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Questions about quantifiers1

  • Johan van Benthem (a1)

The importance of the logical ‘generalized quantifiers’ (Mostowski [1957]) for the semantics of natural language was brought out clearly in Barwise & Cooper [1981]. Basically, the idea is that a quantifier phrase QA (such as “all women”, “most children”, “no men”) refers to a set of sets of individuals, viz. those B for which (QA)B holds. Thus, e.g., given a fixed model with universe E,

where ⟦A⟧ is the set of individuals forming the extension of the predicate “A” in the model. This point of view permits an elegant and uniform semantic treatment of the subject-predicate form that pervades natural language.

Such denotations of quantifier phrases exhibit familiar mathematical structures. Thus, for instance, all A produces filters, and no A produces ideals. The denotation of most A is neither; but it is still monotone, in the sense of being closed under supersets. Mere closure under subsets occurs too; witness a quantifier phrase like few A. These mathematical structures are at present being used in organizing linguistic observations and formulating hypotheses about them. In addition to the already mentioned paper of Barwise & Cooper, an interesting example is Zwarts [1981], containing applications to the phenomena of “negative polarity” and “conjunction reduction”. In the course of the latter investigation, several methodological issues of a wider logical interest arose, and these have inspired the present paper.

In order to present these issues, let us shift the above perspective, placing the emphasis on quantifier expressions per se (“all”, “most”, “no”, “some”, etcetera), viewed as denoting relations Q between sets of individuals.

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I would like to thank Frans Zwarts and Dag Westerståhl for their help and encouragement. The Center for Advanced Study in the Behavioral Sciences at Stanford offered ideal working circumstances during the final phase of this research. Numerous participants in discussions contributed valuable suggestions, notably Jon Barwise. The referee and editor of the Journal provided very useful comments and exhortations.

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[1]Barwise, J. and Cooper, R., Generalized quantifiers and natural language, Linguistics and Philosophy, vol. 4 (1981), pp. 159219.
[2]van Benthem, J., Foundations of conditional logic, preprint, Filosofisch Instituut, Groningen, 1982 (to appear in the Journal of Philosophical Logic).
[3]Feferman, S., Persistent and invariant formulas for outer extensions, Compositio Mathematica, vol. 28 (1969), pp. 2952.
[4]Keenan, E. and Stavi, Y., A semantic characterization of natural language determiners, preprint, Linguistics Department, University of California at Los Angeles, Los Angeles, Cal., 1982.
[5]Mostowski, A., On a generalization of quantifiers, Fundamenta Mathematicae, vol. 44 (1957), pp. 1236.
[6]Zwarts, F., Negatief polaire Uitdrukkingen. I, Glot, vol. 4 (1981), pp. 35132.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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