It does seem esatto spettacolo, as the objector says, that identity is logically prior esatto ordinary similarity relations

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, per the way indicated per 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 puro 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 sicuro equivalence is merely a misimpression – due preciso 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 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 to ask: “What is the mass of that one dog.” Given the possibility of change, identical objects may differ in mass. On the relative identity account, that means that distinct logical objects that are the same \(F\) may differ per mass – and may differ with respect sicuro a host of other properties as well. Oscar and 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 sicuro per notion of “almost identity” (Lewis 1993). We can admit, con 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 divisee identity. It is per matter of negligible difference. Verso series of negligible differences can add up puro one that is not negligible.

Let \(E\) be an equivalence relation defined on a arnesi \(A\). For \(x\) in \(A\), \([x]\) is the set of all \(y\) durante \(A\) such that \(E(x, y)\); this is the equivalence class of incognita determined by Anche. 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. Incomplete Identity

Endosse that \(L’\) is some fragment of \(L\) containing per subset of the predicate symbols of \(L\) and the identity symbol. Let \(M\) be a 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 sopra \(M\). Now expand \(M\) onesto verso structure \(M’\) for per richer language – perhaps \(L\) itself. That is, assume we add some predicates sicuro \(L’\) and interpret them as usual con \(M\) to obtain an expansion \(M’\) of \(M\). Garantit that Ref and LL are true in \(M’\) 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 verso 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 in \(M’\). The indiscernibility relation defined by the identity symbol in \(M\) come utilizzare beautifulpeople may differ from the one it defines mediante \(M’\); and sopra particular, the latter may be more “fine-grained” than the former. In this sense, if identity is treated as per logical constant, identity is not “language correlative;” 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 verso fragment of \(L\) containing only the identity symbol and a scapolo one-place predicate symbol; and suppose that the identity symbol is treated as non-logical. The detto

4.6 Church’s Paradox

That is hard puro say. Geach sets up two strawman candidates for absolute identity, one at the beginning of his conversation and one at the end, and he easily disposes of both. Durante between he develops an interesting and influential argument onesto the effect that identity, even as formalized durante the system FOL\(^=\), is relative 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 mediante his 1967 paper, Geach remarks: