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

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

A version oftype theory is often used to make the semantics of different kinds of expressions explicit. The standard construction defines the set of typesrecursively as follows:

1. e andt are types.
2. Ifa andb are both types, then so is${\displaystyle ⟨a,b⟩}$
3. 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 typese andt, but also acountableinfinity of complex types, some of which include:$${\displaystyle ⟨e,t⟩;⟨t,t⟩;⟨⟨e,t⟩,t⟩;⟨e,⟨e,t⟩⟩;⟨⟨e,t⟩,⟨⟨e,t⟩,t⟩⟩;\dots }$$

• Expressions of typee denote elements of theuniverse of discourse, the set of entities the discourse is about. This set is usually written as${\displaystyle D_e}$. Examples of typee expressions includeJohn andhe.
• Expressions of typet denote atruth value, usually rendered as the set${\displaystyle \{0,1\}}$, where 0 stands for "false" and 1 stands for "true". Examples of expressions that are sometimes said to be of typet aresentences orpropositions.
• Expressions of type${\displaystyle ⟨e,t⟩}$ denotefunctions from the set of entities to the set of truth values. This set of functions is rendered as${\displaystyle D_t^{D_e}}$. Such functions arecharacteristic functions ofsets. 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 denotesets rather than characteristic functions, although, strictly speaking, the latter is more accurate. Examples of expressions of this type arepredicates,nouns and some kinds ofadjectives.
• In general, expressions of complex types${\displaystyle ⟨a,b⟩}$ denote functions from the set of entities of type${\displaystyle a}$ to the set of entities of type${\displaystyle b}$, a construct we can write as follows:${\displaystyle D_b^{D_a}}$.

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

• Type(boy) =${\displaystyle ⟨e,t⟩}$
• Type(sleeps) =${\displaystyle ⟨e,t⟩}$
• Type(every) =${\displaystyle ⟨⟨e,t⟩,⟨⟨e,t⟩,t⟩⟩}$
• Type(every boy) =${\displaystyle ⟨⟨e,t⟩,t⟩}$

and so we can see that the generalized quantifier in our example is of type${\displaystyle ⟨⟨e,t⟩,t⟩}$

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

A useful way to write complex functions is thelambda calculus. For example, one can write the meaning ofsleeps as the following lambda expression, which is a function from an individualx to the proposition thatx sleeps.$${\displaystyle \lambda x.{\mathrm{s}\mathrm{l}\mathrm{e}\mathrm{e}\mathrm{p}}^{\prime }(x)}$$Such lambda terms are functions whose domain is what precedes the period, and whose range are the type of thing that follows the period. Ifx is a variable that ranges over elements of${\displaystyle D_e}$, then the following lambda term denotes theidentity function on individuals:$${\displaystyle \lambda x.x}$$

We can now write the meaning ofevery with the following lambda term, whereX,Y are variables of type${\displaystyle ⟨e,t⟩}$:$${\displaystyle \lambda X.\lambda Y.X\subseteq Y}$$

If we abbreviate the meaning ofboy andsleeps as "B" and "S", respectively, we have that the sentenceevery boy sleeps now means the following:$${\displaystyle (\lambda X.\lambda Y.X\subseteq Y)(B)(S)}$$Byβ-reduction,$${\displaystyle (\lambda Y.B\subseteq Y)(S)}$$and$${\displaystyle B\subseteq S}$$

The expressionevery is adeterminer. Combined with anoun, it yields ageneralized quantifier of type${\displaystyle ⟨⟨e,t⟩,t⟩}$.

Ageneralized quantifier GQ is said to bemonotone increasing (also calledupward entailing) if, for every pair of setsX andY, the following holds:

if${\displaystyle X\subseteq Y}$, then GQ(X)entails GQ(Y).

The GQevery boy is monotone increasing. For example, the set of things thatrun fast is a subset of the set of things thatrun. Therefore, the first sentence belowentails the second:

1. Every boy runs fast.
2. Every boy runs.

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

If${\displaystyle X\subseteq Y}$, 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.

1. No boy runs.
2. No boy runs fast.

The lambda term for thedeterminerno is the following. It says that the two sets have an emptyintersection.$${\displaystyle \lambda X.\lambda Y.X\cap Y=\mathrm{\varnothing }}$$Monotone decreasing GQs are among the expressions that can license anegative polarity item, such asany. Monotone increasing GQs do not license negative polarity items.

1. Good: No boy hasany money.
2. Bad: *Every boy hasany money.

A GQ is said to benon-monotone if 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.

1. Exactly three students ran.
2. 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 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)determinerexactly three is the following. It says that thecardinality of theintersection between the two sets equals 3.$${\displaystyle \lambda X.\lambda Y.|X\cap Y|=3}$$

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

1. Every boy sleeps.
2. Every boy is a boy who sleeps.

It has been proposed thatall determiners—in every natural language—are conservative.^[2] The expressiononly is not conservative. The following two sentences are not equivalent. But it is, in fact, not common to analyzeonly as adeterminer. Rather, it is standardly treated as afocus-sensitiveadverb.

1. Only boys sleep.
2. Only boys are boys who sleep.

• Scope (formal semantics)
• Lindström quantifier
• Branching quantifier

• Stanley Peters; Dag Westerståhl (2006).Quantifiers in language and logic. Clarendon Press.ISBN 978-0-19-929125-0.
• Antonio Badia (2009).Quantifiers in Action: Generalized Quantification in Query, Logical and Natural Languages. Springer.ISBN 978-0-387-09563-9.
• Wągiel M (2021).Subatomic quantification(pdf). Berlin: Language Science Press.doi:10.5281/zenodo.5106382.ISBN 978-3-98554-011-2.

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

• Semantics
• Formal semantics (natural language)
• Quantifier (logic)

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