Probably Possible

Beliefs represent the world—or so goes received wisdom. I believe that Canberra is the capital of Australia iff I represent the world as being such. I don’t believe that Singapore lies north of the equator iff I don’t represent the world as being such.

If belief is all-or-nothing, then either we represent the world as being such-and-such, or we don’t. But Frank Ramsey, according to Richard Jeffrey, has “sucked the marrow” out of such a notion of belief. The correct way to think about belief, according to Jeffrey, is to think of it as coming in degrees. Assuming that this is right, how should we (re-)read the claim that beliefs represent the world?

We should not read the claim as saying that whenever my degree of belief in p is x, I represent the world as being such that it is probable to degree x that p. Suppose my degree of belief in my existing is 1. It seems that I represent the world as being such that I exist, and not that it is probable to the highest degree that I exist. Also, whether by `probable’, we mean `objectively probable’, `epistemically probable’, or `subjectively probable’, etc., a child may have a degree of belief in p without having any concept of probability. (And if by `probable’, we mean `subjectively probable’, then as wo pointed out to me, we will get into a self-referential loop.)

One might think that we can reduce talk about all-or-nothing belief to talk about degrees of belief. The threshold view of all-or-nothing belief (TVB) says that we have an all-or-nothing belief that p iff our degree of belief in p meets a certain threshold. If we have an all-or-nothing belief that p iff we represent the world as being such that p, then from TVB, we get the threshold view of representation (TVP), which says that we represent the world as being such that p iff our degree of belief in p meets a certain threshold.

The above argument for TVP seems attractive at first glance, but it falters on grounds on which TVB falters. Suppose I hold one out of a million tickets in a fair lottery in which there’s exactly one winner. Although my degree of belief in my ticket being the losing ticket is very high (0.999999), I might not believe outright that I’ll lose. Let there be as many tickets in the lottery as you wish—the point is that so long as the threshold for belief falls short of 1, there’ll be a counterexample to TVB. But if the threshold for belief is 1, then it seems that we hardly have any beliefs, for to have a degree of belief of 1 in p would mean that one is disposed to bet everything that one values on p, and we are hardly ever willing to make such bets.

Also, as Mark Kaplan points out in his Decision Theory as Philosophy, TVB makes nonsense of Descartes’s injunction to refrain from believing what one doubts. If TVB is right, and the threshold for belief is less than 1, then so long as my degree of belief in p meets the threshold, I cannot but believe p. It’s also worth pointing out that if the threshold for belief is 1, then Descartes’s injunction is pointless: I cannot but believe only what I don’t doubt.

To meet such objections, one might suggest that the threshold for belief varies with context. Perhaps, in a context in which the lottery is salient to me, the threshold for belief is 1, whereas in other contexts, the threshold for belief is slightly lower. Such a contexualised version of TVB (CTVB) might tell us something about how we use the word `belief’, but it does not sit easy with the view that beliefs represent the world. If we have an all-or-nothing belief that p iff we represent the world as being such that p, then from CTVB, we get the contextualised threshold view of representation (CTVP), which says that we represent the world as being such that p iff our degree of belief in p meets a certain threshold that varies with context. But whether I represent the world as being such that p doesn’t depend on context in the way that CTVP predicts. My store of information relevant to whether p needn’t change when what is salient changes.

Perhaps, if the correct way to think about belief is in terms of degrees of belief, we should not try to argue for TVP via TVB. Perhaps, we should just set aside all talk of all-or-nothing belief, and see whether TVP stands on its own merits. But does it? For example, does the lottery case pose any problem for TVP?

Suppose that the relevant threshold is 0.95. By TVP, since my degree of belief in my ticket losing is 0.999999, I represent the world as being such that my ticket will lose. But must I represent the world in such a way? Intuitively, no. We could imagine that I act in ways that suggest that I don’t represent the world as being such that the ticket will lose—for example, I might keep the ticket until the result of the draw is known. Also, suppose that to represent the world as being such that p is to carve out the space of doxastic possibilities into two regions—one with possibilities consistent with p, and one with possibilities inconsistent with p. Since my degree of belief in my ticket winning is not 0, it’s doxastically possible for me that it wins, and my store of information about the world should somehow reflect this. Hence, the region of possibilities consistent with my representation of the world should consist of a possibility in which my ticket will win. But if so, then I don’t represent the world as being such that my ticket will lose.

Might the upshot of all this be that representation, like belief, comes in degrees? Might it be that the higher my degree of belief in p, the more strongly I represent the world as being such that p? If to represent the world is to store information about the world, then the idea that representation comes in degrees makes sense only to the extent that the idea that the storage of information about the world comes in degrees makes sense. And it’s not clear to me how to make sense of the latter. Either one stores certain information about the world, or one doesn’t.

But perhaps, there are ways to store information about the world while somehow registering one’s confidence in the accuracy of the information. On one view, to represent the world as being such that p is to somehow store the information that p in one’s belief box. But perhaps, instead of there being one belief box, there are many belief boxes—one for each degree of belief. So if my degrees of belief in p, q, and r are 0.8, 0.8, and 0.3 respectively, p and q will be stored in belief box 0.8, and r in belief box 0.3. But then, this leads to a proliferation of belief boxes. Or perhaps, p, q, and r are all stored in a single belief box, but each belief is somehow tagged with a number indicating its strength (metaphorically speaking!). But then, one’s belief box is going to be chock-full of beliefs. Unfortunately, even if talk of belief boxes is meant to be understood metaphorically, I’m not sure what to make of either a proliferation of belief boxes, or chock-full belief boxes. Anyway, it might be worth looking at how various theories of how belief represents the world fare when we shift the focus from all-or-nothing belief to degrees of belief.

(Thanks to wo for discussion.)

Posted by wenghong on 2nd May 2008 under Belief, Credences, Mind. Leave a response?