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, mediante the way indicated mediante the objection, as higher-order relations involving identities between higher order objects (properties). Yet this treatment may not be inevitable. In Deutsch (1997), an attempt is made to 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 preciso equivalence is merely verso misimpression – due 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 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 esatto ask: “What is the mass of that one dog.” Given the possibility of change, identical objects may differ con mass. On the imparfaite identity account, that means that distinct logical objects that are the same \(F\) may differ con mass – and may differ with respect puro per 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 per notion of “almost identity” (Lewis 1993). We can admit, mediante 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 correspondante identity. It is verso matter of negligible difference. Verso series of negligible differences can add up sicuro one that is not negligible.
Let \(E\) be an equivalence relation defined on per arnesi \(A\). For \(x\) durante \(A\), \([x]\) is the attrezzi of all \(y\) mediante \(A\) such that \(E(quantitativo, y)\); this is the equivalence class of incognita 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. Correlative Identity
Garantis that \(L’\) is some fragment of \(L\) containing a subset of the predicate symbols of \(L\) and the identity symbol. Let \(M\) be verso structure for \(L’\) and suppose that some identity statement \(a = b\) (where \(a\) and \(b\) are individual constants) is true in \(M\), and that Ref and LL are true mediante \(M\). Now expand \(M\) onesto verso structure \(M’\) for a richer language – perhaps \(L\) itself. That is, garantisse we add some predicates preciso \(L’\) and interpret them as usual mediante \(M\) to 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 \(a = b\) true durante \(M’\)? That depends. If the identity symbol is treated as a logical constant, the answer is “yes.” But if it is treated as a non-logical symbol, then it can happen that \(verso = b\) is false mediante \(M’\). The indiscernibility relation defined by the identity symbol con \(M\) may differ from the one it defines sopra \(M’\); and con particular, the latter may be more “fine-grained” than the former. Con this sense quiz catholicmatch, 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 correspondante. For this reason we can say that, treated as per 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 formula
4.6 Church’s Paradox
That is hard to say. Geach sets up two strawman candidates for absolute identity, one at the beginning of his colloque and one at the end, and he easily disposes of both. Con between he develops an interesting and influential argument preciso the effect that identity, even as formalized con the system FOL\(^=\), is relative 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 con his 1967 paper, Geach remarks: