predicate with a free variable of type , the proposition
which is intended to be true if and only if is true for at least one object of type .
Note that it is quite possible that may be provable (in a given context) yet cannot be proved for any term of type that can actually be constructed in that context. Therefore, we cannot define the quantifier by taking the idea literally and applying it to terms.
We work in a logic in which we are concerned with which propositions entail which propositions (in a given context); in particular, two propositions which entail each other are considered equivalent.
Let be an arbitrary context and a type in so that is extended by a free variable of type . We assume that we have a weakening rule that allows us to interpret any proposition in as a proposition in . Fix a proposition in , which we think of as a predicate in with the free variable . Then the existential quantification of is any proposition in such that, given any proposition in , we have
It is then a theorem that the existential quantification of , if it exists, is unique. The existence is typically asserted as an axiom in a particular logic, but this may be also be deduced from other principles (as in the topos-theoretic discussion below).
Often one makes the appearance of the free variable in explicit by thinking of as a propositional function and writing instead; to go along with this one usually conflates and . Then the rule appears as follows:
In natural deduction the inference rules for the existential quantifier are given as
In dependent type theory under the identification of propositions as some types, the existential quantifier is given by the bracket type of the dependent sum type. Existential quantifier types in general could be regarded as a particular sort of higher inductive type. In Coq syntax:
Inductive existquant (T:Type) (P:T->Type) : Type :=
| exist : forall (x:T), P x -> existquant T P
| contr0 : forall (p q : existquant T P) p == q
If the dependent type theory has a type of propositions , such as the one derived from a type universe - , then the existential quantification of the family of types indexed by is defined as the dependent function type
By weak function extensionality, the disjunction of two types is a proposition.
The two definitions above are equivalent.
The propositional truncation of a type is equivalent to the following dependent product type
Substituting the dependent sum type for , we have
Given any type , there is an equivalence
and if , then . In addition, for all type families , and , if there is a family of equivalences , then there is an equivalence . All this taken together means that there are equivalences
There is also another definition of the existential quantifier of a family of mere propositions indexed by , as the dependent pushout type of the -indexed family of functions from the dependent product type to the type .
This is the same as the other two definitions because every mere proposition is a subtype of the unit type, and the existential quantification of is the -indexed union of all the as subtypes of the unit types, and the union of all the as subtypes of the unit type is defined to be, the dependent pushout type of the family of dependent product projection functions from the dependent product type to each for .
The categorical semantics of existential quantification is given by the (-1)-truncation of the dependent sum-construction along the projection morphism that projects out the free variable over which the existental quantifier quantifies.
Notice that the categorical semantics of the context extension from to corresponds to base change/pullback along the product projection , where is the interpretation of the type , and is the interpretation of . In other words, if a statement read in context is interpreted as a subobject of , then the statement read in context is interpreted by pulling back along the projection, obtaining a subobject of .
(Often we have a class of display maps and require to be one of these.) Alternatively, any pullback functor can be construed as pulling back along an object , i.e., along the unique map corresponding to an object in the slice , since we have the identification .
Therefore in terms of the internal logic of a suitable category (with sufficient pullbacks)
the predicate is a (-1)-truncated object of the over-topos ;
a truth value is a (-1)-truncated object of itself.
Existential quantification is essentially the restriction of the extra left adjoint of a canonical étale geometric morphism ,
where is the functor that takes an object to the product projection , where is the dependent sum (i.e., forgetful functor taking to ) that is left adjoint to , and where is the dependent product operation that is right adjoint to .
The dependent sum operation restricts to propositions by pre- and postcomposition with the truncation adjunctions
to give existential quantification over the domain (or context) :
In other words, we obtain the existential quantifier by applying the dependent sum, then -truncating the result. This is equivalent to constructing the image as a subobject of the codomain.
Dually, the direct image functor (dependent product) expresses the universal quantifier. (In this case, it is somewhat simpler, since the dependent product automatically preserves -truncated objects; no additional truncation step is necessary.)
The same makes verbatim sense also in the (∞,1)-logic of any (∞,1)-topos.
This interpretation of existential quantification as the left adjoint to context extension is also used in the notion of hyperdoctrine.
The extential quantifier and context extension is related via Frobenius reciprocity: if does not have as a free variable then
Let Set, let be the set of natural numbers and let be the subset of even natural numbers. This expresses the proposition .
Then the dependent sum
is simply the set of even natural numbers. Since this is inhabited, its (-1)-truncation is therefore the singleton set, hence the truth value true. Indeed, there exists an even natural number!
Notice that before the -truncation the operation remembers not just that there is an even number, but it remembers “all proofs that there is one”, namely every example of an even number.
The observation that substitution forms an adjoint pair/adjoint triple with the existantial/universal quantifiers is due to
William Lawvere, Adjointness in Foundations, (TAC), Dialectica 23 (1969), 281-296
William Lawvere, Quantifiers and sheaves, Actes, Congrès intern, math., 1970. Tome 1, p. 329 à 334 (pdf)
and further developed to the notion of hyperdoctrines in
The definition of the existential quantifier in dependent type theory as the propositional truncation of the dependent sum type is found in:
And the existential quantifier defined from the type of propositions and dependent product types can be found in:
Last revised on September 25, 2024 at 07:27:17. See the history of this page for a list of all contributions to it.