Aristotelian indeterminacy and partial beliefs

I’ve just finished a first draft of the second paper of my research leave---title the same as this post. There’s a few different ways to think about this material, but since I hadn't posted for a while I thought I'd write up something about how it connects with/arises from some earlier concerns of mine.

The paper I’m working on ends up with arguments against standard “Aristotelian” accounts of the open future, and standard supervaluational accounts of vague survival. But one starting point was an abstract question in the philosophy of logic: in what sense is standard supervaluationism supposed to be revisionary? So let's start there.

The basic result---allegedly---is that while all classical tautologies are supervaluational tautologies, certain classical rules of inference (such as reductio, proof by cases, conditional proof, etc) fail in the supervaluational setting.

Now I’ve argued previously that one might plausibly evade even this basic form of revisionism (while sticking to the “global” consequence relation, which preserves traditional connections between logical consequence and truth-preservation). But I don’t think it’s crazy to think that global supervaluational consequence is in this sense revisionary. I just think that it requires an often-unacknowledged premise about what should count as a logical constant (in particular, whether “Definitely” counts as one). So for now let’s suppose that there are genuine counterexamples to conditional proof and the rest.

The standard move at this point is to declare this revisionism a problem for supervaluationists. Conditional proof, argument by cases: all these are theoretical descriptions of widespread, sensible and entrenched modes of reasoning. It is objectionably revisionary to give them up.

Of course some philosophers quite like logical revisionism, and would want to face-down the accusation that there’s anything wrong with such revisionism directly. But there’s a more subtle response available. One can admit that the letter of conditional proof, etc are given up, but the pieces of reasoning we normally call “instances of conditional proof” are all covered by supervaluationally valid inference principles. So there’s no piece of inferential practice that’s thrown into doubt by the revisionism of supervaluational consequence: it seems that all that happens is that the theoretical representation of that practice has to take a slightly more subtle form than one might except (but still quite a neat and elegant one).

One thing I mention in that earlier paper but don’t go into is a different way of drawing out consequences of logical revisionism. Forget inferential practice and the like. Another way in which logic connects with the rest of philosophy is in connection to probability (in the sense of rational credences, or Williamson’s epistemic probabilities, or whatever). As I sketched in a previous post, so long as you accept a basic probability-logic constraint, which says that the probability of a tautology should be 1, and the probability of a contradiction should be 0, then the revisionary supervaluational setting quickly forces you to a non-classical theory of probability: one that allows disjunctions to have probability 1 where each disjunct has probability 0. (Maybe we shouldn't call such a thing "probability": I take it that's terminological).

Folk like Hartry Field have argued completely independently of this connection to Supervaluationism that this is the right and necessary way to handle probabilities in the context of indeterminacy. I’ve heard others say, and argue, that we want something closer to classicism (maybe tweaked to allow sets of probability functions, etc). And there are Dutch Book arguments to consider in favour of the classical setting (though I think the responses to these from the perspective of non-classical probabilities are quite convincing).

I’ve got the feeling the debate is at a stand-off, at least at this level of generality. I’m particularly unmoved by people swapping intuitions about degrees of belief it is appropriate to have in borderline cases of vague predicates, and the like (NB: I don’t think that Field ever argues from intuition like this, but others do). Sometimes introspection suggests intriguing things (for example, Schiffer makes the interesting suggestion that one’s degree of belief in a conjunction of two vague propositions is typically matches one’s degree of belief in the propositions themselves). But I can’t see any real dialectical force here. In my own case, I don’t have robust intuitions about these cases. And if I'm to go on testimonial evidence on others intuitions, it’s just too unclear what people are reporting on for me to feel comfortable taking their word for it. I'm worried, for example, they might just be reporting the phenomenological level of confidence they have in the proposition in question: surely that needn’t coincide with one’s degree of belief in the proposition (thinking of an exam you are highly nervous about, but are fairly certain you will pass… your behaviour may well manifest a high degree of belief, even in the absence of phenomenological trappings of confidence). In paradigm cases of indeterminacy, it’s hard to see how to do better than this.

However, I think in application to particular debates we might be able to make much more progress. Let us suppose that the topic for the day is the open future, construed, minimally, as the claim that while there are definite facts about the past and present, the future is indefinite.

Might we model this indefiniteness supervaluationally? Something like this idea (with possible futures playing the role of precisifications) is pretty widespread, perhaps orthodoxy (among friends of the open future). It’s a feature of MacFarlane’s relativistic take on the open future, for example. Even though he’s not a straightforward supervaluationist, he still has truth-value gaps, and he still treats them in a recognizably supervaluational-style way.

The link between supervaluational consequence and the revisionionary behaviour of partial beliefs should now kick in. For if you know with certainty that some P is neither true nor false, we can argue that you should invest no credence at all in P (or in its negation). Likewise, in a framework of evidential probabilities, P gets no evidential probability at all (nor does its negation).

But think what this says in the context of the open future. It’s open which way this fair coin lands: it could be heads, it could be tails. On the “Aristotelian” truth-value conception of this openness, we can know that “the coin will land heads” is gappy. So we should have credence 0 in it, and none of our evidence supports it.

But that’s just silly. This is pretty much a paradigmatic case where we know what partial belief we have and should have in the coin landing heads: one half. And our evidence gives exactly that too. No amount of fancy footwork and messing around with the technicalities of Dempster-Shafer theory leads to a sensible story here, as far as I can see. It’s just plainly the wrong result. (One doesn't improve matters very much by relaxing the assumptions, e.g. taking the degree of belief in a failure of bivalence in such cases to fall short of one: you can still argue for a clearly incorrect degree of belief in the heads-proposition).

Where does that leave us? Well, you might reject the logic-probability link (I think that’d be a bad idea). Or you might try to argue that supervaluational consequence isn’t revisionary in any sense (I sketched one line of thought in support of this in the paper cited). You might give up on it being indeterminate which way the coin will land---i.e. deny the open future, a reasonably popular option. My own favoured reaction, in moods when I’m feeling sympathetic to the open future, is to go for a treatment of metaphysical indeterminacy where bivalence can continue to hold---my colleague Elizabeth Barnes has been advocating such a framework for a while, and it’s taken a long time for me to come round.

All of these reactions will concede the broader point---that at least in this case, we’ve got an independent grip on what the probabilities should be, and that gives us traction against the Supervaluationist.

I think there are other cases where we can find similar grounds for rejecting the structure of partial beliefs/evidential probabilities that supervaluational logic forces upon us. One is simply a case where empirical data on folk judgements has been collected---in connection with indicative conditions. I talk about this in some other work in progress here. Another which I talk about in the current paper, and which I’m particularly interested in, concerns cases of indeterminate survival. The considerations here are much more involved than in indeterminacy we find in connection to the open future or conditionals. But I think the case against the sort of partial beliefs supervaluationism induces can be made out.

All these results turn on very local issues. None, so far as see, generalizes to the case of paradigmatic borderline cases of baldness and the rest. I think that makes the arguments even more interesting: potentially, they can serve as a kind of diagnostic: this style of theory of indeterminacy is suitable over here; that theory over there. That’s a useful thing to have in one’s toolkit.