In [[linguistics|linguistic]] [[semantics]], a '''generalized quantifier''' is an expression that denotes a [[Property (philosophy)|property]]set of a property, also called a [[higher-order]] propertysets. This is the standard semantics assigned to [[Quantifier (linguistics)|quantified]] [[noun phrase]]s, also called [[determiner phrase]]s, or DP for short. In theFor example below, the DPgeneralized quantifier ''every boy'' says of a property X thatdenotes the [[set theory|set]] of entities that are ''boys'' is a [[subset]]sets of thewhich setevery of entities that have property X. So the following sentence says that the set of boysboy is a subsetmember. of the set of sleepers.
::Every boy sleeps.
:: <math>\{xX \,|\, \{x \,|\, \mbox{x is a boy}\} \subseteq \{x\,|\,xX \mbox{ sleeps}\}</math>
This treatment of quantifiers has been essential in achieving a [[compositionality|compositional]] [[semantics]] for sentences containing quantifiers.<ref> Montague, Richard: 1974, '[http://www.blackwellpublishing.com/content/BPL_Images/Content_store/Sample_chapter/9780631215417/Portner.pdf The proper treatment of quantification in English]',
