## Cian Dorr on Imprecise Credences

In the latest Philosophical Perspectives, Cian Dorr has a very interesting paper about a puzzle about what he calls the Eternal Coin. I hope to write more about the particular puzzle in future posts, but I wanted to mention one thing that comes up in passing about imprecise probabilities. In the course of rejecting a solution to one puzzle in terms of imprecise probabilities, he says

I actually don’t think that imprecise (or unsharp) credences are the solution to the particular problem Cian is interested in here; I think the solution is to say the relevant credences are undefined, not imprecise. But I don’t think this is a compelling objection to imprecise credences either.

It is, I think, pretty easy to say what the behavioural difference is between imprecise credences and sharp credences, even if we accept (as I do!) what Adam and Cian have to say about decision making with imprecise credences. The difference comes up in the context of giving advice and evaluating others’ actions. Let’s say that my credence in p is imprecise over a range of about 0.4 to 0.9, and that I make decisions as if my credence is 0.7. Assume also that I have to make a choice between two options, X and Y, where X has a higher expected return iff p is more likely than not. So I choose X. And assume that you have the same evidence as me, and face the same choice.

On the sharp credences framework, I should advise you to do X, and should be critical of you if you don’t do X. On the imprecise credences framework, I should say that you could rationally make either choice (depending on what other choices you had previously made), and shouldn’t criticise you for making either choice (unless it was inconsistent with other choices you’d previously made).

I don’t want to argue here that it makes sense to separate out the role of credences in decision making from the role they play in advice and evaluation. All I do want to argue here is that once we move beyond decision making, and think about advice and evaluation as well, there is a functional difference between sharp and unsharp credences. So the functionalist argument that there is no new state here collapses.

One other note about this argument. I don’t think of sharp and unsharp credences as different kinds of states, or as states that need to be separately postulated and justified. What I think are the fundamental states are comparative credences. The claim that all credences are sharp then becomes the (wildly implausible) claim that all comparative credences satisfy certain structural properties that allow for a sharp representation. The claim that all credences should be sharp becomes the (still implausible, but not crazy) claim that all comparative credences should satisfy those structural properties. Either way, there’s nothing new about unsharp credences that needs to be justified. What needs to be justified is the ruling out of some structural possibilities that look prima facie attractive.

## What is the Equal Weight View of Disagreement?

I often find it hard to apply the Equal Weight View (EWV) in practice. This makes it my task of counterexample generating a little harder than I feel it should be. I can come up with all sorts of cases where I think EWV gets the wrong result, but then I get worried that EWV doesn’t actually say what I think it says about that case. Here’s one example I was working with.

A and B are peers in the salient sense. They have a long track record of checking each other’s work, and they both get things right a high and equal proportion of the time. There is no external evidence that B is in any way epistemically compromised right now. They both try to work out 14 times 27, and A gets 378, while B gets 368. What should A’s credence be that the right answer is 368?

I think the EWV is committed to the answer being 0.5 or thereabouts. After all, A and B are peers, they are just as likely to get the answer right, and probably one of them did get the answer right. So the EWV-endorsed probability distribution, I would think, is that the answers 378 and 368 both get probability nearly 0.5, and the remainder goes to the possibility that they were both wrong.

This strikes me as implausible, since it is easy for A to see that 368 is the wrong answer by using the rule I’ll call D9.

D9. A number is a multiple of 9 iff the sum of the digits of its base-10 representation is a multiple of 9.

So I think this is a case where EWV is wrong, A shouldn’t assign equal weight to 378 and 368 being the correct answer. I can imagine some people denying this, and saying that 378 and 368 should be given equal weight. But I can also imagine some people denying that EWV really has that consequence.

If you’re an EWV-theorist, do you think EWV entails in this case that A should give equal credence to 378 and 368 being the correct answer? If the case is too vaguely described to answer that, consider some of the following variations.

Variation 1. A doesn’t commit to an answer before checking that it is consistent with D9. So that the answer 378 is consistent with D9 is part of her reason for believing the answer is 378. That means, I think, that Christensen’s Independence principle would rule out her going on to use D9 to conclude that B must have made a mistake.

Variation 2. B has never heard of D9. Perhaps this means A and B aren’t peers, because D9 is some evidence that A has and B lacks.

Variation 3. B doesn’t believe D9. Perhaps that’s because he thinks A is misremembering the rule (It’s really a rule for multiples of 11, not multiples of 9), or perhaps because he thinks there are restrictions on the rule (e.g., it is only guaranteed to work for numbers with an even number of digits).

Variation 4. B denies that all multiples of 27 are multiples of 9.

Variation 5. B denies that his answer is inconsistent with D9, since 3+6+8 = 18, while 3+7+8 = 19, so D9 actually supports his answer, not A’s.

I can sort of see how an EWV theorist would deny that EWV applies in variations 2 and 4, but in all the other cases, it seems to me that EWV implies, incorrectly, that A should give equal credence to 368 and 378 being the correct answer. But maybe that’s just because I haven’t understood EWV correctly. Anyone want to correct my understanding?

# Philosophy Compass

## May 2011

Volume 6, Issue 5, Pages 300–373

### Continental

Transcendental Arguments About Other Minds and Intersubjectivity (pages 300–311)
Matheson Russell and Jack Reynolds
Article first published online: 4 MAY 2011 | DOI: 10.1111/j.1747-9991.2011.00394.x

### Epistemology

Bayesianism I: Introduction and Arguments in Favor, (pages 312–320)
Kenny Easwaran
Article first published online: 4 MAY 2011 | DOI: 10.1111/j.1747-9991.2011.00399.x

Bayesianism II: Applications and Criticisms, (pages 321–332)
Kenny Easwaran
Article first published online: 4 MAY 2011 | DOI: 10.1111/j.1747-9991.2011.00398.x

### History of Philosophy

Reidian Metaethics: Part I, (pages 333–340)
Terence Cuneo
Article first published online: 4 MAY 2011 | DOI: 10.1111/j.1747-9991.2011.00393.x

Reidian Metaethics: Part II, (pages 341–349)
Terence Cuneo
Article first published online: 4 MAY 2011 | DOI: 10.1111/j.1747-9991.2011.00392.x

### Logic & Language

Technical Modal Logic, (pages 350–359)
Marcus Kracht
Article first published online: 4 MAY 2011 | DOI: 10.1111/j.1747-9991.2011.00396.x