(1) George believed(?) that England could avoid nuclear war with France.
[Well, he believed that England could avoid war with France, and that entails (1), so...yes? But intuitively, worlds where England has/avoids nuclear war with France are not in his belief state at all.]
(2) The detective believes(?) that the chauffeur didn't do it.
[Well, he believes the butler did it, and that entails (2)...but the problem is the same. The detective hasn't considered the possibility that the chauffeur did it; he overlooked the chauffeur.]
Solution 1...is Stalnaker's suggestion regarding distinguishing between active and passive beliefs. To do this, we model the agent's belief state as a space of relevant alternatives (the possibilities he recognizes) and carve his set of belief-worlds out of this set. Active beliefs are beliefs true in every world in the belief-world set. Passive beliefs are any beliefs that are entailed by this, but involve entailment via unioning with things outside the set of relevant alternatives. This is the problematic feature of the entailment from "avoid war" to "avoid nuclear war" and the entailment from "butler did it" to "chauffeur didn't do it." In short: George passively but not actively believes that England can avoid nuclear war, and the detective passively but not actively believes that the chauffeur didn't do it.
Problems with Solution 1. Sometimes unioning can be achieved in the belief proposition itself: for example, although George only passively believes "England can avoid nuclear war with France," he actively believes "England can avoid war or nuclear war with France." Likewise, the detective only passively believes that the chauffeur didn't do it, but he actively believes "either the butler did it or the chauffeur did it."
Moreover, "every possibility left open by one's belief state is a fortiori relevant" [since the belief state is a subset of the relevant alternatives set.] In short, there is no distinction between active and passive possibilities (they are all active so long as they aren't ruled out by an agent's belief state.) Yet that seems to be precisely what's at issue in the detective case and in the King George case: the possibility that the chauffeur did it, and the possibility of nuclear conflict, aren't "seen" by the subjects' belief states (even if answers to these questions are entailed by things they already believe). Stalnaker's answer is to exclude e.g. chauffeur-did-it worlds from the detective's belief state. Maybe a better alternative is to include these worlds, but to make them somehow inaccessible.
Solution 2: Subject matter/question/resolution sensitivity.
"A state of belief still determines a set of (maximally specific) possible worlds, but only insofar as it determines a coarser set of possibilities from the partition of logical space in question. We could call this latter set a belief partition. Call a partition of logical space used in this role a modal resolution...Realistic states of belief are relativized to finite resolutions." (Yalcin 4)
"With resolutions of logical space now in the picture, we can try using them to characterize belief content which is, in the desired sense, available."
Actively believed propositions are unions of cells of the agent's belief partition. We shall say these propositions are visible to the resolution. Passively/implicitly believed propositions are ones which are entailed by the belief partition but fail to be a union of some set of cells in the partition. This is formulated in:
"Resolution Sensitivity I. An agent's total state of belief is representable as a partial function from questions to answers." (5)
Let's pause over this for a moment. The function is partial simply because there are some questions which put too fine a partition on W for the agent's state to determine an answer. Ask a question that carves along my parition, and I can give you an answer--even if it's "I don't know." Such a question cuts along my partition but my belief state doesn't rule in any answer.
Now how do we deal with the detective and King George? Well, King George does believe that England can avoid nuclear war with France...but he doesn't actively believe it. Does he actively believe that England can avoid war or can avoid nuclear war with France? No, because the union of an invisible and a visible region gives us an invisible region. Mutatis mutandis for the detective.
However, we still supposedly have the "crazy conjunction problem": for any p, q, r s.t. p iff q & r, if you actively believe p, then you actively believe q&r. Prima facie counterexample: I can believe I have two hands when I lack the concept even number, and when I lack the concept prime number. But if I actively believe I have two hands (rather than one, or three, etc.) then I actively believe that I have an even prime number of hands. The problem is that, while the two propositions are invisible, they intersect to form a visible proposition (and of course my belief worlds are a subset of that proposition.)
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