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In formal semantics, a generalized quantifier (GQ) is an expression that denotes a
set of sets In set theory and related branches of mathematics, a collection F of subsets of a given set S is called a family of subsets of S, or a family of sets over S. More generally, a collection of any sets whatsoever is called a family of sets, set ...
. This is the standard semantics assigned to quantified
noun phrase In linguistics, a noun phrase, or nominal (phrase), is a phrase that has a noun or pronoun as its head or performs the same grammatical function as a noun. Noun phrases are very common cross-linguistically, and they may be the most frequentl ...
s. For example, the generalized quantifier ''every boy'' denotes the set of sets of which every boy is a member: $\$ This treatment of quantifiers has been essential in achieving a compositional semantics for sentences containing quantifiers.

# 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 recursively as follows: #''e'' and ''t'' are types. #If ''a'' and ''b'' are 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 types ''e'' and ''t'', but also a countable infinity of 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 type ''e'' denote elements of the
universe of discourse In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range. Overview The doma ...
, the set of entities the discourse is about. This set is usually written as $D_e$. Examples of type ''e'' expressions include ''John'' and ''he''. *Expressions of type ''t'' denote a
truth value In logic and mathematics, a truth value, sometimes called a logical value, is a value indicating the relation of a proposition to truth, which in classical logic has only two possible values ('' true'' or '' false''). Computing In some pr ...
, usually rendered as the set $\$, 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 $\langle e,t\rangle$ denote functions from the set of entities to the set of truth values. This set of functions is rendered as $D_t^$. Such functions are characteristic functions of 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 predicates, nouns and some kinds of
adjective In linguistics, an adjective ( abbreviated ) is a word that generally modifies a noun or noun phrase or describes its referent. Its semantic role is to change information given by the noun. Traditionally, adjectives were considered one of the ma ...
s. *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: $D_b^$. 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) = $\langle\langle e,t\rangle,\langle \langle e, t\rangle, t\rangle\rangle$ 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 $A\subseteq B$.

# 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''. $\lambda x. \mathrm'(x)$ Such lambda terms are functions whose domain is what precedes the period, and whose range are the type of thing that follows the period. If ''x'' is a variable that ranges over elements of $D_e$, then the following lambda term denotes the
identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, u ...
on individuals: $\lambda x.x$ We can now write the meaning of ''every'' with the following lambda term, where ''X,Y'' are variables of type $\langle e,t\rangle$: $\lambda X.\lambda Y. X\subseteq Y$ 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: $(\lambda X.\lambda Y. X\subseteq Y)(B)(S)$ By β-reduction, $(\lambda Y. B \subseteq Y)(S)$ and $B\subseteq S$ The expression ''every'' is a determiner. Combined with a noun, it yields a ''generalized quantifier'' of type $\langle\langle e,t\rangle,t\rangle$.

# 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 $X\subseteq Y$, then GQ(''X'') entails 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 entails the second: #Every boy runs fast. #Every boy runs.

### Monotone decreasing GQs

A GQ is said to be monotone decreasing (also called
downward entailing In linguistic semantics, a downward entailing (DE) propositional operator is one that constrains the meaning of an expression to a lower number or degree than would be possible without the expression. For example, "not," "nobody," "few people," "a ...
) if, for every pair of sets ''X'' and ''Y'', the following holds: :If $X\subseteq Y$, 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 ''no'' is the following. It says that the two sets have an empty
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
. $\lambda X.\lambda Y. X\cap Y= \emptyset$ 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 doesn't entail the second. The fact that the number of students that ran is exactly three doesn't 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't 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 ''exactly three'' is the following. It says that the
cardinality In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = \ contains 3 elements, and therefore A has a cardinality of 3. Beginning in the late 19th century, this concept was generalize ...
of the
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, thei ...
between the two sets equals 3. $\lambda X.\lambda Y. , X\cap Y, =3$

## Conservativity

A determiner D is said to be ''conservative'' if 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 that ''all'' determinersin every natural languageare conservative. 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. Rather, it is standardly treated as a focus-sensitive
adverb An adverb is a word or an expression that generally modifies a verb, adjective, another adverb, determiner, clause, preposition, or sentence. Adverbs typically express manner, place, time, frequency, degree, level of certainty, etc., answering ...
. #Only boys sleep. #Only boys are boys who sleep.

*
Scope (formal semantics) In formal semantics, the scope of a semantic operator is the semantic object to which it applies. For instance, in the sentence "''Paulina doesn't drink beer but she does drink wine''," the proposition that Paulina drinks beer occurs within the sco ...
*
Lindström quantifier In mathematical logic, a Lindström quantifier is a generalized polyadic quantifier. Lindström quantifiers generalize first-order quantifiers, such as the existential quantifier, the universal quantifier, and the counting quantifiers. They were ...
* Branching quantifier