Reply: This is verso good objection. However, the difference between first-order and higher-order relations is relevant here. Traditionally, similarity relations such as incognita and y are the same color have been represented, con the way indicated in the objection, as higher-order relations involving identities between higher order objects (properties). Yet this treatment may not be inevitable. Sopra Deutsch (1997), an attempt is made esatto 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 esatto equivalence is merely per misimpression - due esatto 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 divisee identity is incoherent: “If verso cat and one of its gleeden sito di incontri 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 no sense sicuro 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 mediante mass - and may differ with respect preciso 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 esatto per notion of “almost identity” (Lewis 1993). We can admit, per 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 divisee identity. It is verso matter of negligible difference. A series of negligible differences can add up puro one that is not negligible.
Let \(E\) be an equivalence relation defined on per arnesi \(A\). For \(x\) per \(A\), \([x]\) is the serie of all \(y\) mediante \(A\) such that \(E(quantitativo, y)\); this is the equivalence class of incognita determined by E. The equivalence relation \(E\) divides the batteria \(A\) 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. Imparfaite Identity
Garantis that \(L'\) is some fragment of \(L\) containing verso subset of the predicate symbols of \(L\) and the identity symbol. Let \(M\) be per 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 a structure \(M'\) for a richer language - perhaps \(L\) itself. That is, garantisse we add some predicates puro \(L'\) and interpret them as usual mediante \(M\) sicuro 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 \(per = b\) true in \(M'\)? That depends. If the identity symbol is treated as a 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 mediante \(M'\). The indiscernibility relation defined by the identity symbol durante \(M\) may differ from the one it defines sopra \(M'\); and mediante particular, the latter may be more “fine-grained” than the former. Durante this sense, if identity is treated as per logical constant, identity is not “language incomplete;” whereas if identity is treated as a non-logical notion, it \(is\) language incomplete. For this reason we can say that, treated as a logical constant, identity is ‘unrestricted'. For example, let \(L'\) be per 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 formula
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 discussion 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 correspondante 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 sopra his 1967 paper, Geach remarks:
