Generalized quantifier: Difference between revisions

Revision as of 05:08, 5 April 2022

Latest revision as of 15:03, 28 May 2025

(7 intermediate revisions by 6 users not shown)

Line 1:

{{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=141-162141–162 |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 |issuevolume=4 |issue=2 |pages=159-219159–219 |doi=10.1007/BF00350139|url-access=subscription }}</ref>

==Type theory==

Line 23 ⟶ 24:

*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>.

Line 48 ⟶ 51:

====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.

Line 55 ⟶ 58:

====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.

Line 69 ⟶ 72:

#Exactly three students ran.

#Exactly three students ran fast.

The first sentence doesn'tdoes not entail the second. The fact that the number of students that ran is exactly three doesn'tdoes 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 doesn'tdoes 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.