Abstraction and Identity

In this paper we outline and discuss various solutions to a restricted, but we think, more interesting version of the infamous Caesar Problem. This restricted version, labelled the C-R Problem, occurs in contexts where we have two distinct abstraction principles: and want to settle cross-sortal identity claims of the form: Both abstraction principles, however, are silent with regard to this identity - a special instance of the Caesar Problem. In what follows, we outline two distinct strategies to resolve the C-R problem. The first strategy decides such cross-abstraction identities in terms of whether or not the equivalence relations appearing on the right hand side of the abstraction principles are identical, while the second strategy settles such identities by appeal to the relevant equivalence classes. We then focus our discussion on the latter approach and offer three ways of implementing this strategy. Ultimately, we argue that this strategy fails, as each attempt to appeal to equivalence classes faces unsurmountable difficulties.