 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 \(Q_3\), in exactly the same arrangement. For the sake of illustration, we adopt the rule that only replacement of (at most) a 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 \(Q_1\) in \(P\) and replacing \(Q_1\) by \(P_1\) in \(Q\), 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 \(P_2\) in \(P^1\) by \(Q_2\), to 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 to be entirely the wrong result. Intuitively, \(P\) and \(Q^3\) are the same, as are \(Q\) and \(P^3\). For \(P\) and \(Q^3\) are composed of exactly the same parts put together in exactly the same way, and similarly for \(Q\) and \(P^3\). Furthermore, \(Q_3 (P_3)\) can be viewed as simply the result of taking \(P (Q)\) apart and putting it back together in a slightly different location. And this last difference can be eliminated by switching the locations of \(P^3\) and \(Q^3\) as a last step in the process.