 The first thing to notice about this example is that E cannot be the I-predicable of such a theory, since \(E\) is defined in terms of identity (look at the right side of R). It is ‘=’ that must serve as the \(I\)-predicable, and it renders distinct ordered pairs of integers discernible. The moral is that not all equivalence relations can be drafted to do the job of identity, even given a limited ideology. There is, indeed, a plausible argument that any equivalence relation presupposes identity — not necessarily in the direct way illustrated by (R), but indirectly, nonetheless (see §6). Moreover, from the standpoint of general mathematics, once we have (R), we have the (infinite) equivalence classes it determines and the partition it induces. These are inescapable. Even from a more limited viewpoint, it seems that once we have enough set theory to give us ordered pairs of integers and the ability to define (R), we get the partition it induces as well.