Standard Quantification Theory in the Analysis of English


Standard first-order logic plus quantifiers of all finite orders (“SFOLω”) faces four well-known difficulties when used to characterize the behavior of certain English quantifier phrases. All four difficulties seem to stem from the typed structure of SFOLω models. The typed structure of SFOLω models is in turn a product of an asymmetry between the meaning of names and the meaning of predicates, the element-set asymmetry. In this paper we examine a class of models in which this asymmetry of meaning is removed. The models of this class permit definitions of the quantifiers which allow a desirable flexibility in fixing the domain of quantification. Certain SFOLω type restrictions are thereby avoided. The resulting models of English validate all of the standard first-order logical truths and are free of the four deficiencies of SFOLω models.

This is a preview of subscription content, log in to check access.


  1. 1.

    Barwise, J. and Cooper, R.: Generalized quantifiers and natural language, Linguistics and Philosophy 4 (1981), 159-219.

    Google Scholar 

  2. 2.

    Barwise, J. and Perry, J.: Situations and Attitudes, MIT Press, Cambridge, MA, 1983.

    Google Scholar 

  3. 3.

    Bealer, G.: Quality and Concept, Clarendon Press, Oxford, 1982.

    Google Scholar 

  4. 4.

    Boolos, G.: For every A there is a B, Linguistic Inquiry 12 (1981), 465-466.

    Google Scholar 

  5. 5.

    Boolos, G.: To be is to be a value of a variable (or to be some values of some variables), J. Philos. 81 (1984), 430-449.

    Google Scholar 

  6. 6.

    Chierchia, G. and Turner, R.: Semantics and property theory, Linguistics and Philosophy 11 (1988), 261-302.

    Google Scholar 

  7. 7.

    Cocchiarella, N.: Properties as individuals in formal ontology, Noûs 6 (1972), 165-187.

    Google Scholar 

  8. 8.

    Frege, G.: The Basic Laws of Arithmetic: Exposition of the System, translated and edited by Montgomery Furth, University of California Press, Berkeley and Los Angeles, 1967.

    Google Scholar 

  9. 9.

    Keenan, E. L. and Faltz, L. M.: Boolean Semantics for Natural Language, Reidel, Dordrecht, 1985.

    Google Scholar 

  10. 10.

    Menzel, C.: A complete type-free "second-order" logic and its philosophical foundations, Report No. CSLI-86-40, Center for the Study of Language and Information, Stanford University, 1986.

  11. 11.

    Mostowski, A.: On a generalization of quantifiers, Fund. Math. 44 (1957), 12-36.

    Google Scholar 

  12. 12.

    Schönfinkel, M.: Ñber die Bausteine der mathematischen Logik, Math. Ann. 92 (1924), 305-316, translated as "On the building blocks of mathematical logic" in Jean van Heijenoort (ed.), From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, 1967, pp. 357-366.

    Google Scholar 

  13. 13.

    Sher, G.: The Bounds of Logic: A Generalized Viewpoint, MIT Press, Cambridge, MA, 1991.

    Google Scholar 

  14. 14.

    Stone, M. H.: The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40 (1936), 31-111.

    Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Donaho, S. Standard Quantification Theory in the Analysis of English. Journal of Philosophical Logic 31, 499–526 (2002).

Download citation

  • quantification theory
  • quantifiers
  • semantics
  • type-free logic
  • type theory