Relative Identity and Cardinality1

Because cardinality-statements bear close logical connections to identity-statements, it has been argued that Frege's treatment of cardinality, in which cardinality is "relative to " a concept, implies Geach's treatment of identity, in which identity is "relative to " a predicate. If this is right, then those holding a broadly Fregean account of cardinality must accept Geach's very controversial position on identity. The purpose of this paper is to argue that Frege's position does not imply Geach's, but is instead inconsistent with it. Despite superficial similarities between the two doctrines, the problems with Geach's account give no reason to reject Frege's. Peter Geach famously holds that there is no such thing as absolute identity. There are rather, as Geach sees it, a variety of relative identity relations, each essentially connected with a particular monadic predicate. Though we can strictly and meaningfully say that an individual a is the sam