Reply: This is verso good objection. However, the difference between first-order and higher-order relations is relevant here. Traditionally, similarity relations such as è ebonyflirt gratis x and y are the same color have been represented, per the way indicated sopra 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 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 show that the impression that identity is prior esatto equivalence is merely verso misimpression – due 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 relative 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 giammai sense esatto ask: “What is the mass of that one dog.” Given the possibility of change, identical objects may differ in mass. On the correspondante identity account, that means that distinct logical objects that are the same \(F\) may differ sopra mass – and may differ with respect esatto a host of other properties as well. Oscar and Oscar-minus are distinct physical objects, and therefore distinct logical objects. Distinct physical objects may differ in mass.
Objection 8: We can solve the paradox of 101 Dalmatians by appeal preciso a notion of “almost identity” (Lewis 1993). We can admit, con 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 imparfaite identity. It is per 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 \(x\) in \(A\), \([x]\) is the batteria of all \(y\) in \(A\) such that \(E(incognita, y)\); this is the equivalence class of quantitativo 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. Correlative Identity
Garantisse that \(L’\) is some fragment of \(L\) containing per subset of the predicate symbols of \(L\) and the identity symbol. Let \(M\) be verso structure for \(L’\) and suppose that some identity statement \(per = b\) (where \(a\) and \(b\) are individual constants) is true mediante \(M\), and that Ref and LL are true sopra \(M\). Now expand \(M\) to verso structure \(M’\) for a richer language – perhaps \(L\) itself. That is, garantisse we add some predicates to \(L’\) and interpret them as usual per \(M\) preciso obtain an expansion \(M’\) of \(M\). Assume that Ref and LL are true durante \(M’\) and that the interpretation of the terms \(a\) and \(b\) remains the same. Is \(verso = b\) true mediante \(M’\)? 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 in \(M’\). The indiscernibility relation defined by the identity symbol mediante \(M\) may differ from the one it defines mediante \(M’\); and durante particular, the latter may be more “fine-grained” than the former. Mediante this sense, if identity is treated as a logical constant, identity is not “language correspondante;” whereas if identity is treated as a non-logical notion, it \(is\) language relative. For this reason we can say that, treated as verso logical constant, identity is ‘unrestricted’. For example, let \(L’\) be a fragment of \(L\) containing only the identity symbol and verso solo 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 preciso 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 sicuro the effect that identity, even as formalized con 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: