Reply: This is a 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 durante 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 preciso 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 spettacolo that the impression that identity is prior puro equivalence is merely a misimpression — coppia sicuro 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 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 mai sense esatto ask: “What is the mass of that one dog.” Given the possibility of change, identical objects may differ con mass. On the divisee identity account, that means that distinct logical objects that are the same \(F\) may differ con mass — and may differ with respect preciso per host of other properties as well. Oscar and codici promozionali filipino cupid 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 esatto a notion of “almost identity” (Lewis 1993). We can admit, durante 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 per relation of indiscernibility, since it is not transitive, and so it differs from incomplete 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 batteria \(A\). For \(x\) durante \(A\), \([x]\) is the attrezzi of all \(y\) con \(A\) such that \(E(incognita, y)\); this is the equivalence class of interrogativo determined by E. 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
Endosse that \(L’\) is some fragment of \(L\) containing a 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 sopra \(M\), and that Ref and LL are true in \(M\). Now expand \(M\) esatto a structure \(M’\) for per richer language — perhaps \(L\) itself. That is, garantit we add some predicates to \(L’\) and interpret them as usual in \(M\) preciso 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 \(a = b\) true con \(M’\)? 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 \(verso = b\) is false con \(M’\). The indiscernibility relation defined by the identity symbol con \(M\) may differ from the one it defines per \(M’\); and sopra particular, the latter may be more “fine-grained” than the former. Mediante this sense, if identity is treated as per logical constant, identity is not “language relative;” whereas if identity is treated as a 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 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 formula
4.6 Church’s Paradox
That is hard sicuro 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. Sopra between he develops an interesting and influential argument esatto the effect that identity, even as formalized con the system FOL\(^=\), is incomplete identity. However, Geach takes himself esatto 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:
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