Subject-relative safety and nested safety.

The paper I was going to post took off from very interesting recent work by John Hawthorne and Maria Lasonen that creates trouble from the interaction of safety constraints and a plausible looking principle about chance and close possibilities. The major moving part is a principle that tells you (roughly) that whenever a proposition is high-chance at w,t, then some world compatible with the proposition is  a member of the “safety set” relevant to any subject’s knowledge at w,t (the HCCP principle).

It’s definitely worth checking out Williamson’s reply to H&L. There’s lots of good stuff in it. Two relevant considerations: he formulates a version of safety in the paper that is subject-relativized (one of the “outs” in the argument that H&L identify, and defends this against the criticisms they offer). And he rejects the HCCP principle. The basic idea is this: take some high-but-not-1-chance proposition that’s intuitively known e.g. the ball is about to hit the floor. And consider a world in which this scenario is duplicated many times—enough so that the generalization “some ball fails to hit the floor” is high-chance (though false). Each individual conjunct seems epistemically on a par with the original. But by HCCP, there’s some failure-to-hit world in the safety set, which means at least one of the conjuncts is unsafe and so not known.

Rejecting HCCP is certainly sufficient to get around the argument as stated. But H&L explicitly mention subject-relativization of safety sets as a different kind of response, *compatible* with retaining HCCP. The idea I take it is that if safety sets (at a given time) can vary,  *different* “some ball hitting floor” possibilities could be added to the different safety sets, satisfying HCCP but not necessarily destroying any of the distributed knowledge claims.

I see the formal idea, which is kind of neat. The trouble I have with this is that I’ve got very little grip at all as to *how* subject-relativization would get us out of the H&L trouble. How can particular facts about subjects change what’s in the safety set?

I’m going to assume the safety set (for a subject, at a given time and place) is always a Lewisian similarity sphere—that is, for some formal similarity ordering of worlds, the safety sphere is closed downwards under “similarity to actuality”.  I’ll also assume that *similarity* isn’t subject-relative, though for all I’ll say it could vary e.g. with time. The assumptions are met by Lewis’s accout of counterfactual similarity—in fact, for him similarity isn’t time-relative either—but many other theories can also agree with this.

The assumption that the safety set is always a similarity sphere (in the minimal sense) seems a pretty reasonable requirement, if we’re to justify the gloss of a safety set as a set of the “sufficiently close worlds”.

But just given the minimal gloss, we can get some strong results: in particular, that safety sets for different subjects at a single time will be nested in one another (think of them as “spheres around actuality”–given minimal formal constraints, Lewis articulates, the “spheres” are nested, as the name suggests).

Suppose we have n subjects in an H&L putative “distributed knowledge” case as described earlier. Now take the minimal safety set M among those n subjects. This exists and is a subset of the safety sets of all the others, by nesting. And by HCCP, it has to include a failure-to-hit possibility within it. Say the possibility that’s included in M features ball k failing to hit. But this means that that possibility is also in the safety set relevant to the kth person’s belief that their ball *will* hit the ground, and so their actual belief is unsafe and can’t count as knowledge—exactly the situation that relativizing to subjects was supposed to save us from!

The trouble is, the sort of rescue of distributed knowledge sketched earlier relies on the thought that safety sets for subjects at a time might be “petal shaped”—overlapping, but not nested in one another. But thinking of them as “similarity spheres”, where similarity is not subject relative, simply doesn’t allow this.

Now, this doesn’t close off this line of inquiry. Perhaps we *should* make similarity itself relative to subjects or locations (if so, then we definitely can’t use Lewis’s “Time’s arrow” sense of similarity). Or maybe we could relax the formal restrictions on similarity that allow us to derive nesting (If worlds can be incomparable in terms of closeness to actuality, we get failures of nesting—weakening Lewis’s formal assumptions in this way weakens the associated logic of counterfactuals to Pollock’s SS). But I do think that it’s interesting that the kind of subject-relativity of closeness that might be motivated by e.g. interest-relative invariantism about knowledge (the idea that how “close” the worlds to be in the safety set  depends on the interests etc of the knower) simply don’t do enough to get us out of the H&L worries.  We need a much more thorough-going relativization if we’re going to make progress here.

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