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