 Assume that \(L'\) is some fragment of \(L\) containing a subset of the predicate symbols of \(L\) and the identity symbol. Let \(M\) be a structure for \(L'\) and suppose that some identity statement \(a = b\) (where \(a\) and \(b\) are individual constants) is true in \(M\), and that Ref and LL are true in \(M\). Now expand \(M\) to a structure \(M'\) for a richer language — perhaps \(L\) itself. That is, assume we add some predicates to \(L'\) and interpret them as usual in \(M\) to obtain an expansion \(M'\) of \(M\). Assume that Ref and LL are true in \(M'\) and that the interpretation of the terms \(a\) and \(b\) remains the same. Is \(a = b\) true in \(M'\)? That depends. If the identity symbol is treated as a logical constant, the answer is “yes.” But if it is treated as a non-logical symbol, then it can happen that \(a = b\) is false in \(M'\). The indiscernibility relation defined by the identity symbol in \(M\) may differ from the one it defines in \(M'\); and in particular, the latter may be more “fine-grained” than the former. In this sense, if identity is treated as a logical constant, identity is not “language relative;” whereas if identity is treated as a non-logical notion, it \(is\) language relative. For this reason we can say that, treated as a logical constant, identity is ‘unrestricted’. For example, let \(L'\) be a fragment of \(L\) containing only the identity symbol and a single one-place predicate symbol; and suppose that the identity symbol is treated as non-logical. The formula