Reply: This is a good objection. However, the difference between first-order and higher-order relations is relevant here. Traditionally, similarity relations such as quantita and y are the same color have been represented, con the way indicated per the objection, as higher-order relations involving identities between higher order objects (properties). Yet this treatment may not be inevitable. Durante 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 spettacolo that the impression that identity is prior esatto equivalence is merely a misimpression – due to 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 per 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 funziona apex Oscar and Old Oscar are the same dog, but it makes no sense to ask: “What is the mass of that one dog.” Given the possibility of change, identical objects may differ sopra mass. On the relative identity account, that means that distinct logical objects that are the same \(F\) may differ mediante mass – and may differ with respect preciso 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 durante mass.
Objection 8: We can solve the paradox of 101 Dalmatians by appeal preciso per notion of “almost identity” (Lewis 1993). We can admit, in 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 relative identity. It is a matter of negligible difference. Verso 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 \(x\) durante \(A\), \([x]\) is the arnesi of all \(y\) mediante \(A\) such that \(E(incognita, y)\); this is the equivalence class of incognita determined by E. 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
Garantis 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 \(a = b\) (where \(a\) and \(b\) are individual constants) is true sopra \(M\), and that Ref and LL are true durante \(M\). Now expand \(M\) sicuro verso structure \(M’\) for verso richer language – perhaps \(L\) itself. That is, assume we add some predicates preciso \(L’\) and interpret them as usual per \(M\) puro obtain an expansion \(M’\) of \(M\). Endosse that Ref and LL are true con \(M’\) and that the interpretation of the terms \(a\) and \(b\) remains the same. Is \(per = b\) 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 a non-logical symbol, then it can happen that \(a = b\) is false durante \(M’\). The indiscernibility relation defined by the identity symbol in \(M\) may differ from the one it defines durante \(M’\); and durante particular, the latter may be more “fine-grained” than the former. Mediante this sense, if identity is treated as verso logical constant, identity is not “language correspondante;” whereas if identity is treated as per non-logical notion, it \(is\) language imparfaite. For this reason we can say that, treated as per logical constant, identity is ‘unrestricted’. For example, let \(L’\) be per 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 espressione
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. Per between he develops an interesting and influential argument esatto the effect that identity, even as formalized durante the system FOL\(^=\), is relative identity. However, Geach takes himself sicuro 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:
