Thresholds and credence

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.