Generalized quantifier

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. 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 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: Expressions of type e denote elements of the universe of discourse, the set of entities the discourse is about. This set is usually written as . Examples of type e expressions include John and he. Expressions of type t denote a truth value, 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 denote functions from the set of entities to the set of truth values. This set of functions is rendered as . 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. In general, expressions of complex types denote functions from the set of entities of type to the set of entities of type , a construct we can write as follows: . We can now assign types to the words in our sentence above (Every boy sleeps) as follows.