Absolute and Relative Identity

On the classical, or Fregean, view of identity it is an equivalence relation satisfying Leibniz’s Law (so-called), i.e. if “=” expresses identity, the schema “(x)(y)(x = y.⊐.Fx ⊐ Fy)” is valid. Now these formal properties are sufficient to ensure that within any theory expressible by means of a fixed stock of (one- or many-place) predicates, quantifiers and truth-functional connectives, any two predicates which can be regarded as expressing identity will be extensionally equivalent. These formal properties are not, however, sufficient to ensure that a two-place predicate does express identity within a given theory. For it may simply be that the descriptive resources of the theory are not rich enough to distinguish items between which the equivalence relation expressed by the predicate holds.1