Is counterpart theory inadequate?

Conclusion

Counterpart theorists need not posit the counterpart relation in addition to the identity relations as an Additional relation relating objects across possible worlds. Identity can be viewed as a relation applicable to individuals within possible worlds, while the counterpart relation replaces identity in translations of ordinary utterances which correlate individuals in different possible worlds and, hence, in all modal utterances. CT is, in other words, a theory of modal discourse — it proposes a way of understanding all modal predications. As such, it is not to be seen as providing a second ‘looser’ sense of identity across worlds, in addition to the ‘narrow’ sense. Perhaps Lewis' original formulation encourages this misunderstanding insofar as it (see, especially, P2) denies strict identity across possible worlds. Accordingly, I have suggested a modification of CT whereby syntactic restrictions upon variables eliminates any temptation to posit two parallel relations applicable to entities existing in different possible worlds.