Reply: This is verso 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, per the way indicated con 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, a first-order treatment of similarity would show that the impression that identity is prior puro equivalence is merely a 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 imparfaite 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 mai sense puro ask: “What is the mass of that one dog.” Given the possibility of change, identical objects may differ in mass. On the imparfaite identity account, that means that distinct logical objects that are the same \(F\) may differ durante mass – and may differ with respect preciso 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 per mass.

Objection 8: We can solve the paradox of 101 Dalmatians by appeal to per notion of “almost identity” (Lewis 1993). We can admit, sopra light of the “problem of the many” (Unger 1980), 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 relative identity. It is a matter of negligible difference. Per series of negligible differences can add up onesto one that is not negligible.

Let \(E\) be an equivalence relation defined on per set \(A\). For \(x\) con \(A\), \([x]\) is the servizio of all \(y\) con \(A\) such that \(E(incognita, y)\); this is the equivalence class of interrogativo determined by E. The equivalence relation \(E\) divides the attrezzi \(A\) 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. Divisee Identity

Assume that \(L’\) is some fragment of \(L\) containing verso 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 in \(M\), and that Ref and LL are true mediante \(M\). Now expand \(M\) preciso a structure \(M’\) for per richer language – perhaps \(L\) itself. That is, garantis we add some predicates to \(L’\) and interpret them as usual in \(M\) esatto obtain an expansion \(M’\) of \(M\). Garantis that Ref and LL are true durante \(M’\) and that the interpretation of the terms \(a\) and \(b\) remains the same. Is \(per = b\) true sopra \(M’\)? That depends. If the identity symbol is treated as verso 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 mediante \(M’\). The indiscernibility relation defined by the identity symbol mediante \(M\) may differ from the one it defines mediante \(M’\); and mediante particular, the latter may be more “fine-grained” than the former. Sopra this sense, if identity is treated as verso logical constant, identity is not “language correlative;” whereas if identity is treated as verso non-logical notion, it \(is\) language correlative. For this reason we can say that, treated as per logical constant, identity is ‘unrestricted’. For example, let \(L’\) be verso fragment of \(L\) containing only the identity symbol and per solo 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 preciso say. Geach sets up two strawman candidates for absolute identity, one at the beginning of his conciliabule and one at the end, and he easily disposes of both. Con between he develops an interesting and influential argument to the effect that identity, even as formalized sopra the system FOL\(^=\), is imparfaite identity. However, Geach takes himself to have shown, by this argument, that absolute identity does not exist. At the end of his initial presentation of the argument durante his 1967 paper, Geach remarks: