Quantifiers in natural language correspond to words such as every, some, most, few, and many others.

The Semantics of Determiners

What is the semantics of expressions like every and most? An answer to this question emerged in the early 1980s, in work of Jon Barwise and Robin Cooper (1981), James Higginbotham and Robert May (1981), Edward L. Keenan and Jonathan Stavi (1986), Johan van Benthem (1986), Dag Westerståhl (1985), and many others.

The basic idea of how to interpret quantified expressions comes from Gottlob Frege (1879). Frege observed that the familiar quantifiers ∀ (everything) and ∃ (something) can be thought of, in Frege's terms, as second-level concepts. Let us call whatever gives the interpretation of an expression its semantic value. Assuming an extensional and set-theoretic framework, we my assign predicates sets of individuals as their semantic values. Frege's idea can then be recast as saying that the semantic values...