Probability in the Absence of Uncertainty
Let us now turn to an application of the decision theoretic framework that we have set up here. What follows is something of a digression; readers who wish to move on may skip to chapter 10.
In section 7.3.1, I noted that the Dutch book argument for probabilism is hard to get off the ground because it relies heavily on a fairly strong connection between belief and a specific form of physical behaviour (betting behaviour). However, decision theory gives us a much more general connection between belief and preferences (where the latter are understood as being attitudes that determine, but are not exhausted by, rational betting behaviour). In this setting a more general argument for probabilism is available.Indeed, we will argue that even anti-probabilists, of the sort considered in chapter 7, can and ought to accept a sort of ersatz probabilism. The thesis that you should be uncertain about the vague, and that your uncertainties should be governed by the probability calculus, can be given a purely decision theoretic interpretation in which they come out true, even if one's official theory of credences is non- probabilistic.
Let's begin by considering an anti-probabilist objection to the classical account of vagueness-related belief: If borderline propositions do not have truth values (as philosophers have sometimes supposed), there is no relevant proposition whose truth we are uncertain about, so it seems to be incorrect to describe this as a case where we are uncertain about something. Similarly, if there is no fact about whether Harry is bald, and I know this, then there's nothing to be uncertain about so we shouldn't be asking how confident one should be about whether Harry is bald. If I know and am completely confident that it's borderline whether Harry is bald, one might object, then any assignment of confidence to the proposition that Harry is bald would be epiphenomenal.
Unlike, say, a degree of confidence about whether it will rain tomorrow—which might inform my decision to bring an umbrella—any confidence about Harry's baldness would be practically inert. IfI did have a credence in that proposition I could have pretty much any credence I like without negative consequences, and there would be no way to test what my credences were.This objection rests on a couple of fallacies. First, this theorist is using the word ‘true’ in a way that prevents one from substituting p for ‘p is true', whilst also equating uncertainty about whether or not p with uncertainty about whether p is true or false. Bythis theorist’s lights, these are not the same thing, and it remains far from clear why the lack of a truth value should preclude uncertainty about whether or not p. More to the point, the objection appears to rest on a strong version of the practical irrelevance thesis. According to the considerations in the previous sections of this chapter, one’s credences in the vague are not epiphenomenal: one can test a person’s credences in the vague by observing their behaviour and indiscriminately changing your credences in vague propositions can result in actions with negative consequences, even if they leave your credences in the precise the same.
Now my opponent in chapter 7 could concede this point and still insist that there’s no sense to be made of uncertainty when one considers there to be no fact of the matter. That is to say, the following concession is perfectly compatible with their view: Although being genuinely uncertain in the face of vagueness is a mistake, vagueness-related attitudes can still rationalize distinctive practical behaviour.
In this section, I want to explore this kind of response. I will argue that, even if this theorist is right about there being no such thing as ‘genuine’ uncertainty due to vagueness, there will be some probabilistic notion in the vicinity of uncertainty that informs our practical reasoning.
In particular, I shall argue that there is some kind of mental state, or mental feature that supervenes on your mental states, that does satisfy the axioms of the probability calculus, and moreover governs rational behaviour in the way we thought credences were supposed to. Whether we want to go one step further and call this thing a measure of‘genuine uncertainty’ can be left up for debate. For most purposes, however, this conclusion is strong enough.To illustrate the idea, let me draw an analogy with another debate in the literature. An extremely natural way to interpret the standard formalism of quantum mechanics (without augmenting it or taking it to be a partial description of the fundamental facts) is to treat the classical macroscopic reality around us—us and the things we observe— as constantly splitting into multiple ‘branches’ (see Everett [43]). So, for example, when I flip a coin which has a genuine chance of landing heads and of landing tails, we can expect that in reality there will be one branch in which the coin lands heads and another in which it lands tails. For the time being, let us set aside discussion of the merits of this idea, and assume that this is a correct description of the fundamental universe. According to this picture, there’s a straightforward difficulty for making sense of probability: I know exactly what’s going to happen in the coin flipping case— there’s going to be (with certainty) a branch in which the coin lands heads and another branch in which it lands tails. Since all the potential outcomes actually occur on this picture, it’s impossible to make sense of uncertainty about which outcome will occur.
This was a long-standing obstacle to this interpretation of quantum mechanics, and, for many, it was a decisive objection: the power of quantum theory comes from its ability to predict the chances of things happening, after all, so there is no place for an interpretation that cannot make sense of probability.
However, David Deutsch [31] showed that, even if there'snoroomfor uncertaintyonthiskindofhypothesis, onecan still make sense of probability by looking at rational action.[143] More importantly for that project, he was able to show that a person who acts rationally (and knows some of the relevant quantum theory) will act as though she was uncertain about the outcomes and, moreover, uncertain to the degrees that quantum theory predicts (i.e. as given by the Born rule).Note that if there's no such thing as genuine uncertainty in the many worlds interpretation of quantum mechanics, or indeed, in the case of vagueness, then there's a prima facie puzzle about how to interpret the decision theory, stated in terms of value functions, that I outlined. For example, Averaging explicitly appeals to credences, and if there is no real uncertainty in the cases of interest, then this appeal is problematic. This is solved in Deutsch's framework by dispensing with talk of values and credences, and just talking about an agent's preferences over various acts—things that have a direct behavioural interpretation. In our framework, we can take these to be given by our suppositional preferences over propositions, as defined in section 9.1. While our preferences over action propositions have the clearest behavioural interpretation, even our preferences over other propositions can manifest themselves as dispositional behavioural properties: counterfactuals like ‘if I were in a position to make A or B true, I would make A true' can ground many of your suppositional preferences.
Just as Deutsch was able to do in the many worlds case, we can reconstruct talk of probability, without taking it for granted that people are uncertain in the face of vagueness. In order to do this one must make a few assumptions about the agent's preference relation: they must obey some natural constraints that seem natural for a preference relation to be rational.
One's preferences ought to be irreflexive, asymmetric, and transitive, for instance. You cannot prefer A to B, and B to C without preferring A to C, you cannot prefer A over itself and if you prefer A to B you shouldn't prefer B to A. (Similarly obvious things can be said about the relations being ‘preferable to or as preferable as', ‘as preferable as'.)Another constraint is that preferences be linear. Any two propositions are either as preferable as one another, or one is more preferable than the other. This principle is perhaps more controversial, although as in our discussion in section 7.3.2, its non- obvious appearance doesn't seem to be distinctive to vagueness. One can also state a version of Averaging purely in terms preference, without invoking credences. A straightforward consequence of Averaging is that if E1 and E2 partition A into two,
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