Abstract: This article offers a logical, linguistic, and philosophical account of modern quantification theory. Contrasting the standard approach to quantifiers (according to which logical quantifiers are defined by enumeration) with the generalized approach (according to which quantifiers are defined systematically), the article begins with a brief history of standard quantifier theory and identifies some of its logical, linguistic, and philosophical strengths and weaknesses. It then proceeds to a brief history of generalized quantifier theory and explains how it overcomes the weaknesses of the standard theory. One of the main philosophical advantages of the generalized theory is its philosophically informative criterion of logicality. The paper describes the work done so far in this theory, highlights some of its central logical results, offers an overview of its main linguistic contributions, and discusses its philosophical significance.
Comment: This paper is adequate for an advanced course on philosophy of logic or for a specialised course on quantification. It provides a presentation and a comparison of two different conceptions of quantification: standard modern quantification and generalised quantification. Interestingly, it presents the virtues and drawbacks of each of them from three different points of view: logical, linguistic and philosophical. Moreover, it puts special emphasis on the theme of which quantifiers are to count as logical by focusing on the criterion of logicality which is available for the generalized conception. It presupposes some knowledge of predicate logic as well as of set theory.