In formal 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.[1] [2]
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:
\langlea,b\rangle
Given this definition, we have the simple types e and t, but also a countable infinity of complex types, some of which include:
De
\{0,1\}
\langlee,t\rangle
De | |
D | |
t |
\langlea,b\rangle
a
b
Da | |
D | |
b |
We can now assign types to the words in our sentence above (Every boy sleeps) as follows.
\langlee,t\rangle
\langlee,t\rangle
\langle\langlee,t\rangle,\langle\langlee,t\rangle,t\rangle\rangle
\langle\langlee,t\rangle,t\rangle
\langle\langlee,t\rangle,t\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\subseteqB
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
De
We can now write the meaning of every with the following lambda term, where X,Y are variables of type
\langlee,t\rangle
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:By β-reduction,and
The expression every is a determiner. Combined with a noun, it yields a generalized quantifier of type
\langle\langlee,t\rangle,t\rangle
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\subseteqY
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
X\subseteqY
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 sentences entails the other.
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 students ran fast, so the number of students that did that can be smaller than 3. Conversely, the second sentence does 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 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 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.