Remarks on Free Quantifier Variables

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 _i⊆I(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

$$X_i = \left\{ {b \in I\left| {\left\langle {I,R_1 , \ldots ,R_m } \right\rangle \left| { = A_i \left( {x_i ,y} \right)\left[ {ba} \right]} \right.} \right.} \right\}.$$