7. Jim believes that 2+2=4
8. Jim believes that if a function f is continuous on the interval (a,b) and if F is a function whose derivative is F on the interval (a,b) then the integral from a to b of f-of-x with respect to x equals F-of-b minus F-of-a

So why not use whatever structured proposition Millians use to explain the apparent difference between 1 and 2 to explain the apparent difference between 7 and 8? I think there are two answers. One is that there is a clear sense in which the ‘that’-clauses in 7 and 8 involve claims about different things. To the extent we can respect this, we should. The second is that it seems clear that Jim might have what it takes for 7 to be true, given his cognitive sophistication, while lacking what it takes for 8 to be true. Neither of these points apply to 1 and 2 however. But the issue, I thought, was whether something stops us from extending proposals dealing with 1 and 2 to 5 and 6 or 7 and 8. And I think there is: aboutness and cognitive sophistication. That isn’t a statement of the problem; it’s the reason why we should not extend Millian structured propositionalist strategies to deal with 1 and 2 to more coarse-grained theories of propositions.

I’m not sure I know what you mean by ‘aboutness’. I can understand someone saying that propositions are about different things if they have different constituents, assuming that they have constituents. That seems right and helps with (5) and (6) but not (1) and (2). But that’s just what we’ve been talking about.

I share the feeling that we ought to distinguish necessarily equivalent claims, for instance that’s a good reason to think that propositions aren’t functions from worlds to truth-values. That feeling might be sharpened a little by putting it in terms of these claims being about different things. Is that what you were getting at? That still sounds to me like statement of the problem rather than something that helps to solve it.

Purely pragmatic explanations fail: see David Braun’s “Cognitive Significance, Attitude Ascriptions, and Ways of Believing”. But there are other options for proponents of the view.

The main reason, I think, for not extending Braun’s or Salmon’s proposal to 5 and 6 has to do with aboutness. Intuitively, propositions that are logically or necessarily equivalent are not therefore about the same things. We can and should respect that when theorizing about propositions and our attitudes toward them. Another related point is that it seems that one could have the belief that Christine won but not that she neither lost nor failed to complete since one might lack the cognitive sophistication or relevant relations to the community to grasp the latter, but that is not an impediment to grasping the former.