It is interesting to sort views of personal identity over time based on the answer to:
- Are the grounds of personal identity over time capable of continuous variation between cases of identity and cases of non-identity?
Theories that answer "Yes" require either vague diachronic identity or an arbitrary transition (something like: it's the same person if and only if at least exactly this number of neurons are had in common between the state now and the state one second ago). Moreover, the transition would have to be metaphysically necessary, since we are talking about the grounds of personal identity: we can't have two worlds with the same grounds but different identity facts.
I do not think an arbitrary metaphysically necessary transition here is plausible. What about vague diachronic personal identity? I think that is impossible. For I think the only sources of vagueness in whether I am suffering a horrific pain are the vagueness in "horrific" and in "pain". But if there was vague diachronic identity, then one could have a token case of definitely horrific pain and still have vagueness as to whether I am suffering it. And that is absurd. Moreover, we have the well-known Leibniz's Law argument against vague identity (a does not have the property of being vaguely identical with a; if b has the property of being vaguely identical with a, then by Leibniz's Law, b just isn't a, plain and simple).