Reply: This is a good objection. However, the difference between first-order and higher-order relations is relevant here. Traditionally, similarity relations such as incognita and y are the same color have been represented, sopra the way indicated sopra the objection, as higher-order relations involving identities between higher order objects (properties). Yet this treatment may not be inevitable. Con Deutsch (1997), an attempt is made onesto treat similarity relations of the form ‘\(x\) and \(y\) are the same \(F\)’ (where \(F\) is adjectival) as primitive, first-order, purely logical relations (see also Williamson 1988). If successful, verso first-order treatment of similarity would esibizione that the impression that identity is prior esatto equivalence is merely verso misimpression – paio sicuro the assumption that the usual higher-order account of similarity relations is the only option.

Objection 6: If on day 3, \(c’ = s_2\), as the text asserts, then by NI, the same is true on day 2. But the text also asserts that on day 2, \(c = s_2\); yet \(c \ne c’\). This is incoherent.

Objection 7: The notion of divisee identity is incoherent: “If verso cat and one of its proper parts are one and the same cat, what is the mass of that one cat?” (Burke 1994)

Reply: Young Oscar and Old Oscar are the same dog, but it makes giammai sense puro ask: “What is the mass of that one dog.” Given the possibility of change, identical objects may differ per mass. On the relative identity account, that means that distinct logical objects that are the same \(F\) may differ durante mass – and may differ with respect puro a host of other properties as well. Oscar and Oscar-minus are distinct physical objects, and therefore distinct logical objects. Distinct physical objects may differ mediante mass.

Objection 8: We can solve the paradox of 101 Dalmatians by appeal puro a notion of “almost identity” (Lewis 1993). We can admit, con light of the “problem of the many” (Unger 1980) Come eliminare l’account crossdresser heaven, that the 101 dog parts are dogs, but we can also affirm that the 101 dogs are not many; for they are “almost one.” Almost-identity is not per relation of indiscernibility, since it is not transitive, and so it differs from imparfaite identity. It is verso matter of negligible difference. Per series of negligible differences can add up sicuro one that is not negligible.

Let \(E\) be an equivalence relation defined on per batteria \(A\). For \(x\) con \(A\), \([x]\) is the arnesi of all \(y\) per \(A\) such that \(E(incognita, y)\); this is the equivalence class of x determined by E. The equivalence relation \(E\) divides the batteria \(A\) into mutually exclusive equivalence classes whose union is \(A\). The family of such equivalence classes is called ‘the partition of \(A\) induced by \(E\)’.

## 3. Correlative Identity

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

## 4.6 Church’s Paradox

That is hard onesto say. Geach sets up two strawman candidates for absolute identity, one at the beginning of his conversation and one at the end, and he easily disposes of both. Mediante between he develops an interesting and influential argument preciso the effect that identity, even as formalized in the system FOL\(^=\), is incomplete identity. However, Geach takes himself sicuro have shown, by this argument, that absolute identity does not exist. At the end of his initial presentation of the argument per his 1967 paper, Geach remarks: