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# Remarks on Free Quantifier Variables

• Jouko Väänänen
Chapter
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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## Copyright information

© D. Reidel Publishing Company, Dordrecht, Holland 1979

## Authors and Affiliations

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