Impossible worlds and partial belief

Abstract

One response to the problem of logical omniscience in standard possible worlds models of belief is to extend the space of worlds so as to include impossible worlds. It is natural to think that essentially the same strategy can be applied to probabilistic models of partial belief, for which parallel problems also arise. In this paper, I note a difficulty with the inclusion of impossible worlds into probabilistic models. Under weak assumptions about the space of worlds, most of the propositions which can be constructed from possible and impossible worlds are in an important sense inexpressible; leaving the probabilistic model committed to saying that agents in general have at least as many attitudes towards inexpressible propositions as they do towards expressible propositions. If it is reasonable to think that our attitudes are generally expressible, then a model with such commitments looks problematic.

Comparativism and the Measurement of Partial Belief

“…I have raised this objection only to acknowledge it, and then set it aside. In other works, I have argued that letting Ω include logically impossible worlds creates special problems in the probabilistic context (Elliott 2019a), and will in fact severely undermine the comparativist's analogy with the measurement of length rather than support it (Elliott 2019b). I won't repeat those arguments here, but let me add two further points.…”

“…See, for example, Cozic (2006) and Halpern and Pucella (2011). Elliott (2019a) shows that for the fully general result to hold, Ω + needs to include not only logically impossible worlds, but also 'incomplete' worlds-i.e., worlds that leave some matters unspecified. So perhaps comparativists might maintain the measurement analogy if they let propositions be characterised as sets of possible and impossible/incomplete worlds.…”

‘Ramseyfying’ Probabilistic Comparativism

“…That seems easy enough, but I do not think that this is a viable strategy for the comparativist to adopt. I'll set out the reasons for this very briefly, since most of the relevant issues are discussed at length in (Elliott 2019b). The problem is that once Ω includes enough impossible worlds for the strategy to work (roughly: for any impossibility, there's an impossible world that verifies it), then most subsets of Ω will be meaningless and consequently not representative of any proper contents of belief.…”

“…Where Ω is the space of classically possible worlds, B ⊆ ℘(Ω), and Cr :B → [0, 1],then if Ω + is a rich enough extension of Ω into the space of impossible worlds, there's a probability function Cr + on an algebra of sets B + ⊆ ℘(Ω + ) such that Cr + assigns x to the subset of Ω + that verifies ϕ iff Cr assigns x to the subset of Ω that verifies ϕ. See(Cozic, 2006),(Halpern and Pucella, 2011), and(Elliott 2019b).…”