In a few weeks time (31st March-5th April) I’m going to be visiting the Arche research centre in St Andrews, and giving a series of talks. I studied at Arche for my PhD, so it’ll be really good to go back and see what’s going on.

The talks I’m giving relate to the material on indeterminacy and probability (in particular, evidential probability or partial belief). The titles are as follows:

- Indeterminacy and partial belief I: The open future and future-directed belief.
- Indeterminacy and partial belief II: Conditionals and conditional belief.
- Indeterminacy and partial belief III: Vague survival and de se belief.

A lot of these are based around exploring the consequences of the view that if p is indeterminate, and one knows this (or is certain of it) then one shouldn’t invest any probability in p. In the case of the open future, of conditionals, and in vague survival—for rather different reasons in each case—this seems highly problematic.

But why should you believe that key principle about how attitudes to indeterminacy constrain attitudes to p? The case I’ve been focussing on up till now has concerned a truth-value gappy position on indeterminacy. With a broadly classical logic governing the object language, one postulates truth-value gaps in indeterminate cases. There’s then an argument directly from this to the sort of revisionism associated with supervaluationist positions in vagueness. And from there, and a certain consistency requirement on rational partial belief (or evidence) we get the result. The consistency requirement is simply the claim, for example, that if q follows from p, one cannot rationally invest more confidence in p than one invests in q (given, of course, that one is aware of the relevant facts).

The only place I appeal to what I’ve previously called the “Aristotelian” view of indeterminacy (truth value gaps but LEM retained) is in arguing for the connection between attitudes to determinately p and attitudes to p. But I’ve just realized something that should have been obvious all along—which is that there’s a quick argument to something similar for someone who thinks determinacy is marked by a rejection of excluded middle. Assume, to begin with, that the paracompletist nonclassicist will think in borderline cases, characteristically, one should reject the relevant instance of excluded middle. So if one is fully convinced that p is borderline, one should utterly reject pv~p.

It’s dangerous to generalize about non-classical systems, but the ones I’m thinking of all endorse the claim p|-pvq—i.e. disjunction introduction. So in particular, an instance of excluded middle will follow from p.

But if we utterly reject pv~p in a borderline case (assign it credence 0), then by the probability-logic link we should utterly reject (assign credence 0) anything from which it follows.

In particular, we should assign credence 0 to p. And by parallel reasoning, we should assign credence 0 to ~p.

[Edit: there’s a question, I think, about whether the non-classicist should take us to utterly reject LEM in a borderline case (i.e. degree of partial belief=0). The folklore non-classicist, at least, might suggest that on her conception degrees of truth should be expert functions for partial beliefs—i.e. absent uncertainty about what the degrees of truth are, one should conform the partial beliefs to the degrees of truth. Nick J. J. Smith has a paper where he works out a view that has this effect, from what I can see. It’s available here and is well worth a read. If a paradigm borderline case for the folklore nonclassicist is one where degree of truth of p, not p and pv~p are all 0.5, then one’s degree of belief in all of them should be 0.5. And there’s no obvious violation of the probability-logic link here. (At least in this specific case. The logic will have to be pretty constrained if it isn’t to violate probability-logic connection somewhere).]

If all this is correct, then I don’t need to restrict myself to discussing the consequences of the Aristotelian/supervaluation sort of view. Everything will generalize to cover the nonclassical cases—and will cover both the folklore nonclassicist and the no interpretation nonclassicist discussed in the previous cases (here’s a place where there’s convergence).

[A folklore classicist might object that for them, there isn’t a unique “logic” for which to run the argument. If one focuses on truth-preservation, one gets say a Kleene logic; if one focuses on non-falsity preservation, one gets an LP logic. But I don’t think this thought really goes anywhere…]