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In linguistic semantics, a generalized quantifier (GQ) is an expression that denotes a set of sets. This is the standard semantics assigned to quantified noun phrases. 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.
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:
Given this definition, we have the simple types e and t, but also a countable infinity of complex types, some of which include:
We can now assign types to the words in our sentence above (Every boy sleeps) as follows.
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]\displaystyle{ A\subseteq B }[/math].
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.
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]\displaystyle{ D_e }[/math], then the following lambda term denotes the identity function on individuals:
We can now write the meaning of every with the following lambda term, where X,Y are variables of type [math]\displaystyle{ \langle e,t\rangle }[/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:
The expression every is a determiner. Combined with a noun, it yields a generalized quantifier of type [math]\displaystyle{ \langle\langle e,t\rangle,t\rangle }[/math].
A generalized quantifier GQ is said to be monotone increasing, also called upward entailing, just in case, for any two sets X and Y the following holds:
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:
A GQ is said to be monotone decreasing, also called downward entailing just in case, for any two sets X and Y, the following holds:
An example of a monotone decreasing GQ is no boy. For this GQ we have that the first sentence below entails the second.
The lambda term for the determiner no is the following. It says that the two sets have an empty intersection.
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.
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 two sentences entail the other.
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 of the intersection between the two sets equals 3.
A determiner D is said to be conservative if the following equivalence holds:
For example, the following two sentences are equivalent.
It has been proposed that all natural language determiners (i.e. in every language) are conservative (Barwise and Cooper 1981). 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.