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The Hierarchy Theorem for Generalized Quantifiers

Lauri Hella, Kerkko Luosto and Jouko Väänänen
The Journal of Symbolic Logic
Vol. 61, No. 3 (Sep., 1996), pp. 802-817
DOI: 10.2307/2275786
Stable URL: http://www.jstor.org/stable/2275786
Page Count: 16
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The Hierarchy Theorem for Generalized Quantifiers
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Abstract

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.

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