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{{Short description|Expression denoting a set of sets in formal semantics}}
In [[formal semantics (natural language)|formal semantics]], a '''generalized quantifier''' ('''GQ''') is an expression that denotes a [[set of sets]].  This is the standard semantics assigned to [[Quantifier (logic)|quantified]] [[noun phrase]]s. For example, the generalized quantifier ''every boy''  denotes the set of sets of which every boy is a member:
<math display="block">\{X \mid \forall x (x \text{ is a boy} \to x \in X) \}</math>

This treatment of quantifiers has been essential in achieving a [[compositionality|compositional]] [[semantics]] for sentences containing quantifiers.<ref>{{cite book |last1=Montague |first1=Richard |author-link1=Richard Montague |date=1974 |url=http://www.blackwellpublishing.com/content/BPL_Images/Content_store/Sample_chapter/9780631215417/Portner.pdf |chapter=The proper treatment of quantification in English |title=Philosophy, Language, and Artificial Intelligence |series=Studies in Cognitive Systems |volume=2
|editor1-last=Kulas |editor1-first=J. |editor2-last=Fetzer |editor2-first=J.H. |editor3-last=Rankin |editor3-first=T.L. |pages=141–162 |publisher=Springer, Dordrecht |doi=10.1007/978-94-009-2727-8_7|isbn=978-94-010-7726-2 }}</ref><ref name=Barwise>{{cite journal |last1=Barwise |first1=Jon |author-link1=Jon Barwise |last2=Cooper |first2=Robin |date=1981 |title=Generalized quantifiers and natural language |url=https://link.springer.com/article/10.1007%2FBF00350139 |journal=Linguistics and Philosophy |volume=4 |issue=2 |pages=159–219 |doi=10.1007/BF00350139|url-access=subscription }}</ref>

==Type theory==
A version of [[type theory]] is often used to make the semantics of different kinds of expressions explicit. The standard construction defines the set of types [[recursion|recursively]] as follows:
#''e'' and ''t'' are types.
#If ''a'' and ''b'' are both types, then so is <math>\langle a,b\rangle</math>
#Nothing is a type, except what can be constructed on the basis of lines 1 and 2 above.

Given this definition, we have the simple types ''e'' and ''t'', but also a [[countable]] [[infinity]] of complex types, some of which include:
<math display="block">\langle e,t\rangle;\qquad \langle t,t\rangle;\qquad \langle\langle e,t\rangle, t\rangle; \qquad\langle e,\langle e,t\rangle\rangle; \qquad \langle\langle e,t\rangle,\langle \langle e, t\rangle, t\rangle\rangle;\qquad \ldots</math>

*Expressions of type ''e'' denote elements of the [[universe of discourse]], the set of entities the discourse is about. This set is usually written as <math>D_e</math>.  Examples of type ''e'' expressions include ''John'' and ''he''. 
*Expressions of type ''t'' denote a [[truth value]], usually rendered as the set <math>\{0,1\}</math>, where 0 stands for "false" and 1 stands for "true".   Examples of expressions that are sometimes said to be of type ''t'' are ''sentences'' or ''propositions''.
*Expressions of type <math>\langle e,t\rangle</math> denote [[Function (mathematics)|functions]] from the set of entities to the set of truth values. This set of functions is rendered as <math>D_t^{D_e}</math>. Such functions are [[Indicator function|characteristic functions]] of [[Set (mathematics)|sets]].   They map every individual that is an element of the set to "true", and everything else to "false."  It is common to say that they denote ''sets'' rather than characteristic functions, although, strictly speaking, the latter is more accurate. Examples of expressions of this type are [[predicate (grammar)|predicates]], [[noun]]s and some kinds of [[adjective]]s. 
*In general, expressions of complex types <math>\langle a,b\rangle</math> denote functions from the set of entities of type <math>a</math> to the set of entities of type <math>b</math>, a construct we can write as follows: <math>D_b^{D_a}</math>.

We can now assign types to the words in our sentence above (Every boy sleeps) as follows.
*Type(boy) = <math>\langle e,t\rangle</math>
*Type(sleeps) = <math>\langle e,t\rangle</math>
*Type(every) = <math>\langle\langle e,t\rangle,\langle \langle e, t\rangle, t\rangle\rangle</math>
*Type(every boy) = <math>\langle\langle e,t\rangle,t\rangle</math>
and so we can see that the generalized quantifier in our example is of type <math>\langle\langle e,t\rangle,t\rangle</math>

Thus, every denotes a function from a ''set'' to a function from a set to a truth value.  Put differently, it denotes a function from a set to a set of sets.  It is that function which for any two sets ''A,B'', ''every''(''A'')(''B'')= 1 if and only if <math>A\subseteq B</math>.

