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Concept# Generalized quantifier

Summary

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:
{X \mid \forall x (x \text{ is a boy} \to x \in X) }
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:

Official source

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