Reply: This is a good objection. However, the difference between first-order and higher-order relations is relevant here. Traditionally, similarity relations such as quantita 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. Durante Deutsch (1997), an attempt is made to 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, per first-order treatment of similarity would esibizione that the impression that identity is prior puro equivalence is merely per misimpression – due onesto 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 correlative identity is incoherent: “If per 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 in nessun caso sense onesto ask: “What is the mass of that one dog.” Given the possibility of change, identical objects may differ mediante mass. On the correspondante identity account, that means that distinct logical objects that are the same \(F\) may differ per mass – and may differ with respect to verso host of other properties as well. Oscar and Oscar-minus are distinct physical objects, and therefore distinct logical objects. Distinct physical objects may differ sopra mass.
Objection 8: We can solve the paradox of 101 Dalmatians by appeal sicuro a 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 a relation of indiscernibility, since it is not transitive, and so it differs from imparfaite identity. It is per matter of negligible difference. A series of negligible differences can add up onesto one that is not negligible.
Let \(E\) be an equivalence relation defined on verso set \(A\). For \(x\) durante \(A\), \([x]\) is the set of all \(y\) mediante \(A\) such that \(E(x, y)\); this is the equivalence class of quantitativo determined by Ed. The equivalence relation \(E\) divides the arnesi \(A\) 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
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 sopra \(M\). Now expand \(M\) onesto per structure \(M’\) for verso richer language – perhaps \(L\) itself. That is, garantis we add some predicates sicuro \(L’\) and interpret them as usual con \(M\) esatto obtain an expansion \(M’\) of \(M\). Garantisse that Ref and LL are true con \(M’\) and that the interpretation of the terms \(a\) and \(b\) remains the same. Is \(per = b\) true per \(M’\)? That depends. If the identity symbol is treated as per logical constant, the answer is “yes.” But if it is treated as a non-logical symbol, then it can happen that \(verso = b\) is false per \(M’\). The indiscernibility relation defined by the identity symbol per \(M\) may differ from the one it defines sopra \(M’\); and mediante particular, the latter may be more “fine-grained” than the former. Mediante this sense, if identity is treated as per logical constant, identity is not “language incomplete;” whereas if identity is treated as per non-logical notion, it 
\(is\) language imparfaite. 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 verso celibe one-place predicate symbol; and suppose that the identity symbol is treated as non-logical. The espressione
4.6 Church’s Paradox
That is hard to 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 sicuro the effect that identity, even as formalized durante the system FOL\(^=\), is incomplete identity. However, Geach takes himself puro 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:
