 Chandler’s transitivity argument can be adapted to produce a puzzle that is like those discussed in Sections 4.1–4.2 above in that it involves the danger of ‘bare identities’, a puzzle that Salmon (1982) has called ‘The Four Worlds Paradox’. To illustrate the puzzle: suppose that the actual world (w1) contains a bicycle, a, that is (actually) originally composed of A1 + B1 + C1, and suppose that there is a possible world, w5, containing a bicycle, b (not identical with a), that is originally composed (in w5) of A2 + B2 + C1 (where A1 ≠ A2 and B1 ≠ B2). Then, it seems, the application of the tolerance principle to each of a and b may generate two further possible worlds, in one of which (w6) there is a bicycle with the original composition A1 + B2 + C1 that is identical with a, and in the other of which (w7) there is a bicycle with the original composition A1 + B2 + C1 that is identical with b. Since there need apparently be no further difference between the intrinsic features of w6 and w7 on which this difference in identities could depend, we appear to have a case of bare identities. This ‘Four Worlds Paradox’ is like Chandler’s original transitivity puzzle in that it does not seem that an appeal to individual essences could solve it without conflicting with the tolerance principle. If that is so, the proponent of transworld identity (as opposed to counterpart theory) appears to be left with two options consistent with the transitivity of identity: the denial of the tolerance principle, and the acceptance of bare identities. (But cf. Dorr, Hawthorne, and Yli-Vakkuri 2021.) It may be argued, however, that the acceptance of bare identities can be made more palatable in this case by the adoption of a non-transitive accessibility relation between possible worlds. (See Salmon 1982, 230–252; and, for discussion, Roca-Royes 2016. For a defence of the employment of counterpart theory to solve the Four Worlds Paradox, see Forbes 1985, Ch. 7. For discussion of a radical response that retains the tolerance principle and yet avoids bare identities, but only at the cost of claiming that two bicycles could completely coincide in one possible world, simultaneously sharing all their parts, see Roca-Royes 2016, discussing Williamson 1990. On the relevance to the Four Worlds Paradox to Kripke’s principle of the necessity (essentiality) of origin for artefacts, see also Robertson 1998 and Hawthorne and Gendler 2000.)