Suppose we have some doxastic or epistemic status—say, belief or knowledge—that involves a credence threshold, such as that to count as believing *p*, you need to assign a credence of, say, at least 0.9 to *p*. I used to think that propositions that meet the threshold are apt to have credences distributed somewhat uniformly between the threshold or 1. But now I think this may be completely wrong.

**Toy model:** A perfectly rational agent has a probability space with *N* options and assigns equal credence to each option. There are 2^{N} propositions (up to logical equivalence) that can be formed concerning the *N* options, e.g., “option 1 or option 2 or option 3”, one for each subset of the *N* options.

Given the toy model, for a threshold that is not too close to 0.5, and for a moderately large *N* (say, 10 or more), most of the 2^{N} propositions that meet the threshold condition meet it just barely. The reason for that is this. A proposition can be identified with a subset of {1, ..., *N*}. The probability of the proposition is *k*/*N* where *k* is the number of elements in the subset. For any integer *k* between 0 and *N*, the number of propositions that have probability *k*/*N* will then be the binomial coefficient *k*!(*N* − *k*)!/*N*!. But when we look at this as a function of *k*, it will have roughly a normal distribution with standard deviation *σ* = *N*^{1/2}/2 and center at *N*/2, and that distribution decays very fast, so most of the propositions that have probability at least *k*/*N* will have probability pretty close to *k*/*N* if *k*/*N* − 1/2 is significantly bigger than 1/*N*^{1/2}.

I should have some graphs here, but it’s a really busy week.

## 2 comments:

I would think off the top of my head that propositions would have credences distributed between 0 and 1. Somehow. Maybe normally, maybe bimodally if we give ourselves a little credit. But the point is that the threshold is not an interesting factor in the distribution. You choose a threshold for some other reason. Then there is no particular reason to think that the distribution between the threshold and 1 is mathematically interesting.

I think that matches what you are saying in the post.

Right. My simple model in the post predicts that proposition credences have something close to a normal distribution centered on 1/2.

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