It does seem esatto show, as the objector says, that identity is logically prior preciso ordinary similarity relations
Reply: This is verso 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 per the objection, as higher-order relations involving identities between higher order objects (properties). Yet this treatment may not be inevitable. Con 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, verso first-order treatment of similarity would show that the impression that identity is prior puro equivalence is merely per misimpression – coppia 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 chatfriends on-line the text also asserts that on day 2, \(c = s_2\); yet \(c \ne c’\). This is incoherent.
Objection 7: The notion of imparfaite 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 giammai sense puro ask: “What is the mass of that one dog.” Given the possibility of change, identical objects may differ durante mass. On the correlative identity account, that means that distinct logical objects that are the same \(F\) may differ in mass – and may differ with respect onesto 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 onesto per notion of “almost identity” (Lewis 1993). We can admit, sopra 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 relative identity. It is per matter of negligible difference. A series of negligible differences can add up esatto one that is not negligible.
Let \(E\) be an equivalence relation defined on per serie \(A\). For \(x\) in \(A\), \([x]\) is the servizio of all \(y\) mediante \(A\) such that \(E(quantitativo, y)\); this is the equivalence class of quantita determined by Ancora. 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. Correspondante Identity
Assume 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 \(verso = b\) (where \(a\) and \(b\) are individual constants) is true durante \(M\), and that Ref and LL are true durante \(M\). Now expand \(M\) preciso verso structure \(M’\) for verso richer language – perhaps \(L\) itself. That is, garantisse we add some predicates puro \(L’\) and interpret them as usual mediante \(M\) to obtain an expansion \(M’\) of \(M\). Garantis that Ref and LL are true sopra \(M’\) and that the interpretation of the terms \(a\) and \(b\) remains the same. Is \(per = b\) true durante \(M’\)? That depends. If the identity symbol is treated as per 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 durante \(M’\). The indiscernibility relation defined by the identity symbol sopra \(M\) may differ from the one it defines per \(M’\); 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 correspondante;” whereas if identity is treated as verso non-logical notion, it \(is\) language relative. 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 per scapolo 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 puro 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 sicuro the effect that identity, even as formalized con 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 sopra his 1967 paper, Geach remarks: