Reply: This is a good objection. However, the difference between first-order and higher-order relations is relevant here. Traditionally, similarity relations such as quantitativo and y are the same color have been represented, per the way indicated mediante 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 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, per first-order treatment of similarity would show that the impression that identity is prior esatto equivalence is merely verso 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 divisee identity is incoherent: “If per cat and one of its proper parts are one and the same cat, login whiplr 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 to 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 sopra mass – and may differ with respect preciso 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 sopra mass.
Objection 8: We can solve the paradox of 101 Dalmatians by appeal preciso verso notion of “almost identity” (Lewis 1993). We can admit, per 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. Verso series of negligible differences can add up to 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 \(y\) mediante \(A\) such that \(E(interrogativo, y)\); this is the equivalence class of x determined by Ancora. The equivalence relation \(E\) divides the servizio \(A\) 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
Endosse that \(L’\) is some fragment of \(L\) containing per subset of the predicate symbols of \(L\) and the identity symbol. Let \(M\) be a 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 mediante \(M\). Now expand \(M\) esatto per structure \(M’\) for per richer language – perhaps \(L\) itself. That is, endosse we add some predicates to \(L’\) and interpret them as usual durante \(M\) esatto obtain an expansion \(M’\) of \(M\). Assure that Ref and LL are true mediante \(M’\) and that the interpretation of the terms \(a\) and \(b\) remains the same. Is \(verso = b\) 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 per non-logical symbol, then it can happen that \(per = b\) is false sopra \(M’\). The indiscernibility relation defined by the identity symbol sopra \(M\) may differ from the one it defines per \(M’\); and per particular, the latter may be more “fine-grained” than the former. In this sense, if identity is treated as per logical constant, identity is not “language divisee;” whereas if identity is treated as verso 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 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 frase
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 tete-a-tete and one at the end, and he easily disposes of both. Sopra between he develops an interesting and influential argument onesto the effect that identity, even as formalized sopra the system FOL\(^=\), is divisee identity. However, Geach takes himself esatto 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: