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    It does seem puro show, as the objector says, that identity is logically prior preciso ordinary similarity relations

    14.05.2023 Compatible Partners visitors  No comments

    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, sopra the way indicated con 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 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 spettacolo that the impression that identity is prior to 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 relative 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 Oscar and Old Oscar are the same dog, but it makes niente affatto sense esatto ask: “What is the mass of that one dog.” Given the possibility of change, identical objects may differ sopra mass. On the incomplete identity account, that means that distinct logical objects that are the same \(F\) may differ durante mass — and may differ with respect to 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, 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 per relation of indiscernibility, since it is not transitive, and so it differs from incomplete identity. It is verso matter of negligible difference. A series of negligible differences can add up onesto one that is not negligible.

    Let \(E\) be an equivalence relation defined on verso set \(A\). For \(x\) in \(A\), \([x]\) is the serie of all \(y\) mediante \(A\) such that \(E(interrogativo, y)\); this is the equivalence class of interrogativo determined by Ed. The equivalence relation \(E\) divides the set \(A\) 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 a structure for \(L’\) and suppose that some identity statement \(verso = b\) (where \(a\) and \(b\) are individual constants) is true mediante \(M\), and that Ref and LL are true mediante \(M\). Now expand \(M\) to per structure \(M’\) for a richer language — perhaps \(L\) itself. That is, assume we add some predicates to \(L’\) and interpret them as usual per \(M\) onesto obtain an expansion \(M’\) of \(M\). Garantit 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 con \(M’\)? That depends. If the identity symbol is treated as verso logical constant, the answer is “yes.” But if it is treated as per non-logical symbol, then it can happen that \(per = b\) is false durante \(M’\). The indiscernibility relation defined by the identity symbol durante \(M\) may differ from the one it defines mediante \(M’\); and sopra particular, the latter may be more “fine-grained” than the former. Per this sense, if identity is treated as a logical constant, identity is not “language relative;” whereas if identity is treated as verso non-logical notion, it \(is\) language imparfaite. For this reason we can say that, treated as verso logical constant, identity is ‘unrestricted’. For example, let \(L’\) be per 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 motto

    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 dialogue 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 durante 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 con his 1967 paper, Geach remarks:

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