The tyranny of the objective
A widely held view in philosophy is that ordinary information and ordinary belief are concerned with "objective" propositions whose truth-value doesn't vary between perspectives or locations within a world.
Some hold that all genuine content is objective, and that the appearance of counterexamples is an illusion that can somehow be explained away. (See, e.g., Stalnaker 1981, Magidor 2015, or Cappelen and Dever 2013.) Even those who accept that there is genuinely perspectival or self-locating information tend to treat it as a special case that requires special rules for integration with ordinary, non-perspectival information. (See, e.g., Bostrom 2002, Meacham 2008, Moss 2012, Titelbaum 2013, Builes 2020, or Isaacs, Hawthorne, and Russell 2022).
I think of this as the tyranny of the objective. (Compare Chalmers 1998.)
In my view, all ordinary belief, and all ordinary information, is perspectival. Our senses tell us how things are here and now, around us. By scientific experiments and observations, we can find out more about our solar system or about the biology of organisms on our planet. When we learn such facts, we may also learn objective facts: by coming to know that it is raining, I also come to know that it is raining somewhere in the history of the world. But this objective belief is unusual and derivative. Ordinary confirmation is always confirmation of perspectival hypotheses by perspectival evidence.
Lewis 1979 explained how this can be modelled formally. We simply need to replace the uncentred worlds of traditional confirmation theory with centred worlds.
The clearest sign that something is amiss with the objectivist mainstream is that it can't account for elementary facts about reasoning with perspectival information. As Isaacs, Hawthorne, and Russell 2022, 252 put it: "All the precise theories we know of face very serious objections." I agree.
Of course, dropping the objectivist starting point doesn't automatically solve the difficult puzzles discussed in Bostrom 2002 or Isaacs, Hawthorne, and Russell 2022. But I think it sets the ground for a solution, and explains where certain arguments go wrong.
To see what I mean, let's have a closer look at Isaacs, Hawthorne, and Russell 2022.
The paper explores whether our evidence supports the hypothesis that there are many universes. In the main part of the paper, the authors (henceforth, IRH) assume that centred credences are derived from uncentred priors by special rules. In Appendix B, IRH consider the possibility of starting with centred priors, but they argue that this doesn't affect the conclusion that our evidence supports the multiverse hypothesis.
Concretely, IRH prove two theorems. The theorems are complicated, but a simple example illustrates the key moves.
Let H1 be the hypothesis that there is exactly one universe, and H2 the hypothesis that there are two universes. Assume that each universe has a fixed chance p of being inhabited. Assume that p < 0.5. For simplicity, let's assume that an inhabited universe contains exactly one centre from which it is observed. The evidence that is received at such a centre is "local" insofar as it doesn't reveal anything about what might be the case in other universes. But it reveals (among other things) that this universe is inhabited.
Does such evidence support H2 over H1?
To answer this question, we need some assumptions about the (centred) priors.
Let Pr be a rational prior credence function. Let I=1 be the hypothesis that there is exactly one inhabited universe. Since the chance of any universe being inhabited is p, we might expect that
For (1), the idea is that if there's just one universe, and any universe has a fixed chance p of being inhabited, then the probability of this one universe being inhabited is p.
For (2), we assume that there are two universes, U1 and U2. Each has an independent chance p of being inhabited. There are two ways for there to be exactly one inhabited universe: U1 is inhabited and U2 isn't, or U2 is inhabited and U1 isn't. Each scenario has probability p(1-p). So the total probability of I=1 is 2p(1-p).
Now let E be our evidence. Plausibly,
The idea here is that our evidence E is not made any more or less probable by the presence of a second, uninhabited universe.
Since p < 0.5, it follows (by a little maths) that Pr(E | H2) > Pr(E | H1). And so E supports H2 over H1.
IRH's "Theorem 3" generalizes this result.
The point I want to make is that this line of reasoning is highly dubious if we take centred priors seriously.
Let's return to the priors. We have to make a choice: Should the prior Pr assign positive probability only to inhabited points, or can it also assign positive probability to uninhabited points?
This is a somewhat arcane theoretical question, since any evidence will immediately rule out uninhabited points anyway.
Suppose we decide that Pr assigns positive probability only to inhabited points. Then (1) is false. Given that there is exactly one universe, the prior probability that this universe is inhabited must be 1, not p. (2) is also false. In general, on this approach we can't assume that the prior probabilities align with the chances.
We can hold on to (1) and (2) only if we allow uninhabited points to have positive prior probability. But if we do that, we should give up (3).
To see why, let <E,-> be a two-universe world in which E is true at the first universe and the second universe is uninhabited. Let <E> be a one-universe world in which E is true. If Pr assigns positive probability to both locations in <E,-> then Pr(E | <E,->) is less than 1, while Pr(E | <E>) is 1. So Pr(E | H2 ∧ I=1) < Pr(E | H1 ∧ I=1).
There is no plausible view on which (1)-(3) are all true. So we don't get "Theorem 3". Nor do we get the stronger "Theorem 4".
IRH assume that the prior Pr assigns positive probability only to inhabited points, except in worlds that are entirely uninhabited: here, the prior assigns positive probability to a "dummy" centre. This is formally consistent and makes it possible to accept (1)-(3), but it is an entirely implausible account of rational priors.