==Typed lambda calculus==
A useful way to write complex functions is the [[lambda calculus]]. For example, one can write the meaning of ''sleeps'' as the following lambda expression, which is a function from an individual ''x'' to the proposition that ''x sleeps''.
<math display="block">\lambda x. \mathrm{sleep}'(x)</math>
Such lambda terms are functions whose domain is what precedes the period, and whose range are the type of thing that follows the period. If ''x'' is a variable that ranges over elements of <math>D_e</math>, then the following lambda term denotes the [[identity function]] on individuals:
<math display="block">\lambda x.x</math>

We can now write the meaning of ''every'' with the following lambda term, where ''X,Y'' are variables of type <math>\langle e,t\rangle</math>:
<math display="block">\lambda X.\lambda Y. X\subseteq Y</math>

If we abbreviate the meaning of ''boy'' and ''sleeps'' as "''B''" and "''S''", respectively, we have that the sentence ''every boy sleeps'' now means the following:
<math display="block">(\lambda X.\lambda Y. X\subseteq Y)(B)(S)</math>
By [[Lambda calculus#β-reduction|β-reduction]],
<math display="block">(\lambda Y. B \subseteq Y)(S)</math>
and
<math display="block">B\subseteq S</math>

The expression ''every'' is a [[determiner (linguistics)|determiner]]. Combined with a [[noun]], it yields a ''generalized quantifier'' of type <math>\langle\langle e,t\rangle,t\rangle</math>.

==Properties==
===Monotonicity===
====Monotone increasing GQs====
A ''generalized quantifier'' GQ is said to be [[monotone increasing]] (also called [[upward entailing]]) if, for every pair of sets ''X'' and ''Y'', the following holds:
:if <math>X\subseteq Y</math>, then GQ(''X'') [[Entailment|entail]]s GQ(''Y'').
The GQ ''every boy'' is monotone increasing. For example, the set of things that ''run fast'' is a subset of the set of things that ''run''. Therefore, the first sentence below [[Entailment|entail]]s the second:
#Every boy runs fast.
#Every boy runs.

====Monotone decreasing GQs====
A GQ is said to be [[monotone decreasing]] (also called [[downward entailing]]) if, for every pair of sets ''X'' and ''Y'', the following holds:
:If <math>X\subseteq Y</math>, then GQ(''Y'') entails GQ(''X'').
An example of a monotone decreasing GQ is ''no boy''. For this GQ we have that the first sentence below entails the second.
#No boy runs.
#No boy runs fast.
The lambda term for the [[determiner (linguistics)|determiner]] ''no'' is the following. It says that the two sets have an empty [[Intersection (set theory)|intersection]].
<math display="block">\lambda X.\lambda Y. X\cap Y= \emptyset</math>
Monotone decreasing GQs are among the expressions that can license a [[negative polarity item]], such as ''any''. Monotone increasing GQs do not license negative polarity items.
#Good: No boy has '''any''' money.
#Bad: *Every boy has '''any''' money.

====Non-monotone GQs====
A GQ is said to be ''non-monotone'' if it is neither monotone increasing nor monotone decreasing. An example of such a GQ is ''exactly three boys''. Neither of the following sentences entails the other.
#Exactly three students ran.
#Exactly three students ran fast.
The first sentence does not entail the second. The fact that the number of students that ran is exactly three does not entail that each of these students ''ran fast'', so the number of students that did that can be smaller than 3. Conversely, the second sentence does not entail the first.  The sentence ''exactly three students ran fast'' can be true, even though the number of students who merely ran (i.e. not so fast) is greater than 3.

The lambda term for the (complex) [[determiner (linguistics)|determiner]] ''exactly three'' is the following. It says that the [[cardinality]] of the [[Intersection (set theory)|intersection]] between the two sets equals 3.
<math display="block">\lambda X.\lambda Y. |X\cap Y|=3</math>

===Conservativity===
{{Further|Conservativity}}
A determiner D is said to be ''conservative'' if the following equivalence holds:
<math display="block">D(A)(B) \leftrightarrow D(A)(A\cap B)</math>
For example, the following two sentences are equivalent.
#Every boy sleeps.
#Every boy is a boy who sleeps.

It has been proposed that ''all'' determiners{{emdash}}in every natural language{{emdash}}are conservative.<ref name=Barwise />  The expression ''only'' is not conservative. The following two sentences are not equivalent.  But it is, in fact, not common to analyze ''only'' as a [[determiner (linguistics)|determiner]]. Rather, it is standardly treated as a [[focus-sensitive]] [[adverb]].
#Only boys sleep.
#Only boys are boys who sleep.

==See also==
*[[Scope (formal semantics)]]
*[[Lindström quantifier]]
*[[Branching quantifier]]

==References==
<references />

==Further reading==
* {{cite book|author1=[[Stanley Peters]]|author2=Dag Westerståhl|title=Quantifiers in language and logic|year=2006|publisher=Clarendon Press|isbn=978-0-19-929125-0}}
* {{cite book|author=Antonio Badia|title=Quantifiers in Action: Generalized Quantification in Query, Logical and Natural Languages|year=2009|publisher=Springer|isbn=978-0-387-09563-9}}
* {{Cite book
| vauthors=Wągiel M
| title =   Subatomic quantification
| place = Berlin
| publisher = Language Science Press
| date = 2021
| format = pdf
| url = http://langsci-press.org/catalog/book/317
| doi =10.5281/zenodo.5106382
| doi-access=free
| isbn = 978-3-98554-011-2
}}

==External links==
*Dag Westerståhl, 2011. '[http://plato.stanford.edu/entries/generalized-quantifiers/ Generalized Quantifiers]'.  [[Stanford Encyclopedia of Philosophy]].

{{Formal semantics}}
[[Category:Semantics]]
[[Category:Formal semantics (natural language)]]
[[Category:Quantifier (logic)]]