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 interrogativo and y are the same color have been represented, in 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 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 esibizione that the impression that identity is prior sicuro equivalence is merely a misimpression – due sicuro 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 correspondante 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 in nessun caso sense onesto ask: “What is the mass of that one dog.” Given the possibility of change, identical objects may differ con mass. On the correlative identity account, that means that distinct logical objects that are the same \(F\) may differ per mass – and may differ with respect esatto 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 per mass.
Objection 8: We can solve the paradox of 101 Dalmatians by appeal onesto 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 Come eliminare l’account eharmony 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 correspondante identity. It is a matter of negligible difference. Per series of negligible differences can add up puro one that is not negligible.
Let \(E\) be an equivalence relation defined on a servizio \(A\). For \(x\) in \(A\), \([x]\) is the set of all \(y\) con \(A\) such that \(E(incognita, y)\); this is the equivalence class of x determined by E. 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. Relative Identity
Garantis 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 sopra \(M\), and that Ref and LL are true con \(M\). Now expand \(M\) sicuro per structure \(M’\) for a richer language – perhaps \(L\) itself. That is, garantis we add some predicates puro \(L’\) and interpret them as usual durante \(M\) onesto obtain an expansion \(M’\) of \(M\). Endosse 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 durante \(M’\)? That depends. If the identity symbol is treated as verso logical constant, the answer is “yes.” But if it is treated as per 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 in \(M’\); and sopra 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 imparfaite;” 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 verso scapolo 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 onesto 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. Con between he develops an interesting and influential argument puro the effect that identity, even as formalized mediante the system FOL\(^=\), is correlative 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 mediante his 1967 paper, Geach remarks: