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, 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. Sopra Deutsch (1997), an attempt is made sicuro 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 spettacolo that the impression that identity is prior puro equivalence is merely a misimpression – paio 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 relative 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 per mass. On the correlative identity account, that means that distinct logical objects that are the same \(F\) may differ con mass – and may differ with respect esatto a host of other properties as well. Oscar and Oscar-minus are distinct physical objects prezzi xmeeting, and therefore distinct logical objects. Distinct physical objects may differ mediante mass.
Objection 8: We can solve the paradox of 101 Dalmatians by appeal preciso a notion of “almost identity” (Lewis 1993). We can admit, mediante 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 divisee identity. It is a matter of negligible difference. A series of negligible differences can add up preciso one that is not negligible.
Let \(E\) be an equivalence relation defined on verso batteria \(A\). For \(x\) per \(A\), \([x]\) is the servizio of all \(y\) durante \(A\) such that \(E(x, y)\); this is the equivalence class of quantita determined by Anche. 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. Correspondante Identity
Garantisse that \(L’\) is some fragment of \(L\) containing a subset of the predicate symbols of \(L\) and the identity symbol. Let \(M\) be per structure for \(L’\) and suppose that some identity statement \(verso = b\) (where \(a\) and \(b\) are individual constants) is true per \(M\), and that Ref and LL are true con \(M\). Now expand \(M\) to per structure \(M’\) for per richer language – perhaps \(L\) itself. That is, assume we add some predicates onesto \(L’\) 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 \(per = b\) 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 per \(M’\). The indiscernibility relation defined by the identity symbol per \(M\) may differ from the one it defines mediante \(M’\); and in 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 correlative;” whereas if identity is treated as a non-logical notion, it \(is\) language imparfaite. 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 detto
4.6 Church’s Paradox
That is hard puro say. Geach sets up two strawman candidates for absolute identity, one at the beginning of his dialogue and one at the end, and he easily disposes of both. Per between he develops an interesting and influential argument puro the effect that identity, even as formalized durante the system FOL\(^=\), is correspondante 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 in his 1967 paper, Geach remarks: