A puzzle about belief reports
Consider a long list S1...Sn of sentences such that (a) each Si is trivially equivalent to its predecessor and successor (if any), and (b) S1 is not trivially equivalent to Sn.
For example, S1 might be a complicated mathematical or logical statement, and S1...Sn a process of slowly transforming S1 into a simpler expression. For another example, S1...Sn might be statements in different languages, where each Si qualifies as a direct translation of its neighbor(s) but S1 is not a direct translation of Sn.
Next imagine a series of belief reports R1...Rn (in possibly different languages) whose complement sentences are S1...Sn respectively. So, if Si is in English, then Ri has the form 'x believes that Si'.
Since the equivalence between S1 and Sn is far from obvious, one can easily think of cases in which R1 is true and Rn false (or vice versa), for a fixed subject x.
The puzzle is that R1...Rn then looks a lot like a sorites series. For each Si with i<n, one might think that if one can truly attribute to an agent a belief with complement Si, then one can also truly attribute to her a belief with complement S{i+1}.
Informally, the point is that in ordinary contexts, belief reports are not sensitive to minor variations in the complement sentence. If you believe that Caesar conquered Gaul, you also believe that Gaul was conquered by Caesar; if you believe that it's raining one can also report, in French, that tu crois qu'il pleut. But a lot of minor variations can add up to a major variation, and belief attributions are sensitive to major variations. You can believe that 237 * 78 = 18286 without believing that 0 = 1.
The puzzle is not that 'believe' is vague. That is hardly surprising. The puzzle is that 'X and Y are minor variations' is intransitive while 'X and Y attribute the same belief' is transitive. So the following principle must be false:
Replacing the complement sentence in a belief report by a minor variation or translation does not affect what belief is attributed.
Since we can't say that, what can we say instead?
One option is to say that beliefs are much more fine-grained than we might have thought, so that different belief reports practically never attribute the same belief: believing that Caesar conquered Gaul is not the same as believing that Gaul was conquered by Caesar; the belief attributed in English with 'believes that it rains' cannot be attributed in French; and so on. Each element of R1...Rn thus attributes a belief with a different content. The puzzle can then be resolved by assuming that 'believe' is vague, so that it is indeterminate whether or not the agent believes some of contents attributed in between R1 and Rn.
But this looks unsatisfactory. Can't we say that at least some minor variations or translations in the complement sentence make no difference to the attributed belief? We surely can. For example, we could say that R1 and R2 attribute the same belief, but not R2 and R3. Or that R1 through R10 all attribute the same belief, but not R10 and R11. Note the trade-off between these proposals. If we want to maintain that minor variations in complement sentences often don't correspond to different beliefs, we have to say that there are longish sub-sequence of R1...Rn all members of which attribute the same belief, while their immediate neighbours outside the subsequence do not. This also looks implausible. After all, the difference between S1 and S10 is much bigger than the difference between S10 and S11; yet only the latter difference is supposed to correspond to a genuine difference in belief!
I think the best answer is to return to a version of the first option. On this view, 'believe' semantically expresses a relation R between a subject and a very finely individuated entity -- the complement sentence itself, or a combination of the sentence and its truth-conditions, or something like that. But R isn't a very natural relation, and we shouldn't think of it as the "belief relation". Rather, there's a diverse list of factors that determine whether, in a given context, one can truly utter 'x believes that S'. Among them are, very roughly: (i) Is x disposed to assent to S, or a close translation of S? (ii) Is x disposed to act in a way characteristic of ordinary people who are disposed to assent to S? (iii) Is x in a state whose function is to occur under conditions which, together with certain facts we currently take for granted, entail the truth of S? And so on. There are many ways to make these questions precise, and to balance them against each other. None of them is once and for all privileged by our linguistic conventions. This is why belief reports are (context-sensitive) and vague.