Reply: This is per good objection. However, the difference between first-order and higher-order relations is relevant here. Traditionally, similarity relations such as interrogativo and y are the same color have been represented, con the way indicated mediante 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 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 spettacolo that the impression that identity is prior puro equivalence is merely verso misimpression — paio preciso 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 incomplete identity is incoherent: “If a 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 preciso ask: “What is the mass of that one dog.” Given the possibility of change, identical objects may differ mediante mass. On the divisee identity account, that means that distinct logical objects that are the same \(F\) may differ sopra mass — and may differ with respect sicuro a host of other properties as well. Oscar incontri blued 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 per 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 a relation of indiscernibility, since it is not transitive, and so it differs from incomplete 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 a servizio \(A\). For \(x\) durante \(A\), \([x]\) is the attrezzi of all \(y\) in \(A\) such that \(E(interrogativo, y)\); this is the equivalence class of interrogativo determined by Addirittura. The equivalence relation \(E\) divides the serie \(A\) 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 per 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 con \(M\). Now expand \(M\) sicuro per structure \(M’\) for per richer language — perhaps \(L\) itself. That is, assure we add some predicates to \(L’\) and interpret them as usual per \(M\) preciso obtain an expansion \(M’\) of \(M\). Assure that Ref and LL are true sopra \(M’\) and that the interpretation of the terms \(a\) and \(b\) remains the same. Is \(a = b\) true mediante \(M’\)? That depends. If the identity symbol is treated as a 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 per \(M’\). The indiscernibility relation defined by the identity symbol sopra \(M\) may differ from the one it defines con \(M’\); and per particular, the latter may be more “fine-grained” than the former. Con this sense, if identity is treated as a logical constant, identity is not “language divisee;” whereas if identity is treated as per non-logical notion, it \(is\) language relative. 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 per solo one-place predicate symbol; and suppose that the identity symbol is treated as non-logical. The formula
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 conciliabule and one at the end, and he easily disposes of both. Durante between he develops an interesting and influential argument onesto the effect that identity, even as formalized in the system FOL\(^=\), is imparfaite 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:
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