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 boyThis treatment of quantifiers has been essential in achieving a compositional semantics for sentences containing quantifiers.

Model theory

Definition

In first-order logic, there are two quantifiers. They have a fixed meaning in model-theoretic semantics (that is, set-theoretic semantics) of first-order logic, as follows.

Given a first-order language, a model of the language, and an interpretation is the universe.

denotes the power set operation.

This may appear somewhat circular, as set theory is usually formalized in a first-order logic (as in ZFC set theory). However, if one takes such a set theory as a given base, then one can build other first-order logics over this base set theory. This is the typical perspective taken in model theory.

themselves as symbols that are being modelled. This is similar to how equality itself interpreted as a binary relation symbol in first-order logic with equality.

may quantify over k.

can be regarded as a k, a quantifier over k variables can be regarded as a predicate for k.

It is modelled by an n. This general definition a generalized quantifier definition is sometimes called a Lindström quantifier.

, then it is monadic, otherwise it is polyadic.

Examples

Of type ⟨1⟩:

• , etc.

• meaning "there exists at most n.

• .

• .

Of type ⟨2⟩:

• meaning "is a well-ordering is a well-ordered partially ordered set. Notably, well-ordering is not axiomatizable in standard first-order logic, thus showing that we have expanded the power of the logical language.

  . For example, the infinite Ramsey theorem, and draw an edge between any pair of points, and color each edge from one of a finite number of colors, then there exists an infinite clique is colored with the i.

Of type ⟨n⟩:

• , such that any size-n.

Of type ⟨1, 1⟩:

• . Similarly, "Some", "Not any", and "Not all" are of type ⟨1, 1⟩. In this way, the 4 types of sentences in term logic are naturally expressed in first-order logic with generalized quantifiers.

• Similarly, "at least 5", "exactly 3", "an even number of", "there are more than", .

• The Härtig quantifier, meaning "equally many".

Operations

Quantifiers can be combined and modified to create more quantifiers, using operations upon quantifiers.

Relativization: An n can be relativizedIteration.

.

quantifier.

Resumption, it can be resumed, using the fact that a k is a formula that is interpreted over a plain model.

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:

1. e and t are types.

2. If a and 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 types e and t, but also a countable infinity

• Expressions of type e denote elements of the universe of discourse. Examples of type e expressions include John and he.

• Expressions of type t denote a truth value, 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.

• denote functions. 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 adjectives.

• .

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.

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 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, then the following lambda term denotes the identity function

We can now write the meaning of every with the following lambda term, where X,Y

If we abbreviate the meaning of boy and sleeps as "B" and "S", respectively, we have that the sentence every boy sleeps By β-reduction

The expression every is a determiner. Combined with a noun, it yields a generalized quantifier.

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:

1. , then GQ(X) entailss 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 entailss the second:

1. Every boy runs fast.

2. Every boy runs.

Monotone decreasing GQs

A GQ is said to be monotone decreasing (also called downward entailing) if, for every pair of sets X and Y, the following holds:

1. , 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.

1. No boy runs.

2. No boy runs fast.

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.

1. Good: No boy has any money.

2. 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.

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 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

Conservativity

A determiner D is said to be conservative For example, the following two sentences are equivalent.

1. Every boy sleeps.

2. 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.

1. Only boys sleep.

2. Only boys are boys who sleep.

See also

• Scope (formal semantics)

• Lindström quantifier

• Branching quantifier

References

Further reading

External links

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