Abstract
Introduction We shall study (generalized) quantifiers in the framework introduced by Barwise and Cooper in [ 1 ], the logical investigation of which has been continued in van Ben them [3] and Keenan and Stavi [2]. The present paper, although self-contained, is in the spirit of [3]. Its main characteristic is a systematic use of a method of proof introduced by van Benthem, which is based on a representation of quantifiers as relations on the natural numbers. With this method we give simplified proofs of a number of van Benthem's results and prove some new ones. In particular, the number-theoretic representation proves convenient for studying monotonicity behavior of quantifiers, and our main result is a characterization of the first-order definable quantifiers in terms of monotonicity.
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