The heuristic of averaging beliefs is clearly poor. We all accept that in principle we should agree on the same truth (subject only to differences in irreducible priors) after we share our evidence and update on the union; if we just average the beliefs of some people (expert or otherwise), we under- or over-count some evidence. But good luck finding a willing expert partner for such an exercise - one both capable of really doing it, and generous enough to go to the expense.
You seem to suggest a slightly more subtle heuristic for combining conflicting experts' views, but it's still imperfect.
For example, suppose there are contested facts X and Y. Suppose all 3 experts agree that X->Q and Y->Q, but 1 holds (X and not Y), 1 (Y and not X), and the other (not X and not Y). I claim that this is effectively a 2/3 vote for Q, even though there's a 2/3 vote against both X and Y, although of course as a practical matter such wild disagreement amongst "experts" makes me suspicious of their credentials :) I think this is acceptable even if X is christianity, Y is islam, and Q is "some sort of afterlife". I just wouldn't make the mistake of doing no evaluation of the evidence for X and Y myself.
Here's where your heuristic would work: 1/3 expert holds "A and A->G", 1/3 holds "B and B->G", and 1/3 denies all 4 statements. This should be interpreted at best as a 1/3 vote for G (maybe you think it's no evidence at all?)
I do like your heuristic.
I would say it requires "A and A->G and not B" and "B and B->G and not A"
such wild disagreement amongst "experts" makes me suspicious of their credentials
I think that's part of what I'm trying to quantify here. when there's little direct evidence(or we don't understand it ourselves), and a lot of thinking, experts are pretty much defined by the opinions of other experts. If we want to guess at the reliability of their conclusions, the only track record we have is how often other experts agree with them.
If a majority of experts agree on an issue, a rationalist should be prepared to defer to their judgment. It is reasonable to expect that the experts have superior knowledge and have considered many more arguments than a lay person would be able to. However, if experts are split into camps that reject each other's arguments, then it is rational to take their expert rejections into account. This is the case even among experts that support the same conclusion.
If 2/3's of experts support proposition G , 1/3 because of reason A while rejecting B, and 1/3 because of reason B while rejecting A, and the remaining 1/3 reject both A and B; then the majority Reject A, and the majority Reject B. G should not be treated as a reasonable majority view.
This should be clear if A is the koran and B is the bible.
Positions that fundamentally disagree don't combine in dependent aspects on which they agree. On the contrary, If people offer lots of different contradictory reasons for a conclusion (even if each individual has consistent beliefs) it is a sign that they are rationalizing their position.
An exception to this is if experts agree on something for the same proximal reasons. If pharmacists were split into camps that disagreed on what atoms fundamentally were, but agreed on how chemistry and biology worked, then we could add those camps together as authorities on what the effect of a drug would be.
If we're going to add up expert views, we need to add up what experts consider important about a question and agree on, not individual features of their conclusions.
Some differing reasons can be additive: Evolution has support from many fields. We can add the analysis of all these experts together because the paleontologists do not generally dispute the arguments of geneticists.
Different people might justify vegetarianism by citing the suffering of animals, health benefits, environmental impacts, or purely spiritual concerns. As long as there isn't a camp of vegetarians that claim it does not have e.g. redeeming health benefits, we can more or less add all those opinions together.
We shouldn't add up two experts if they would consider each other's arguments irrational. That's ignoring their expertise.
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