Abstract
I consider two puzzles in which an agent undergoes a sequence of decision problems. In both cases it is possible to respond rationally to any given problem yet it is impossible to respond rationally to every problem in the sequence, even though the choices are independent. In particular, although it might be a requirement of rationality that one must respond in a certain way at each point in the sequence, it seems it cannot be a requirement to respond as such at every point for that would be to require the impossible.
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It seems natural to suppose that backwards supertasks, such as this one, are possible if the ordinary kind are. For example, presumably Zeno’s Achilles performs such a supertask every time he moves, in much the same way as he performs a forwards supertask.
Or at least, something that can immediately be converted into “hedons” before the next choice.
One might put it paradoxically: suppose that Alice and Ecila are intrinsic duplicates at 11 am. Between 12 pm and 1 pm Alice will undergo one of these decision sequences, while Ecila does not, and never will. Ecila has the right dispositional properties to be rational, yet Alice does not, since there will be some instance of (1) she does not satisfy.
One might think the deontic Barcan formula fails for more mundane reasons. The interesting thing about these puzzles is that they provide counterexamples to the principle that a conjunction of rational requirements is a rational requirement. This is much weaker since it remains valid even over the class of variable domain Kripke models. Similar remarks apply also to the Barrett–Arntzenius puzzle.
I would like to thank Frank Arntzenius, Cian Dorr and Kenny Easwaran for helpful discussion on the issues relating to this paper.
References
Barrett, J., & Arntzenius, F. (1999). An infinite decision puzzle. Theory and Decision, 46(1), 101–103.
Yablo, S. (2000). A reply to new Zeno. Analysis, 60(2), 148–151.
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Bacon, A. A paradox for supertask decision makers. Philos Stud 153, 307–311 (2011). https://doi.org/10.1007/s11098-010-9496-1
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DOI: https://doi.org/10.1007/s11098-010-9496-1