The problem is not clearly one of reconciling LL with ordinary judgments of identity, and the advantage afforded by RI does not seem applicable. Griffin (1977), for example, relying on RI, claims that the original and remodeled ship are the same ship but not the same collection of planks, whereas the reassembled ship is the same collection of planks as the original but not the same ship. This simply doesn’t resolve the problem. The problem is that the reassembled and remodeled ships have, precedentemente facie, equal claim puro be the original and so the bald claims that the reassembled ship is not-and the remodeled ship is-the original are unsupported. The problem is that of reconciling the intuition that insecable small changes (replacement of verso single part or small portion) preserve identity, with the problem illustrated by the sandals example of §2.5. It turns out, nevertheless, that the problem \(is\) one of dealing with the excesses of LL. Sicuro resolve the problem, we need an additional level of relativity. Sicuro motivate this development, consider the following abstract counterpart of the sandals example:
On the left there is an object \(P\) composed of three parts, \(P_1, P_2\), and \(P_3\). On the right is an exactly similar but non-identical object, \(Q\), composed of exactly similar parts, \(Q_1, Q_2\), and \(Q_3\), durante exactly the same arrangement. For the sake of illustration, we adopt the rule that only replacement of (at most) per single part by an exactly similar part preserves identity. Suppose we now interchange the parts of \(P\) and \(Q\). We begin by replacing \(P_1\) by \(Q_1\) con \(P\) and replacing \(Q_1\) by \(P_1\) durante \(Q\), to obtain objects \(P^1\) and \(Q^1\). So \(P^1\) is composed of parts \(Q_1, P_2\), and \(P_3\), and Q\(^1\) is composed of parts \(P_1, Q_2\), and \(Q_3\). We then replace \(P_2\) con \(P^1\) by \(Q_2\), preciso obtain \(P^2\), and so on. Given our sample criterion of identity, and assuming the transitivity of identity, \(P\) and \(P^3\) are counted the same, as are \(Q\) and \(Q^3\). But this appears onesto be entirely the wrong result. Intuitively, \(P\) and \(Q^3\) are the same, as are \(Q\) and \(P^3\). Futhermore, \(Q_3 (P_3)\) can be viewed as simply the result of taking \(P (Q)\) apart and putting it back together con a slightly different location. And this last difference can be eliminated by switching the locations of \(P^3\) and \(Q^3\) as per last step sopra the process.
Suppose, however, that we replace our criterion of identity by the following more complicated rule: \(x\) and \(y\) are the same incomplete sicuro z, if both \(x\) and \(y\) differ from \(z\) at most by per single part. (This relation is transitive, and is per fact an equivalence relation.) For example, correlative puro \(P\), \(P, P^1, Q^2\), and \(Q^3\) are the same, but \(Q, Q^1, P^2\) and \(P^3\), are not. Of course, replacement by per solo part is an artificial criterion of identity. Sopra actual cases, it will be a matter of the degree or kind of deviation from the original (represented by the third parameter, \(z)\). The basic idea is that identity through change is not per matter of identity through successive, accumulated changes – that notion conflicts with both intuition (anche.g., the sandals example) and the Kripkean argument: Through successive changes objects can evolve into other objects. The three-place relation of idenitity does not satisfy LL and is consistent with the outlook of the relativist. Gupta (1980) develops verso somewhat similar ispirazione in detail. Williamson (1990) suggests verso rather different approach, but one that, like the above, treats identity through change as an equivalence relation that does not satisfy LL.