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Ramsey Theory Is Needed for Solving Definability Problems of Generalized Quantifiers

  • Kerkko Luosto
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1754)

Abstract

In recent years, generalized quantifiers (see [H3]) have received quite a lot of novel interest because of their applications to computer science and linguistics. Their definability theory has made considerable progress during the last decade, which will be the subject of the next section. The proofs of many of these results often use results of Ramsey theory, such as theorems of van derWaerden and Folkman, and yet, the answers to some of the definability problems seem obvious from the outset. This raises the natural question whether Ramsey theory is really needed in the proofs (cf. [vBW]) or whether easier ways of proof might be discovered. The purpose of this paper is to argue in favour of the former and to convince the reader of the cruciality of Ramsey theory for quantifier definability theory.

Keywords

Relative Rank Relation Symbol Boolean Combination Ramsey Theory Congruence Closure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Ca1.
    Xavier Caicedo: Maximality and interpolation in abstract logics (back-and-forth techniques). Doctoral Dissertation, University of Maryland, 1978, 146 pp.Google Scholar
  2. Ca2.
    Xavier Caicedo: Definability properties and the congruence closure. Archive for Mathematical logic 30 (1990), 231–240.zbMATHCrossRefMathSciNetGoogle Scholar
  3. Co.
    Luis Jaime Corredor: El reticulo de las logicas de primer orden con cuantificadores cardinales. Revista Colombiana de Matemáticas XX (1986), 1–26.MathSciNetGoogle Scholar
  4. FSV.
    J. Flum, M. Schiehlen and J. Väänänen: Quantifiers and congruence closure. Studia Logica 62 (1999), 315–340.zbMATHCrossRefMathSciNetGoogle Scholar
  5. G. S. C. Garavaglia: Relative strength of Malitz quantifiers. Notre Dame Journal of Formal Logic 19 (1978), 495–503.zbMATHCrossRefMathSciNetGoogle Scholar
  6. H1.
    Lauri Hella: Definability hierarchies of generalized quantifiers. Annals of Pure and Applied Logic 43 (1989), 235–271.zbMATHCrossRefMathSciNetGoogle Scholar
  7. H2.
    Lauri Hella: Logical hierarchies in PTIME. Information and Computation 129 (1996), 1–19.zbMATHCrossRefMathSciNetGoogle Scholar
  8. H3.
    Lauri Hella: Generalized quantifiers in finite model theory. Manuscript, 17 pp.Google Scholar
  9. HL.
    L. Hella and K. Luosto: Finite generation problem and n-ary quantifiers. In M. Krynicki, M. Mostowski and L. W. Szczerba (eds.): Quantifiers: Logics, Models and Computation, Kluwer Academic Publishers 1995, 63–104.Google Scholar
  10. K. H. Jerome Keisler: Logic with the quantifier “there exist uncountably many”. Annals of Mathematical Logic 1 (1970), 1–93.CrossRefGoogle Scholar
  11. KV.
    P. Kolaitis and J. Väänänen: Generalized quantifiers and pebble games on finite structures. Annals of Pure and Applied Logic 74 (1995), 23–75.zbMATHCrossRefMathSciNetGoogle Scholar
  12. Li.
    Per Lindström: Personal communication. Via Dag Westerstähl, 1993.Google Scholar
  13. Lu.
    Kerkko Luosto: Hierarchies of monadic generalized quantifiers. To appear in The Journal of Symbolic Logic.Google Scholar
  14. LT.
    K. Luosto and Jerzy Tyszkiewicz: On resumptions of monadic generalized quantifiers. Manuscript, 6 pp.Google Scholar
  15. NV.
    J. Nešetřil and Jouko Väänänen: Combinatorics and quantifiers. Commentationes Mathematicae Universitatis Carolinae 37 (1996), 433–443.MathSciNetzbMATHGoogle Scholar
  16. S.
    Saharon Shelah: The theorems of Beth and Craig in abstract model theory, III: Δ-logics and infinitary logics. Israel Journal of Mathematics 69,no.2 (1990), 193–213.zbMATHMathSciNetCrossRefGoogle Scholar
  17. V1.
    Jouko Väänänen: A hierarchy theorem for Lindström quantifiers. In M. Furberg, T. Wetterström and C. Åberg (eds.): Logic and Abstraction. Acta Philosophica Gothoburgesia 1 (1986), 317–323.Google Scholar
  18. V2.
    Jouko Väänänen: Unary quantifiers on finite models. Journal of Logic, Language and Information 6 (1997), 275–304.zbMATHCrossRefMathSciNetGoogle Scholar
  19. vBW.
    J. van Benthem and D. Westerstáhl: Directions in generalized quantifier hierarchy. Studia Logica 55 (1995), 389–419.zbMATHCrossRefMathSciNetGoogle Scholar
  20. vW.
    B. L. van der Waerden: Beweis einer Baudetschen Vermutung. Nieuw Arch. Wisk. 15 (1927), 212–216.Google Scholar
  21. W. Dag Westerstáhl: Quantifiers in natural language. A survey of some recent work. In M. Krynicki, M. Mostowski and L. W. Szczerba (eds.): Quantifiers: Logics, Models and Computation, Kluwer Academic Publishers 1995, 359–408.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Kerkko Luosto
    • 1
  1. 1.Department of MathematicsUniversity of HelsinkiYliopistonkatu 5Finland

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