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) “$\exists$” is a quantifier used to express, given a predicate $\phi$ with a free variable $x$ of type $T$, the proposition
which is intended to be true if and only if $\phi a$ is true for at least one object $a$ of type $T$.
Note that it is quite possible that $\exists\, x\colon T, \phi x$ may be provable (in a given context) yet $\phi a$ cannot be proved for any term $a$ of type $T$ 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 $\Gamma$ be an arbitrary context and $T$ a type in $\Gamma$ so that $\Delta \coloneqq \Gamma, x\colon T$ is $\Gamma$ extended by a free variable $x$ of type $T$. We assume that we have a weakening rule that allows us to interpret any proposition $Q$ in $\Gamma$ as a proposition $Q[\hat{x}]$ in $\Delta$. Fix a proposition $P$ in $\Delta$, which we think of as a predicate in $\Gamma$ with the free variable $x$. Then the existential quantification of $P$ is any proposition $\exists\, x\colon T, P$ in $\Gamma$ such that, given any proposition $Q$ in $\Gamma$, we have
It is then a theorem that the existential quantification of $P$, 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 $P$ explicit by thinking of $P$ as a propositional function and writing $P(x)$ instead; to go along with this one usually conflates $Q$ and $Q[\hat{x}]$. Then the rule appears as follows:
In type theory under the identification of propositions as types, the existential quantifier is given by the bracket type of the dependent sum type.
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 $Q$ to $Q[\hat{x}]$ corresponds to base change/pullback along the product projection $\sigma(T) \times A \to A$, where $\sigma(T)$ is the interpretation of the type $T$, and $A$ is the interpretation of $\Gamma$. In other words, if a statement $Q$ read in context $\Gamma$ is interpreted as a subobject of $A$, then the statement $Q$ read in context $\Delta = \Gamma, x \colon T$ is interpreted by pulling back along the projection, obtaining a subobject of $\sigma(T) \times A$.
(Often we have a class of display maps and require $f$ to be one of these.) Alternatively, any pullback functor $f^\ast\colon Set/A \to Set/B$ can be construed as pulling back along an object $X = (f \colon B \to A)$, i.e., along the unique map $!\colon X \to 1$ corresponding to an object $X$ in the slice $Set/A$, since we have the identification $Set/B \simeq (Set/A)/X$.
Therefore in terms of the internal logic of a suitable category $\mathcal{E}$ (with sufficient pullbacks)
the predicate $\phi$ is a (-1)-truncated object of the over-topos $\mathcal{E}/X$;
a truth value is a (-1)-truncated object of $\mathcal{E}$ itself.
Existential quantification is essentially the restriction of the extra left adjoint of a canonical étale geometric morphism $X_\ast$,
where $X^\ast$ is the functor that takes an object $A$ to the product projection $\pi \colon X \times A \to X$, where $X_! = \Sigma_X$ is the dependent sum (i.e., forgetful functor taking $f \colon A \to X$ to $A$) that is left adjoint to $X^\ast$, and where $X_\ast = \Pi_X$ is the dependent product operation that is right adjoint to $X^\ast$.
The dependent sum operation restricts to propositions by pre- and postcomposition with the truncation adjunctions
to give existential quantification over the domain (or context) $X$:
In other words, we obtain the existential quantifier by applying the dependent sum, then $(-1)$-truncating the result. This is equivalent to constructing the image as a subobject of the codomain.
Dually, the direct image functor $\forall$ (dependent product) expresses the universal quantifier. (In this case, it is somewhat simpler, since the dependent product automatically preserves $(-1)$-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 $\psi$ does not have $x$ as a free variable then
Let $\mathcal{E} =$ Set, let $X = \mathbb{N}$ be the set of natural numbers and let $\phi \coloneqq 2\mathbb{N} \hookrightarrow \mathbb{N}$ be the subset of even natural numbers. This expresses the proposition $\phi(x) \coloneqq IsEven(x)$.
Then the dependent sum
is simply the set $2 \mathbb{N} \in Set$ 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 $(-1)$-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
Bill Lawvere, Adjointness in Foundations, (TAC), Dialectica 23 (1969), 281-296
Bill 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