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Remarks on Free Quantifier Variables

  • Jouko Väänänen
Part of the Synthese Library book series (SYLI, volume 122)

Abstract

We consider the logic L q, introduced by Thomason [8], which is obtained from L ωω by adding monadic quantifier variables. Thus if A 1,…,A n are formulae of L q, x 1,…,x n individual variables and Q a quantifier variable of type <n>, then
$$Q{x_1} \ldots {x_n}{A_1}({x_1}) \ldots {A_n}({x_n})$$
is a formula of L q. An interpretation of L q is obtained from an interpretation of L ωω by assigning to the quantifier variables mondaic generalized quantifiers in the sense of Lindström [6]. Thus a quantifier variable Q of type <n> is assigned in a given domain I a class Q of structures of the form <I,X,…,X n> such that X iI(i=1…n) and every isomorphic copy of a structure in Q is also in Q. In this assignment
$$ \left\langle {I,R_1 , \ldots ,R_m } \right\rangle \left| { = Qx_1 \ldots x_m A_1 \left( {x_1 ,y} \right) \ldots A_n \left( {x_n ,y} \right)\left[ a \right]} \right. $$
iff
$$ \left\langle {I,X_1 , \ldots ,X_n } \right\rangle \in 2 $$
where
$$X_i = \left\{ {b \in I\left| {\left\langle {I,R_1 , \ldots ,R_m } \right\rangle \left| { = A_i \left( {x_i ,y} \right)\left[ {ba} \right]} \right.} \right.} \right\}.$$

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References

  1. [1]
    J. Barwise, ‘The Hanf number of second order logic’, J. Symb. Logic 37 (1972), 588–594.CrossRefGoogle Scholar
  2. [2]
    J. L. Bell and A. B. Slomson, Models and Ultraproducts: An Introduction, North-Holland, Amsterdam, 1969, pp. ix+322.Google Scholar
  3. [3]
    S. Feferman, ‘Applications of many sorted interpolation theorems’, Proceedings of Symposia in Pure Mathematics XXV, AMS, Providence, Rhode Island, 1974, 205–224.Google Scholar
  4. [4]
    G. Fuhrken, ‘A remark on the Härtig-quantifier’, Z. Math. Logik u. Grundl. Math. 18 (1972), 227–228.CrossRefGoogle Scholar
  5. [5]
    K. Härtig, ‘Uber einen Quantifikator mit zwei Wirkungsbereichen’, Colloquy on the Foundations of Mathematics, Mathematical Machines and their Applications, Tihany (Hungary) 1964, Budapest/Paris, 1965, 31–36.Google Scholar
  6. [6]
    P. Lindström, ‘First order logic with generalised quantifiers’, Theoria 32 (1966), 186–195.Google Scholar
  7. [7]
    R. Montague, ‘Reductions of higher order logic’, in J. W. Addison, L. Henkin and A. Tarski, eds., The Theory of Models, North-Holland, Amsterdam, 1967, 265–273.Google Scholar
  8. [8]
    R. Thomason, ‘A system of logic with free variables ranging over quantifiers (abstract)’, J. Symb. Logic 31 (1966), 700.Google Scholar
  9. [9]
    M. Yasuhara, ‘Incompleteness of Lν languages’, Fund. Math. 66 (1969), 147–152.Google Scholar
  10. [10]
    M. Yasuhara, ‘Syntactical and semantical properties of generalised quantifiers’, J. Symb. Logic 31 (1966), 617–632.CrossRefGoogle Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht, Holland 1979

Authors and Affiliations

  • Jouko Väänänen
    • 1
  1. 1.Department of MathematicsUniversity of ManchesterUK

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