 Suppose for the moment that we do not assign any special interpretation to the identity symbol. We treat it like any other two place predicate. Let \(M\) be a structure for \(L\) and assume that Ref and LL are true in \(M\). Call the relation defined in \(M\) by the conjunction of Ref and LL ‘indiscernibility’ (see Enderton 2000, for the definition of definability in a structure). There are three important points to note about the relationship between indiscernibility, and the relation \(I(A,x,y)\). First, indiscernibility need not be the relation \(I(A,x,y)\) (where \(A\) is the domain of the structure). It might be an equivalence relation \(E\) having the property that for some elements \(u,v\), of the domain, \(E(u,v)\) holds, although \(I(A,u,v)\) fails. Secondly, there is no way to “correct for” this possibility. There is no sentence or set of sentences that could be added to the list beginning with Ref and LL that would guarantee that indiscernibility coincides with \(I(A,x,y)\). This fact is usually expressed by saying that identity is not a first-order or “elementary” relation. (For a proof, see Hodges 1983.) However, in a language such as set theory (as usually interpreted) or second-order logic, in which there is a quantifier ‘all \(X\)’ permitting quantification over all subsets of a given set, \(I(A,x,y)\) is definable (Freund 2019).