Generalized quantifier

Informal semantics,ageneralized quantifier(GQ) is an expression that denotes aset of sets.This is the standard semantics assigned toquantified noun phrases.For example, the generalized quantifierevery boydenotes the set of sets of which every boy is a member: $\{X\mid \forall x(x{\text{ is a boy}}\to x\in X)\}$

This treatment of quantifiers has been essential in achieving acompositional semanticsfor sentences containing quantifiers.^[1]^[2]

Type theory

A version oftype theoryis often used to make the semantics of different kinds of expressions explicit. The standard construction defines the set of typesrecursivelyas follows:

eandtare types.
Ifaandbare both types, then so is $\langle a,b\rangle$
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 typeseandt,but also acountable infinityof complex types, some of which include: $\langle e,t\rangle;\qquad \langle t,t\rangle;\qquad \langle \langle e,t\rangle,t\rangle;\qquad \langle e,\langle e,t\rangle \rangle;\qquad \langle \langle e,t\rangle,\langle \langle e,t\rangle,t\rangle \rangle;\qquad \ldots$

Expressions of typeedenote elements of theuniverse of discourse,the set of entities the discourse is about. This set is usually written as $D_{e}$ .Examples of typeeexpressions includeJohnandhe.
Expressions of typetdenote atruth value,usually rendered as the set $\{0,1\}$ ,where 0 stands for "false" and 1 stands for "true". Examples of expressions that are sometimes said to be of typetaresentencesorpropositions.
Expressions of type $\langle e,t\rangle$ denotefunctionsfrom the set of entities to the set of truth values. This set of functions is rendered as $D_{t}^{D_{e}}$ .Such functions arecharacteristic functionsofsets.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 denotesetsrather than characteristic functions, although, strictly speaking, the latter is more accurate. Examples of expressions of this type arepredicates,nounsand some kinds ofadjectives.
In general, expressions of complex types $\langle a,b\rangle$ denote functions from the set of entities of type $a$ to the set of entities of type $b$ ,a construct we can write as follows: $D_{b}^{D_{a}}$ .

We can now assign types to the words in our sentence above (Every boy sleeps) as follows.

Type(boy) = $\langle e,t\rangle$
Type(sleeps) = $\langle e,t\rangle$
Type(every) = $\langle \langle e,t\rangle,\langle \langle e,t\rangle,t\rangle \rangle$
Type(every boy) = $\langle \langle e,t\rangle,t\rangle$

and so we can see that the generalized quantifier in our example is of type $\langle \langle e,t\rangle,t\rangle$

Thus, every denotes a function from asetto 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 setsA,B,every(A)(B)= 1 if and only if $A\subseteq B$ .

Typed lambda calculus

A useful way to write complex functions is thelambda calculus.For example, one can write the meaning ofsleepsas the following lambda expression, which is a function from an individualxto the proposition thatx sleeps. $\lambda x.\mathrm {sleep} '(x)$ Such lambda terms are functions whose domain is what precedes the period, and whose range are the type of thing that follows the period. Ifxis a variable that ranges over elements of $D_{e}$ ,then the following lambda term denotes theidentity functionon individuals: $\lambda x.x$

We can now write the meaning ofeverywith the following lambda term, whereX,Yare variables of type $\langle e,t\rangle$ : $\lambda X.\lambda Y.X\subseteq Y$

If we abbreviate the meaning ofboyandsleepsas "B"and"S",respectively, we have that the sentenceevery boy sleepsnow means the following: $(\lambda X.\lambda Y.X\subseteq Y)(B)(S)$ Byβ-reduction, $(\lambda Y.B\subseteq Y)(S)$ and $B\subseteq S$

The expressioneveryis adeterminer.Combined with anoun,it yields ageneralized quantifierof type $\langle \langle e,t\rangle,t\rangle$ .

Properties

Monotonicity

Monotone increasing GQs

Ageneralized quantifierGQ is said to bemonotone increasing(also calledupward entailing) if, for every pair of setsXandY,the following holds:

if

X\subseteq Y

,then GQ(X)entailsGQ(Y).

The GQevery boyis monotone increasing. For example, the set of things thatrun fastis a subset of the set of things thatrun.Therefore, the first sentence belowentailsthe second:

Every boy runs fast.
Every boy runs.

Monotone decreasing GQs

A GQ is said to bemonotone decreasing(also calleddownward entailing) if, for every pair of setsXandY,the following holds:

If

X\subseteq Y

,then GQ(Y) entails GQ(X).

An example of a monotone decreasing GQ isno 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 thedeterminernois the following. It says that the two sets have an emptyintersection. $\lambda X.\lambda Y.X\cap Y=\emptyset$ Monotone decreasing GQs are among the expressions that can license anegative polarity item,such asany.Monotone increasing GQs do not license negative polarity items.

Good: No boy hasanymoney.
Bad: *Every boy hasanymoney.

Non-monotone GQs

A GQ is said to benon-monotoneif it is neither monotone increasing nor monotone decreasing. An example of such a GQ isexactly 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 studentsran fast,so the number of students that did that can be smaller than 3. Conversely, the second sentence does not entail the first. The sentenceexactly three students ran fastcan 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)determinerexactly threeis the following. It says that thecardinalityof theintersectionbetween the two sets equals 3. $\lambda X.\lambda Y.|X\cap Y|=3$

Conservativity

A determiner D is said to beconservativeif the following equivalence holds: $D(A)(B)\leftrightarrow D(A)(A\cap B)$ For example, the following two sentences are equivalent.

Every boy sleeps.
Every boy is a boy who sleeps.

It has been proposed thatalldeterminers—in every natural language—are conservative.^[2]The expressiononlyis not conservative. The following two sentences are not equivalent. But it is, in fact, not common to analyzeonlyas adeterminer.Rather, it is standardly treated as afocus-sensitive adverb.

Only boys sleep.
Only boys are boys who sleep.

References

^Montague, Richard(1974). "The proper treatment of quantification in English". In Kulas, J.; Fetzer, J.H.; Rankin, T.L. (eds.).Philosophy, Language, and Artificial Intelligence(PDF).Studies in Cognitive Systems. Vol. 2. Springer, Dordrecht. pp. 141–162.doi:10.1007/978-94-009-2727-8_7.ISBN 978-94-010-7726-2.
^^a ^bBarwise, Jon;Cooper, Robin (1981)."Generalized quantifiers and natural language".Linguistics and Philosophy.4(2): 159–219.doi:10.1007/BF00350139.

External links

Dag Westerståhl, 2011. 'Generalized Quantifiers'.Stanford Encyclopedia of Philosophy.

[1] Montague, Richard(1974). "The proper treatment of quantification in English". In Kulas, J.; Fetzer, J.H.; Rankin, T.L. (eds.).Philosophy, Language, and Artificial Intelligence(PDF).Studies in Cognitive Systems. Vol. 2. Springer, Dordrecht. pp. 141–162.doi:10.1007/978-94-009-2727-8_7.ISBN 978-94-010-7726-2.

[Barwise-2] Barwise, Jon;Cooper, Robin (1981)."Generalized quantifiers and natural language".Linguistics and Philosophy.4(2): 159–219.doi:10.1007/BF00350139.

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