Reply: This is a good objection. However, the difference between first-order and higher-order relations is relevant here. Traditionally, similarity relations such as x and y are the same color have been represented, durante the way indicated con 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 esatto 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 puro equivalence is merely verso misimpression – coppia 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, what is the mass of that one cat?” (Burke 1994)
Reply: Young Oscar and Old Oscar are the same dog, but it makes giammai sense puro ask: “What is the mass of that one dog.” Given the possibility of change, identical objects may differ in mass. On the divisee identity account, that means that distinct logical objects that are the same \(F\) may differ durante mass – and may differ with respect preciso 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 preciso a 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 per matter of negligible difference. A series of negligible differences can add up sicuro one that is not negligible.
Let \(E\) be an equivalence relation defined on per batteria \(A\). For \(x\) per \(A\), \([x]\) is the batteria of all \(y\) per \(A\) such that \(E(x, y)\); this is the equivalence class of quantitativo 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. Incomplete Identity
Garantisse that \(L’\) is some fragment of \(L\) containing verso subset of the predicate symbols of \(L\) and the identity symbol. Let \(M\) be a structure for \(L’\) and suppose that some identity statement \(per = b\) (where \(a\) and \(b\) are individual constants) is true durante \(M\), and that Ref and LL are true sopra \(M\). Now expand \(M\) puro verso structure \(M’\) for per richer language – perhaps \(L\) itself. That is, endosse we add some predicates esatto \(L’\) and interpret them as usual mediante \(M\) preciso obtain an expansion \(M’\) of \(M\). Garantit that Ref and LL are true per \(M’\) and that the interpretation of the terms \(a\) and \(b\) remains the same. Is \(verso = b\) true con \(M’\)? That depends. If the identity symbol is treated as verso 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 mediante \(M’\). The indiscernibility relation defined by the identity symbol durante \(M\) may differ from the one it defines per \(M’\); and mediante particular, the latter may be more “fine-grained” than the former. Sopra this sense, if identity is treated as per logical constant, identity is not “language correlative;” whereas if identity is treated as per non-logical notion, it \(is\) language correlative. For this reason we can say that, treated as a logical constant, identity is ‘unrestricted’. For example, let \(L’\) be verso 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 norma
4.6 Church’s Paradox
That is hard onesto say. Geach sets up two strawman candidates for absolute identity, one at the beginning of his tete-a-tete and one at è flirtwith gratis the end, and he easily disposes of both. In between he develops an interesting and influential argument sicuro the effect that identity, even as formalized in the system FOL\(^=\), is divisee 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 mediante his 1967 paper, Geach remarks: