Let I(c, w, p) be an Inaccuracy function. The three arguments are a credence function c, a world w, and a proposition p. Assume for now just that I is truth-directed, so if c(p) is closer to the truth value of p in w, I says that c is less inaccurate.1
1 More formally, identify worlds with functions from propositions to {0, 1}, where 0 is false and 1 is true. I is truth-directed if whenever c1(p) is closer to w(p) than c2(p) is, then I(c1, w, p) < I(c2, w, p).
Extend I so that the third argument can be a finite set of propositions P as follows:
\[ I(c, w, P) = \frac{\sum_{p \in P} I(c, w, p)}{|P|} \]
Roughly speaking, this is to say that I is an additive inaccuracy measure, though not everyone includes the normalisation (i.e., dividing by |P|) when defining what it is for I to be additive, so I want to be a bit careful on the terminology here.2
2 To be sure, when everything is finite, the dividing by |P| doesn’t really matter. But it will become important in the more general case, so I’m being fussy about it here.
Assume now that W is a finite set of worlds. Say O is an ordering of W if it is a subset of P(W) (i.e., the powerset of W) satisfying the following constraints.
\[ \begin{align*} |O| &= |W| \\ W &\in O \\ \emptyset &\notin O \\ \text{If }1 \leq n \leq |W|&, \exists !x \in O: |x| = n \end{align*} \]
Intuitively, O is generated by arranging the elements of W in order, and it is the set consisting of the singleton of the first element in the order, the pair set of the first two elements in the order, the set of the first three elements in the order, and so on up to the full set.
That’s enough definitions; here’s the result I’ve been trying, and failing, to prove.
Assume that for some finite W, and world w in W, the following is true for any ordering O of W.
\[ I(c_1, w, O) < I(c_2, w, O) \]
Question: does this result follow?
\[ I(c_1, w, P(W)) \]
That is, if c1 is less inaccurate than c2 on every ordering of W, is it less inaccurate than c2 on the full algebra based on W?
Here’s a more restricted version of this question. Does this result hold if we also assume that c1 and c2 are ‘coherent’, i.e., are probability functions?
The reason I’m interested in this is that if this result holds, I think it can be used to motivate an argument for countable additivity. When W is countably infinite, P(W) is uncountable, so I(c, w, P(W)) isn’t going to be easily definable. But we can ask the following. Let Ok (for integer k) be the subset of O consisting of sets of size k or smaller. Then we can ask whether c1 order-dominates c2, i.e., whether the following holds for all w, O.
\[ lim_{k \rightarrow \infty} I(c_1, w, O_k) < lim_{k \rightarrow \infty} I(c_2, w, O_k) \]
That’s a very strong constraint. But if c1 is countably additive, and c2 is not, then (I think!) there will be a set W such that c1 does in fact order-dominate c2 with respect to W. And that seems like a sufficient reason to say that c2 is (to use a technical term) bad.
But all this turns on the thought that order-domination is a genuine constraint in the finite case, and I can’t figure out whether that’s true.