 Third, given any structure \(M\) for \(L\) in which Ref and LL are true, there is a corresponding structure \(QM\), the ‘quotient structure’ determined by \(M\), in which indiscernibility does coincide with \(I(A,x,y)\). \(QM\) is obtained in roughly the following way: Let the elements of \(QM\) be the equivalence classes \([x]\), for elements \(x\) of \(M\) determined by indiscernibility in \(M\). If \(F\) is a one-place predicate true in \(M\) of some object \(x\) in \(M\), then define \(F\) to be true of \([x]\) in \(QM\), and similarly for many-place predicates and constants. It can then be shown that any sentence true in \(M\) is true in \(QM\), and vice versa. The existence of quotient structures makes it possible to treat the identity symbol as a logical constant interpreted in terms of \(I(A,x,y)\). There is in fact in general no other way to guarantee that Ref and LL will hold in every structure. (As Quine (1970) points out, however, a finite language will always contain a predicate satisfying Ref and LL in any structure; cf. Hodges 1983.) The alternative, however, is to view FOL with Ref and LL (FOL\(^=)\) as a proper theory in whose models (structures in which Ref and LL hold) there will be an equivalence relation \(E\) such that if \(E(x,y)\) holds, then \(x\) and \(y\) will be indiscernible with respect to the defined subsets of the domain. But we cannot in general assume that every subset of the domain is definable. If the domain is infinite, \(L\) runs out of defining formulas long before the domain runs out of subsets. Nonetheless, a strong metatheorem asserts that any set of formulas that has a model, has a countable (finite or denumerable) model. This means that the difference between indiscernibility and \(I(A,x,y)\) is minimized at least to the extent that, for a sufficiently rich language such as \(L\), the valid formulas concerning indiscernibility (i.e., the formulas true in every model of what is termed below ‘the pure \(L\)-theory with identity’) coincide with the valid formulas concerning \(I(A,x,y)\). (See Epstein 2001 for a sketch of a proof of this fact.) This is not to say, however, that there isn’t a significant difference between identity qua indiscernibility and identity qua \(I(A,x,y)\) (see below). Both points of view — that FOL\(^=\) is a proper theory and that it is a logic — may be found in the literature (Quine 1970). The latter is the more usual view and it will count here as part of the standard account of identity.