Counting delineations

I presented my paper on indeterminacy and conditionals in Konstanz a few days ago. The basic question that paper poses is: if we are highly confident that a conditional is indeterminate, what sorts of confidence in the conditional itself are open to us?

Now, one treatment I’ve been interested in for a while is “degree supervaluationism”. The idea, from the point of view of the semantics, is to replace appeal to a single intended interpretation (with truth=truth at that interpretation) or set of “intended interpretations” (with truth=truth at all of them) with a measure over the set of interpretations (with truth to degree d = being true at exactly measure d of the interpretations). A natural suggestion, given that setting, is that if you know (/are certain) S is true to measure d, then your confidence in S should be d.

I’d been thinking of degree-supervaluationism in this sense, and the more standard set-of-intended-interpretations supervaluationism, as distinct options. But (thanks to Tim Williamson) I realize now that there may be an intermediate option.

Suppose that S= the number 6 is bleh. And we know that linguistic conventions settle that numbers <5 are bleh, and numbers >7 are not bleh. The available delineations of “nice”, among the integers, are ones where the first non-bleh number is 5, 6, 7 or 8. These will count as the “intended interpretations” for a standard supervaluational treatment, so “6 is bleh” will be indeterminate—in this context, neither true nor false.

I’ve discussed in the past several things we could say about rational confidence in this supervaluational setting. But one (descriptive) option I haven’t thought much about is to say that you should proportion your confidence to the number of delineations on which “6 is bleh” comes out true. In the present case, our confidence that 6 is bleh should be 0.5, our confidence that 5 is bleh should come out 0.25, and our confidence that 7 is bleh should come out 0.25.

Notice that this *isn’t* the same as degree-supervaluationism. For that just required some measure or other over the space of interpretations. And even if that was non-zero everywhere apart from ones which place first non-bleh number in 5-8, there are many options available. E.g. we might have a measure that assigns 0.9 to the interpretation which makes 5 the first non-bleh number, and distributes 0.3333… to the others. In other words, the degree-supervaluationist needn’t think that the measure is a measure *of the number of delineations*. I usually think of it (in the finite case), intuitively, as a measure of the “degree of intendedness” of each interpretation. In a sense, the degree-supervaluationists I was thinking of conceive of the measure as telling us to what extent usage and eligibility and other subvening facts favour one interpretation or another. But the kind of supervaluationists we’re now considering won’t buy into that at all.

I should mention that even if, descriptively, it’s clear what proposal here is, it’s less clear how the count-the-delineations supervaluationists would go about justifying the rule for assigning credences that I’m suggesting for them. Maybe the idea is that we should seek some kind of compromise between the credences that would be rational if we took D to be the unique intended interpretation, for each D in our set of “intended interpretations” (see this really interesting discussion of compromise for a model of what we might say—the bits at the end on mushy credence are particularly relevant). And they’ll be some oddities that this kind of theorist will have to adopt—e.g. for a range of cases, they’ll be assigning significant credence to sentences of the form “S and S isn’t true”. I find that odd, but I don’t think it blows the proposal out of the water.

Where might this be useful? Well, suppose you believe in B-theoretic branching time, and are going to “supervaluate” over the various future-branches (so “there will be a sea-battle” will a truth-value gap, since it is true on some but not all). (This approach originates with Thomason, and is still present, with tweaks, in recent relativistic semantics for branching time). “Branches” play the role of “interpretations”, in this setting. I’ve argued in previous work that this kind of indeterminacy about branching futures leads to trouble on certain natural “rejectionist” readings of what our attitudes to known indeterminate p should be. But a count-the-branches proposal seems pretty promising here. The idea is that we should proportion our credences in p to the *number* of branches on which p is true.

Of course, there are complicated issues here. Maybe there are just two qualitative possibilities for the future, R and S. We know R has a 2/3 chance of obtaining, and S a 1/3 chance of obtaining. In the B-theoretic branching setting, an R-branch will exist, and an S-branch will exist. Now, one model of the metaphysics at this point is that we don’t allow qualitatively duplicate future brnaches: so there are just two future-branches in existence, the R one and the S one. On a count-the-branches recipe, we’ll get the result that we should have 1/2 credence that R will obtain. But that conflicts with what the instruction to proportion our credences to the known chances would give us. Maybe R is primitively attached to a “weight” of 2/3—but our count-the-branches recipe didn’t say anything about that.

An alternative is that we multiply indiscernable futures. Maybe there are two, indiscernable R futures, and only one S future. Then apportioning  the credences in the way mentioned won’t get us into trouble. And in general, if we think whenever the chance (at moment m) that p is k, then the proportion of p-futures to non-p-futures is k, then  we’ll have a recipe that coheres nicely with the principal principle.

Let me be clear that I’m not suggesting that we identify chances with numbers-of-branches. Nor am I suggesting that we’ve got some easy route here for justifying the principal principle. The only thing I want to say is that *if* we’ve got a certain match between chances and numbers of future branches, then two recipes for assigning credences won’t conflict.

(I emphasized earlier that count-the-precisifications supervaluationism had less flexibility than degree-supervaluationism where the relevant measure was unconstrained by counting considerations. In a sense, what the above little discussion highlights is that when we move from “interpretations” to “branches” as the locus of supervaluational indeterminacy, this difference in flexibility evaporates. For in the case where that role is played by actually existing futures, then there’s at least the possibility of mutiplying qualitatively indiscernable futures. That sort of maneuver has little place in the original, intended-interpretations settings, since presumably we’ve got an independent fix on what the interpretations are, and we can’t simply postulate that the world gives us intended interpretations in proporitions that exactly match the credences we independently want to assign to the cases.)

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