Skip to main content Accessibility help

The hierarchy theorem for generalized quantifiers

  • Lauri Hella (a1), Kerkko Luosto (a2) and Jouko Väänänen (a3)

The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity type t there is a generalized quantifier of type t which is not definable in the extension of first order logic by all generalized quantifiers of type smaller than t. This was proved for unary similarity types by Per Lindström [17] with a counting argument. We extend his method to arbitrary similarity types.

Hide All
[1]Barwise, J. and Feferman, S. (editors), Model-theoretic logics, Springer-Verlag, 1985.
[2]Cai, J., Fürer, M., and Immerman, N., An optimal lower bound on the number of variables for graph identification, Combinatorial, vol. 12 (1992), pp. 389410.
[3]Caicedo, X., Back-and-forth systems for arbitrary quantifiers, Mathematical logic in Latin America (Arruda, A. I., Chuaqui, R., and da Costa, N.C.A., editors), North-Holland, 1980, pp. 83102.
[4]Dawar, A., Generalized quantifiers and logical reducibilities, Journal of Logic and Computation, vol. 5 (1995), pp. 213226.
[5]Fagin, R., The number of finite relational structures, Discrete Mathematics, vol. 19 (1977), pp. 1721.
[6]Hella, L., Logical hierarchies in PTIME, Information and Computation, a preliminary version appeared in Proceedings of the 7th IEEE symposium on logic in computer science, 1992.
[7]Hella, L., Definability hierarchies of generalized quantifiers, Annals of Pure and Applied Logic, vol. 43 (1989), pp. 235271.
[8]Hella, L. and Sandu, G., Partially ordered connectives and finite graphs, Quantifiers: Logics, models and computation (Krynicki, M., Mostowski, M., and Szczerba, L., editors), vol. II, Kluwer Academic Publishers, 1995, pp. 7988.
[9]Hella, L., Väänänen, J., and Westerståhl, D., Definability ofpolyadic lifts of generalized quantifiers, to appear.
[10]Kolaitis, Ph. and Väänänen, J., Generalized quantifiers and pebble games on finite structures, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 2375.
[11]Krynicki, M., Lachlan, A., and Väänänen, J., Vector spaces and binary quantifiers, Notre Dame Journal of Formal Logic, vol. 25 (1984), pp. 7278.
[12]Lindström, P., First order predicate logic with generalized quantifiers, Theoria, vol. 32 (1966), pp. 186195.
[13]Luosto, K., Hierarchies of monadic generalized quantifiers, to appear.
[14]Nešetřil, J. and Väänänen, J., Combinatorics and quantifiers, Comment. Math. Univ. Carol., to appear.
[15]Väänänen, J., Remarks on generalized quantifiers and second-order logics, Set theory and hierarchy theory (Waszkiewicz, J., Wojciechowska, A., and Zarach, A., editors), vol. 14, Prace Naukowe Instytutu Matematyki Politechniki Wroclawskiej, Wroclaw, 1977, pp. 117123.
[16]Väänänen, J., A hierarchy theorem for Lindström quantifiers, Logic and abstraction (Furberg, M., Wetterström, T., and Åberg, C., editors), vol. 1, Acta Philosophica Gotheburgensia, 1986, pp. 317323.
[17]Westerståhl, D., personal communication.
[18]Väänänen, J., Quantifiers informal and natural languages, Handbook of philosophical logic (Gabbay, D. and Guenther, F., editors), vol. IV, D. Reidel, Dordrecht, 1989, pp. 1131.
[19]Väänänen, J., Iterated quantifiers, Dynamics, polarity and quantification (Kanazawa, M. and Pinon, C., editors), CSLI Publications, Stanford, 1994, pp. 173209.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed