It does seem sicuro esibizione, as the objector says, that identity is logically prior onesto ordinary similarity relations

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, durante the way indicated mediante the objection, as higher-order relations involving identities between higher order objects (properties). Yet this treatment may not be inevitable. Per Deutsch (1997), an attempt is made preciso 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 esibizione that the impression that identity is prior puro equivalence is merely a misimpression – due puro 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 per niente sense puro ask: “What is the mass of that one dog.” Given the possibility of change, identical objects may differ con mass. On the incomplete identity account, that means that distinct logical objects that are the same \(F\) may differ in mass – and may differ with respect puro per 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 preciso verso notion of “almost identity” (Lewis 1993). We can admit, durante 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 verso relation of indiscernibility, since it is not transitive, and so it differs from correlative identity. It is a matter of negligible difference. Per series of negligible differences can add up esatto one that is not negligible.

Let \(E\) be an equivalence relation defined on per arnesi \(A\). For \(x\) con \(A\), \([x]\) is the arnesi of all \(y\) mediante \(A\) such that \(E(interrogativo, y)\); this is the equivalence class of interrogativo determined by Di nuovo. 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. Correspondante Identity

Assure 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 mediante \(M\). Now expand \(M\) preciso verso structure \(M’\) for a richer language – perhaps \(L\) itself. That is, assure we add some predicates puro \(L’\) and interpret them as usual in \(M\) sicuro obtain an expansion \(M’\) of \(M\). Garantit that Ref and LL are true sopra \(M’\) and that the interpretation of the terms \(a\) and \(b\) remains the same. Is \(verso = b\) 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 verso non-logical symbol, then it can happen that \(per = b\) is false durante \(M’\). The indiscernibility relation defined by the identity symbol per \(M\) may differ from the one it defines in \(M’\); and per particular, the latter may be more “fine-grained” than the former. Mediante this sense, if identity is treated as verso logical constant, identity is not “language relative;” whereas if identity is treated as a non-logical notion, it \(is\) language correspondante. 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 solo one-place predicate symbol; and suppose that the identity symbol is treated as non-logical. The norma

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 discussion and one at the end, and he easily disposes of both. Per between he develops an interesting and influential argument preciso the effect that identity, even as formalized con the system FOL\(^=\), is relative identity. However, Geach takes himself preciso have shown, by this argument, that absolute identity does not exist. At the end of his initial presentation of the argument con his 1967 paper, Geach remarks:

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