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\section*{existential quantifier}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{type_theory}{}\paragraph*{{Type theory}}\label{type_theory}

[[!include type theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_logic}{In logic}\dotfill \pageref*{in_logic} \linebreak
\noindent\hyperlink{in_type_theory}{In type theory}\dotfill \pageref*{in_type_theory} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{categorical_semantics}{Categorical semantics}\dotfill \pageref*{categorical_semantics} \linebreak
\noindent\hyperlink{frobenius_reciprocity}{Frobenius reciprocity}\dotfill \pageref*{frobenius_reciprocity} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[logic]], the \emph{existential quantifier} (or \emph{particular quantifier}) ``$\exists$'' is a [[quantifier]] used to express, given a [[predicate]] $\phi$ with a [[free variable]] $x$ of [[type]] $T$, the [[proposition]]

\begin{displaymath}
\exists\, x\colon T, \phi x
  \,,
\end{displaymath}
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 [[proof|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.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_logic}{}\subsubsection*{{In logic}}\label{in_logic}

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 \textbf{existential quantification} of $P$ is any proposition $\exists\, x\colon T, P$ in $\Gamma$ such that, given any proposition $Q$ in $\Gamma$, we have

\begin{itemize}%
\item $\exists\, x\colon T, P \vdash_{\Gamma} Q$ if and only if $P \vdash_{\Gamma, x\colon T} Q[\hat{x}]$.

\end{itemize}
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:

\begin{itemize}%
\item $\exists\, x\colon T, P(x) \vdash_{\Gamma} Q$ if and only if $P(x) \vdash_{\Gamma, x\colon T} Q$.

\end{itemize}
\hypertarget{in_type_theory}{}\subsubsection*{{In type theory}}\label{in_type_theory}

In [[type theory]] under the identification of [[propositions as types]], the existential quantifier is given by the [[bracket type]] of the [[dependent sum type]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{categorical_semantics}{}\subsubsection*{{Categorical semantics}}\label{categorical_semantics}

The [[categorical semantics]] of existential quantification is given by the [[n-truncated object of an (infinity,1)-category|(-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)

\begin{itemize}%
\item a [[type]] $X$ is given by an [[object]] $X \in \mathcal{E}$,


\item the [[predicate]] $\phi$ is a [[(-1)-truncated]] object of the [[over-topos]] $\mathcal{E}/X$;


\item a [[truth value]] is a [[(-1)-truncated]] object of $\mathcal{E}$ itself.



\end{itemize}
Existential quantification is essentially the restriction of the extra [[left adjoint]] of a canonical [[étale geometric morphism]] $X_\ast$,

\begin{displaymath}
(X_!\dashv X^*\dashv X_*):\mathcal{E}/X
    \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}}
    \mathcal{E} 
  \,,
\end{displaymath}
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 [[truncated|truncation]] [[adjunctions]]

\begin{displaymath}
\tau_{\leq -1} \mathcal{E}
  \stackrel{\overset{\tau_{\leq -1}}{\leftarrow}}{\underset{}{\hookrightarrow}}
  \mathcal{E}
\end{displaymath}
to give existential quantification over the domain (or context) $X$:

\begin{displaymath}
\itexarray{
    \mathcal{E}/X
    & 
    \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}}
    &
    \mathcal{E}
    \\
    {}^\mathllap{\tau_{\leq_{-1}}}\downarrow \uparrow
    &&
    {}^\mathllap{\tau_{\leq_{-1}}}\downarrow \uparrow
    \\
    \tau_{\leq_{-1}} 
    \mathcal{E}/X
    &
      \stackrel{\overset{\exists}{\to}}{\stackrel{\overset{}{\leftarrow}}{\underset{\forall}{\to}}}
    &
    \tau_{\leq_{-1}} \mathcal{E}
  }
  \,.
\end{displaymath}
In other words, we obtain the existential quantifier by applying the [[dependent sum]], then $(-1)$-[[truncated|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 \emph{[[hyperdoctrine]]}.

\hypertarget{frobenius_reciprocity}{}\subsubsection*{{Frobenius reciprocity}}\label{frobenius_reciprocity}

The extential quantifier and [[context extension]] is related via [[Frobenius reciprocity]]: if $\psi$ does not have $x$ as a [[free variable]] then

\begin{displaymath}
\exists_x (\phi \wedge \psi)
  \Leftrightarrow
  (\exists_x \phi) \wedge \psi
  \,.
\end{displaymath}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

Let $\mathcal{E} =$ [[Set]], let $X = \mathbb{N}$ be the set of [[natural number]]s 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]]

\begin{displaymath}
(\phi \in Set/{\mathbb{N}})
  \mapsto 
  (\sum_{x\colon X} \phi(x) \in Set)
\end{displaymath}
is simply the set $2 \mathbb{N} \in Set$ of even natural numbers. Since this is [[inhabited]], its [[truncated|(-1)-truncation]] is therefore the singleton set, hence the [[truth value]] \emph{[[true]]}. Indeed, there exists an even natural number!

Notice that before the $(-1)$-truncation the operation remembers not just \emph{that} there is an even number, but it remembers ``all proofs that there is one'', namely every example of an even number.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[possibility]]


\item [[universal quantifier]]


\item [[elimination of quantifiers]]


\item [[generalized quantifier]]



\end{itemize}
[[!include logic symbols -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

The observation that [[substitution]] forms an [[adjoint pair]]/[[adjoint triple]] with the existantial/universal quantifiers is due to

\begin{itemize}%
\item [[William Lawvere]], \emph{[[Adjointness in Foundations]]}, (\href{http://www.emis.de/journals/TAC/reprints/articles/16/tr16abs.html}{TAC}), Dialectica 23 (1969), 281-296


\item [[William Lawvere]] \emph{Quantifiers and sheaves}, Actes, Congr\`e{}s intern, math., 1970. Tome 1, p. 329 \`a{} 334 (\href{http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0329.0334.ocr.pdf}{pdf})



\end{itemize}
and further developed to the notion of [[hyperdoctrines]] in

\begin{itemize}%
\item [[William Lawvere]], \emph{Equality in hyperdoctrines and comprehension schema as an adjoint functor}, Proceedings of the AMS Symposium on Pure Mathematics XVII (1970), 1-14.

\end{itemize}
[[!redirects existential quantifier]] [[!redirects existential quantifiers]] [[!redirects existential quantification]] [[!redirects existential quantifications]] [[!redirects particular quantifier]] [[!redirects particular quantifiers]] [[!redirects particular quantification]] [[!redirects particular quantifications]]

[[!redirects existence]]



\end{document}