Generalized quantifier

A generalized quantifier is a generalization of the universal and existential quantifier of predicate logic to a higher-order concept of a quantifier as a set of sets. This generalization was already implicit in Montague (1974) but made explicit in Barwise & Cooper (1981). Montague (1974) proposed a compositional translation of quantified sentences making use of lambda-abstraction. He showed that a compositional translation of (i) into (ii) is possible if the subject every boy takes the predicate walks, as in (iii):


 * (i)  Every boy walks
 * (ii) All(x) [ boy(x) -> walk(x) ]
 * (iii) Walk translates into walk, every boy translates as lambda P [ All(x) [ boy(x) -> P(x) ]]

The noun phrase every boy denotes a set of properties (or sets), namely the set of properties that every boy has and sentence (i) is true when the property of walking is in this set. Barwise and Cooper showed that this treatment of NPs as sets of sets can be used to assign denotations to those NPs that cannot be represented in first-order logic, like most, or only very cumbersomely, like three. They did not use Montague's intermediate logical language, but a direct interpretation in set-theoretic terms (|| || is the interpretation function):

(iv) || most boys || =  { X subset E : | X intersect || boys || > | || boys || - X | || two boys || =  { X subset E : | X intersect || boys || | >= 2

The interpretation of most boys is the set of sets such that there are more boys that they do contain than don't, and the interpretation of two boys is the set of sets which contain at least two boys.

Link
Utrecht Lexicon of Linguistics