Congrats Elliott! Looks like a nice paper.

(FWIW, I don't give much weight to critical-set views, anyway.)

In section 3. The Drop, you assume biographical identity is determinately-all-or-determinately-nothing, but this doesn't seem very plausible to me. What could a justification for a specific such account even look like, with specific precise cutoffs for a given person? The only I could imagine is someone very sharply going from fully personally identifying to not at all identifying with their past with the additional tiny change. However, I would be surprised if that happened for most people or that we should interpret this as actually giving a determinate sharp cutoff when it would happen.

In my view, the result is going to be fairly continuous, because any of the following reasons (conditioning on any of them) will hold:

1. The facts about identity are continuous or highly graded (whether precise or vague). Emile* is partly Emile, to different degrees for different degrees of change.
   1. I don't know if there's a good way to extend critical-set views for this possibility, though.
2. The facts about identity are actually determinately-all-or-determinately-nothing, but we (or you, or I) have a fairly smooth^[1] credence distribution for the locations of the cutoffs, and taking expected values (or using another approach to moral uncertainty) gives a continuous or highly graded view.
   1. Also, our credence distributions are almost certain to remain fairly smooth, although they can change. I can't imagine being discontinuously confident in a specific precise cutoff. You'd need a special reason to believe in a specific cutoff over others, but I don't expect to uncover one.
3. The facts about identity are vague, and we can treat it like 2 (or 1 and 2), assigning weights fairly smoothly across precisifications.

Plus, if someone is specifically endorsing a critical-range view, then they'll probably be more broadly sympathetic to vagueness, including for identity, anyway. OTOH, if they did something like 3 for the critical range, too, then they'd turn the critical-range view into something like a critical-level view with uncertainty about the location of the critical level, rather than supervaluationism, which requires truth of the predicate on all precisifications.

I think your objections in sections 4 and 5 are good, and probably (?) extend to vague accounts and highly graded accounts of biographical identity.

1. ^{^}

   Probably a few ways to formalize this, but I imagine Lipschitz continuity of the probability distribution over the set of possible cutoffs or over large subsets, and for each "type" of cutoff, a unimodal probability distribution for its location.