April 04, 2005

Belief and Closure

In a 2002 Phil Review paper, discussed here at Certain Doubts, Jeremy Fantl and Matt McGrath argue that justification depends in part on pragmatic factors--Sally can be justified in believing p and Jane not justified in believing p, even though Sally and Jane have exactly the same evidence, because the stakes are higher for Jane than for Sally.

Brian is arguing that the difference doesn't come from pragmatic encroachment on justification; it comes from pragmatic encroachment on belief. To say that someone believes that p tout court is to say that her degree of belief in p passes a certain threshold, and that threshold is determined by pragmatics. So, if Sally and Jane believe that p to exactly the degree that is supported by their evidence, Sally will count as believing that p and Jane will not.

Brian suggests this definition (to a first approximation):

X believes that p iff for all actions A, B, X prefers A to B iff she prefers A & p to B & p


There are some complications to this, to deal with inconsistent agents and to deal with cases where p is of no practical importance, so it isnít perfect as it stands. But I think itís close enough to the truth to run with, at least properly understood

and going on to say that we must let the quantifier range over live practical options for X. (Look below the jump for an example of how this yields pragmatic encroachment on the degree of belief.)

I think Brian's proposed definition of belief leads to closure failures. Now, of course belief isn't closed under logical consequence; I may fail to believe all the logical consequences of my beliefs, because I haven't thought about them. But Brian's definition has (I think) the effect that I can have degrees of belief in p, q, and p & q that are in exact accord with the probability calculus--and yet I can believe p and q, and not believe p & q. That seems bad.

The failure of closure isn't exactly the one you might expect.

Here's the failure you might expect: On a view on which belief is belief to degree n or greater, closure will fail, because the property "having probability of n or greater" isn't closed under logical consequence (if n isn't 1 or 0). So you might say: Take a case in which p has probability 0.9, q has probability 0.9, and the threshold is between 0.9 and 0.81; then closure will fail.

That suggests a case in which there were three relevant pairs of actions, all of which require confidence of at least 0.833 in the relevant proposition for action; say, offering a bet at 5 to 1 odds.

(I'm just recapping my thought process, btw.)

So: assume you believe p and q to degree 0.9, and you believe they're independent, so you believe p & q to degree 0.81.

Action A = offering 5 to 1 odds against p (that is, if p is true you win 1, if p is false you lose 5); B = not doing so.
Action C = offering 5 to 1 odds against q; D = not doing so.
E = offering 5 to 1 odds against p & q; F = not doing so.

You prefer A to B, C to D, and F to E.

But, on Brian's definition, you don't believe either p or q. Because you have to consider whether you prefer E & p to F & p. And you believe that, given p, you'd have a 0.9 probability of winning that bet; which would make it worth offering. So you prefer E & p to F & p, even though you prefer F to E. Hence you don't count as believing p, on Brian's definittion. (q, the same.)

Here's the case where closure fails:

Action A = offering 5 to 1 odds against p; B = not doing so.
Action C = offering 5 to 1 odds against q; D = not doing so.
E = offering 99 to 1 odds against p & q; F = not doing so.

You prefer A to B, C to D, and F to E.

And you also prefer F & p to E & p, and F & q to E & q. You believe that, given p, you'd have a 0.9 probability of winning the bet that p & q. But you need a 0.99 probability in order to offer the bet. So, for all three pairs of actions, your preferences are unchanged by conjoining p to each option. Hence you believe p.

(I just noticed this: It seems as though you are indifferent between A and C, but you prefer A & p to C & p. Perhaps this shows that you don't believe p. But it seems to me that similar examples could be constructed to show that you don't believe just about anything. So I don't think this will rescue the argument.)

Similarly, you believe q.

However, you prefer E & p & q to F & p & q (since p & q guarantees that you win your bet), even though you prefer F to E; so you do not believe p & q. Hence your beliefs aren't and shouldn't be closed under logical consequence.

I was surprised at the way closure fails here--I don't have much more to say about the consequences, except that I have a funny feeling that this means that this way of defining belief isn't going to work at all.

But I should say that I'm extremely sympathetic with Brian's stated project, with the exception that I don't think that pure epistemology is going to turn out to be purely probabilistic:

there is no pragmatics in probabilistic epistemology, and hence no pragmatics in epistemology proper, but plenty of pragmatics in the relationship between probabilistic and non-probabilistic doxastic states, and hence pragmatics in non-probabilistic epistemology.

I think we can find a domain that's purely epistemic, with no pragmatic encroachment whatsoever. That's the domain circumscribed by question 5 here. But the way I want to go about this is not to try to rig things so that traditional epistemological terms such as "knowledge" or "justification" turn out to be free of pragmatic encroachment. Rather, we have to find or invent a concept that is free of pragmatic encroachment--I like "strength of evidence"--retreat within its walls, and throw the rest of the concepts to the pragmatic wolves.

Posted by Matt Weiner at April 4, 2005 09:21 AM

A couple of quick comments. I actually agree with most of the last paragraph. I don't rig things so "knowledge" is free of pragmatic encroachment - indeed it's set up so that knowledge has a pragmatic element. So does having a justified belief, though not having a justified degree of belief x. So I'm close to your view of what we do with pragmatics.

The point about comparing A&p to C&p is interesting. I'd rest a lot more weight on that than you do. I tend to have the following couple of beliefs, that maybe you don't share.

(a) Once we're in the position of being able to state a (non-unitary) numerical degree of belief in p, that's pretty close to not believing p.

(b) In practical situations where we are free to offer whatever odds we like, so things like offering bets (as well as accepting them) become practical options, we have fewer outright beliefs than we otherwise have.

So if that case shows that when those are practical options, we don't believe p and don't believe q, then I can live with that. I treat belief as an approximation to an ideal, and how close an approximation it needs to be is situation dependent.

Having said that, it's a really nice case. I'd played with some examples where I got some mileage out of comparing (something like) E&p to F&p or to A&p, but I hadn't thought of the A/C comparison, which looks like it does a lot of work.

None of this is to say that closure holds on the theory. In fact I think that it doesn't hold across propositions we don't think about. E.g. we can believe every proposition in the book and believe there is a mistake in the book, as long as we don't focus too much on too many of the 'intermediate' conjunctions of claims in the book.

Posted by: Brian Weatherson at April 4, 2005 09:36 AM

Agreed about your first paragraph--didn't mean to suggest that you wanted to free 'knowledge' from pragmatic encroachment. But aren't you trying to isolate the encroachment in 'having a justified belief' in the belief part? I may be misreading your post--maybe that only deals with the McGrath-Fantl case. I'm definitely in agreement about the lack of pragmatic approach on degree of belief(or whatever we want to call it), except I'm not positive that I want to sign on to "being justified in having a degree of belief x" as opposed to "having x degree of support for having a belief."

Part of the reason I didn't rest much weight on A&p vs. C&p was I only thought of it while I was typing up the other argument I'd thought of. I don't think I do agree with (a). As for (b), the intention wasn't that you have the ability to offer any bets--these are the only bets you can offer. Perhaps it would be best if I said you can accept bets at 1-to-5 odds or something.

Also, I was trying to be a good Bayesian and have bets stand in for other options. I might try to think of an analogous case involving bets, but I'm not sure where it leads....

I think you're right about propositions we don't think about--in fact, I think deductive closure for belief will turn out to be soritical in much the same way that I think it is for knowledge (which way that is is TBD). But that won't help us in this case, where all the relevant cases are being thought about.

Posted by: Matt Weiner at April 4, 2005 12:15 PM