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\begin{document}

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\section*{generalized quantifier}

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

\hypertarget{model_theory}{}\paragraph*{{Model theory}}\label{model_theory}

[[!include model theory - contents]]

\hypertarget{linguistics}{}\paragraph*{{Linguistics}}\label{linguistics}

[[!include linguistics - 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{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\textbf{Generalized quantifiers} were introduced into [[model theory]] by the Polish logician A. Mostowski in order to supplement the usual [[existential quantifier|existential]] and [[universal quantifiers]] of [[first-order logic|first-order predicate logic]] with the aim to circumvent shortcomings concerning expressivity and categoricity.

Their use for the [[semantics|semantic]] analysis of natural language determiners in the footsteps of [[Richard Montague]] revolutionized [[linguistics]] in the early 1970s.

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

\begin{defn}
\label{GeneralizedQuantifier}\hypertarget{GeneralizedQuantifier}{}
A \emph{generalized quantifier} or an \emph{interpretation of a quantifier symbol} $Q$ is a mapping $\mu_Q$ from triples of cardinal numbers $\langle \mathfrak{m},\mathfrak{n},\mathfrak{p}\rangle$ such that $\mathfrak{m}+\mathfrak{n}=\mathfrak{p}$ to $\Omega=\{0,1\}$. The satisfaction relation for $Q$ is defined for structures $\mathfrak{A}$ with carrier $A$ and valuation $x$ by

\begin{displaymath}
\begin{aligned}
\mathfrak{A}\models_x(Qv_n)\phi\quad&{iff}\quad \mu_Q(\mathfrak{m},\mathfrak{n},\mathfrak{p})=1\;,\; with
\\
\mathfrak{m}&=card(\{a\in A:\mathfrak{A}\models_{x(n/a)}\phi\})
\\
\mathfrak{n}&=card(\{a\in A:\mathfrak{A}\models_{x(n/a)}\not\phi\})
\\
\mathfrak{p}&=card(A)\;.
\end{aligned}
\end{displaymath}
\end{defn}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The [[universal quantifier]] $\forall$ has $\mu_\forall(\mathfrak{m},\mathfrak{n},\mathfrak{p})=1$ iff $\mathfrak{n}=0$.


\item The quantifier $Q_\alpha$ ``there exist at least $\aleph_\alpha$'' is given by $\mu_{Q_\alpha}(\mathfrak{m},\mathfrak{n},\mathfrak{p})=1$ iff $\mathfrak{m}\geq\aleph_{\alpha}\;$.


\item The quantifier $\mathsf{W}$ ``most'' is given by $\mu_{\mathsf{W}}(\mathfrak{m},\mathfrak{n},\mathfrak{p})=1$ iff $\mathfrak{m}>\mathfrak{n}\;$. Note that this uses the second variable as well, whereas $\forall$ or $Q_\alpha$ depend only on $card(\{a\in A:\mathfrak{A}\models_{x(n/a)}\phi\})$.



\end{itemize}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[model theory]]


\item [[second-order logic]]


\item [[cardinal number]]


\item [[Lawvere distribution]]


\item [[Richard Montague]]



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

Generalized quantifiers were introduced in

\begin{itemize}%
\item A. Mostowski, \emph{On a Generalization of Quantifiers} , Fund. Math. \textbf{44} (1957). (\href{http://matwbn.icm.edu.pl/ksiazki/fm/fm44/fm4412.pdf}{pdf})

\end{itemize}
An early textbook account is in ch.13 of

\begin{itemize}%
\item [[John Bell|J. L. Bell]], A. B. Slomson, \emph{Models and Ultraproducts: An Introduction} , North-Holland Amsterdam 1969. (Dover reprint)

\end{itemize}
Several chapters treat their model-theoretic role in

\begin{itemize}%
\item J. Barwise, [[Solomon Feferman|S. Feferman]] (eds.), \emph{Model-theoretic Logics} , Springer Heidelberg 1985 (freely available online: \href{http://projecteuclid.org/euclid.pl/1235417263#toc}{toc}) .

\end{itemize}
In the context of [[Martin-Löf type theory]] they are discussed by

\begin{itemize}%
\item G. Sundholm, \emph{Constructive Generalized Quantifiers} , Synthese \textbf{79} no.1 (1989) pp.1-12.

\end{itemize}
For the use of generalized quantifiers in natural language semantics see

\begin{itemize}%
\item J. Barwise, R. Cooper, \emph{Generalized Quantifiers and Natural Language} , Linguistics and Philosophy \textbf{4} no.2 (1981) pp.159-219.


\item E. Keenan, J. Stavi, \emph{A semantic characterization of natural language determiners} , Linguistics \& Philosophy \textbf{9} (1986) pp.253–326.


\item D. Westerståhl, \emph{Generalized quantifiers: linguistics meets model theory} , in Aloni, Dekker (eds.), \emph{The Cambridge Handbook of Semantics} , Cambridge UP 2014. (\href{https://www.philosophy.su.se/polopoly_fs/1.165898.1391710202!/menu/standard/file/2013%20CambridgeHandbookGQ.pdf}{draft})



\end{itemize}
Some problems with the standard approach in linguistics are discussed in

\begin{itemize}%
\item H. Ben-Yami, \emph{Generalized Quantifiers, and Beyond} , Logique Et Analyse no.208 (2009) pp.309-326.


\item M. Hackl, \emph{On the Grammar and Processing of Proportional Quantifiers: Most Versus More Than Half} , Natural Language Semantics \textbf{17} no.1 (2009) pp.63-98.



\end{itemize}
A dependent type theoretic analysis of natural language generalized quantifiers is in

\begin{itemize}%
\item T. Fernando, \emph{Conservative generalized quantifiers and presupposition} , Semantics and Linguistic Theory \textbf{XI} (2001) pp.172–91. (\href{http://www.scss.tcd.ie/Tim.Fernando/cgq.pdf}{draft})

\end{itemize}
[[!redirects Generalized quantifier]] [[!redirects Generalized Quantifier]] [[!redirects generalized quantifiers]] [[!redirects Generalized quantifiers]] [[!redirects Generalized Quantifiers]] [[!redirects generalised quantifier]] [[!redirects Generalised quantifier]] [[!redirects Generalised quantifiers]] [[!redirects Generalised Quantifier]]



\end{document}