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May 16th, 2011

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?

Posted by Brian Weatherson in Uncategorized

2 Comments »

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2 Responses to “What is the Equal Weight View of Disagreement?”

  1. Stinky Koala says:

    Disclaimer: I am not by any means an EWV theorist. However, it strikes me that the application of EWV is precisely when (as Brian is careful to set up) there is no other evidence to distinguish the sources of your two possible answers. Further, distinguishing which is actually correct (say using rule D9) is a separate question; EWV seems to only apply to the question of judging who is more likely to be correct given knowledge about A and B.

    This seems to me to partition our estimates up as follows.

    1) If A and B are both peers, and all else is equal, as you say we must estimate the likelihood of A’s answer to be identical to that of B’s answer. We may use a principle of our own (such as D9) to incorporate more information, but this is independent of the application of EWV (rather than other methods) to decide who is likely correct.

    2) If A and B were not truly peers, the above analysis doesn’t apply; or at least it doesn’t apply with the equal weight precision. (Presumably if we can specify the extent to which B is more often correct than A, we get a revised expectation of which answer is correct.) As you say, Brian, this would be the case (to some degree) if A were aware of D9 but B were not.

    3) If A and B truly were peers and we discover that B was wrong and and A was correct, we then revise our estimation that A and B were indistinguishable in reliability.

    I think most of your situations are subsumed by my #2 — there are a variety of ways in which one person can be less reliable than another, but I don’t see that that relates very strongly to EWV. My impression is that EWV is conditional on both parties being equivalently reliable.

    I tend to think about the application of EWV as follows: we presuppose an observer trying to evaluate a scenario which he is not qualified to evaluate, by meta-analyzing the observations of (say) two people who are (to some degree) qualified. This observer is familiar with the qualifications of the two people, and is trying to formulate his evaluation appropriately. (If this is wholly incorrect, please let me know! I am — appropriately to the discussion — no expert. Hopefully I won’t receive contradictory responses.)

    If what I said above is reasonable, our analysis of A’s answer vs. B’s answer using rule D9 would fall outside the scope of EWV.

  2. Richard Chappell says:

    Variation 1 might be escaped on the grounds that there is a significant procedural asymmetry between how A and B reached their answers. That is: A employed a ‘check’ before committing to her answer, whereas B did not. In that case, both agents should presumably share a high prior confidence that if just one of them performs an extra ‘check’ before committing to an answer, that person is more likely right than the person who didn’t check their working (or who didn’t check it using such an apparently reliable method). Or were you thinking that it makes an important difference that the check is employed before rather than after committing to the answer? (That would seem an unattractive feature to build into the view.)

    Variations 3 and 5 strike me as good cases, assuming that B is just as baffled by A’s “obvious mistakes” as A is by B’s (to equalize the phenomenological evidence).

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