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gjm comments on Is Belief in Belief a Useful Concept? - Less Wrong Discussion
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It looks to me as if this conflates two distinctions that, a priori, seem like separate things.
I can see why these might turn out to be related -- e.g., because those voluntarily adopted beliefs are adopted explicitly, using language, which tends to encourage propositional reasoning, whereas implicit unconscious beliefs are handled by fuzzy neural-network things inside our hardware. So maybe the two distinctions do go together.
But the names you've adopted for them are based on the binary-versus-qualitative distinction, whereas (I think) what's important about them -- what determines the role they play in our thought and speech and action -- is the voluntary-versus-automatic distinction. (A person could voluntarily decide to act as though the probability of some important proposition is 75%. That would be better treated as what you're calling a "binary belief", no?)
And it seems to me that if we adopt terminology adapted to the distinction that's actually relevant here -- say, "voluntary" versus "automatic" -- we've gone much of the way back to talking about "belief in belief" again. (Not all the way, but I have another point to make about that which I'll put in another comment because it's logically separate.)
You are probably right about the voluntary or non-voluntary aspects of the behavior being more important here. I can certainly choose to say "the probability of X is 75%" and choose to conform my behavior to that. So maybe the article could be rewritten to emphasize that aspect more. But it would still happen that people have two kinds of assessment of reality which do not necessarily line up completely: a voluntary one and an automatic one. The problem I see with "belief in belief" is that it seems to suggest that people are wrong about what assessment they have. Instead of this, it seems to me that what people are talking about when they say they believe something is precisely the voluntary assessment. When they say "I believe in God", they mean that they are choosing to act -- including in their own minds, insofar as they can control this -- as if God exists. They do not necessarily mean that they have a high automatic assessment of the idea that God exists. I agree that if they did mean the latter, you would sometimes find people who are mistaken about their own assessment. One thing I did not talk about in the article is the fact that people are also more or less consciously aware of the contrast between their automatic assessment and their voluntary assessment. A religious person I know said he would he happy with a 30% chance his religion was true, but he did not mean by that he would act as though it had a 30%; he acts as though it is absolutely true. So he realizes that his voluntary assessment and his automatic assessment do not match.