If we were thinking in a purely Bayesian framework, we would have to say that the two parties have different priors for the probability of the CEF thesis conditional on whatever evidence they cite, since they each look at that evidence and wind up with different posteriors. So Aumann’s agreement result wouldn’t hold.

Then the idea is that this applies to EW itself; we want c to be equal to the weighted sum of what the methods of EW tell you credence in EW should be and what non-EW methods tell you credence in EW should be. That yields the equation I gave. (Actually on second thought I think it yields a different equation that makes c 2/3.)

But this depends on the assumption that, on non-EW methods, the credence I get for EW is 1. You ask, fairly enough, “If I think EW is wrong why shouldn’t my credence in EW be 0?” The answer is that my reasons for thinking EW is wrong may not have conformed to the methods that I ought to use if I think EW is wrong. If EW is wrong then I should just evaluate arguments for myself, without regard to what my peers think. But if I do that, then ex hypothesi I get a credence of 1 in EW, since I find the arguments for it entirely convincing.

Put another way, we could also ask “If I think EW is right why shouldn’t my credence in EW be 1?” If that were right then Brian’s argument wouldn’t go through. The problem is that the methods you use to decide that EW was right weren’t the methods that EW itself prescribes.

(Actually, this assumes that the only alternatives are that we give equal weight to peers and that we give no weight to peers. If we consider a position on which we should give unequal weight to peers, maybe that would change the argument.)

So now you should think, “Well, there’s a 0.5 chance that EW is right and I should just average my credence and his credence — yielding 0.25 — and there’s a 0.5 chance EW is wrong and I should just trust myself — yielding 1.

Am I misreading this. If I think that EW is wrong (as you seem to say in the last sentence above) why would I trust myself and return to my initial credence for EW, which was 1? Why would I put any credence for EW—if I think it’s wrong—other than 0?

So now you should think, “Well, there’s a 0.5 chance that EW is right and I should just average my credence and his credence — yielding 0.25 — and there’s a 0.5 chance EW is wrong and I should just trust myself — yielding 1. So my total credence in EW should be 0.625.” Then you think, “Now there’s a 0.625 chance that I should just average my credence and his credence — yielding 0.3125 — and there’s a 0.375 chance that I should just trust myself — yielding 1. So my total credence in EW should be 0.625 * 0.3125 + 0.375 * 1 = 0.570.” Etc. This sequence converges to a fixed point at 0.586.

As for my point of view, where I’m the person who doesn’t find the arguments for EW convincing at all, there doesn’t seem to be a problem. I have a credence of 0 in EW, so I should do what non-EW tells me to do: evaluate the arguments over EW for myself, ignoring what you think. Again, this yields a credence of 0 in EW. So my credence of 0 in EW is stable.

Matt,

Suppose I find it equally credible that my credence for EW is correct and Your credence for EW is correct. You place no credence in EW. I place complete credence in EW. From my point of view,(though not yours) I should weight the truth of my credence for EW at .5 and I should weight the truth of Your credence for EW at .5. So, shouldn’t it look like .5(1)EW + .5(0)EW = .5? That is, shouldn’t I weight these opposing credences for EW on the assumption alone that EW is correct, since that is what I believe prior to our disagreement over EW? From your point of view, there is a .5 credence that EW is correct, and so a .5 credence that you should move to the midpoint between 0 and 1. There is also a .5 credence that you are right and that EW has no credibility (in that case you should not move from 0). It should then look like .5(.5)EW + .5(0)EW = .25. No?

Mike, that’s pretty much what I’m suggesting you should do. In this case, if you were disregarding your peer (as not-EW counsels), you would give EW a credence of 1. If you give equal weight to your own view and your peer’s (as EW counsels), then you would give EW a credence of half what you actually give it. If we weight these two by your credence in not-EW and EW, we get that the credence in EW (call it c) should be (1 – c) * 1 + c * c/2. (Each term is the credence in not-EW/EW times the credence that not-EW/EW counsels that you give to EW.) The only <1 solution to this equation is 2 – √2 = 0.586.