Duals of knowledge and belief
On the modal analysis of belief, 'S believes that p' is true iff p is true at all possible worlds compatible with S's belief state. So 'believes' is a necessity modal. One might expect there to be a dual possibility modal, a verb V such that 'S Vs that p' is true iff p is true at some worlds compatible with S's belief state. But there doesn't seem to be any such verb in English (or German). Why not?
What do we use if we want to say that something is compatible with someone's beliefs? Suppose at some worlds compatible with Betty's belief state, it is currently snowing. We could express this by "Betty does not believe that it is not snowing". But (for some reason) that's really hard to parse.
Arguably, the most natural choice is: "Betty believes that it might be snowing". Here, the possibility modal 'might' is embedded under the necessity modal 'believes'. Clearly the embedded 'might' is relative to Betty's belief state: "Betty believes that it might be snowing" does not state that Betty believes that for all we know, it might be snowing. So 'might', in effect, serves as the dual of 'believes', but it has to be embedded under 'believes' because we need a transitive verb to indicate the person whose beliefs are compatible with the relevant proposition.
But why does "believes that might" express the dual of belief, rather than a higher-order belief about belief? Because the logic of belief is arguably KD45, and in KD45, □◇p is equivalent to ◇p.
In fact, this is a nice argument in favour of assuming that the logic of belief is (at least) KD45: the assumption explains why "believes that might" is commonly used to express the dual of belief, and why there's no need to introduce a separate verb for the dual.
What about knowledge? There is also no dual for 'knows' in English. But here the situation is different.
First, unlike "Betty does not believe that it is not snowing", "Betty does not know that it is not snowing" is not too hard to understand.
Second, the logic of knowledge is plausibly weaker than KD45, so "knows that might" is plausibly not equivalent to "not knows not". Indeed, "Betty knows that it might be snowing" does suggest that Betty has higher-order knowledge concerning the possibility of snow, rather than simply a first-order knowledge state that is compatible with snow.
So why don't we have a dual for 'knows'? The reason, I suspect, is that absense of knowledge is less unified than absence of belief. There are different reasons why someone might fail to know not-p, and it's useful to have different expressions for the different cases.
One reason why Betty might fail to know that it is not currently snowing is that it is in fact snowing. If it snowing, then Betty can't know that it is not snowing, because knowledge entails truth. But in such a case, the norms of pragmatics imply that instead of '~K~p' we should simply say 'p': it is shorter and more informative.
Another reason why Betty might fail to know that it is not currently snowing is that she fails to believe that it isn't snowing. If knowledge entails belief, then lack of belief entails lack of knowledge. So it might be more informative to use the dual of belief ('believes that might') rather than the dual of knowledge, especially if we also don't know whether it is snowing.
Third, if we don't know whether it is snowing, and we know that Betty doesn't know either, then it is usually better to say that Betty doesn't know whether it is snowing, rather than that she doesn't know that it is not snowing. Again, it's more informative, and not more complicated.
These don't cover all possibilities. Sometimes we may know that it is not snowing, and we want to communicate that Betty is not aware of this fact. In that case, we seem to fall back on 'not knows not': "Betty doesn't know that it is not snowing".
In sum, here's my conjecture:
1. We don't have a designated dual of 'believes' because we already have 'believes that might', which serves the same purpose.
2. We don't have a designated dual of 'knows' because there are usually more informative things to say, and we have the means to say these more informative things.
Cool post!
A thought: I think to really express the compatibility of the above 'possibility modal' in the initial sense you suggest you would need to say both "Betty does not believe that it is not snowing" and "Betty does not believe that it is snowing," or maybe best is "Betty does not believe either that it is snowing nor that it is not snowing." In the way you have it, saying merely "Betty does not believe that it is not snowing," you have to place emphasis on the second 'not' to get the right pragmatic force. This may be part of why it is hard to parse.
I like the idea that 'believes that it might be snowing' ascribes to the believer a belief-compatibility position with respect to the proposition that it is snowing, but I worry that 'believes that it might be snowing' expresses a genuine propositional attitude toward the proposition that it might be snowing. It seems to me that 'believes that it might' expresses an attitude that can indicate the believer's compatibility position, but doesn't out and out express it (instead it expresses the propositional attitude). This is because of the possibility of being in a position where one's belief state is compatible with both p and not p, but one doesn't believe that it might be that p. That is, that one takes no attitude toward the proposition expressed by "it might be the case that p." In such a case, it would be false to say "Betty believes that it might be snowing" even though her belief state is compatible with its snowing/not snowing.
It seems that to say "believes that it might be snowing" truly of Betty, we must know something about Betty's thoughts about the weather. Namely, that she is having thoughts about the weather and finds herself in a position where she isn't committed one way or the other. If she has no thoughts about the weather it would be odd to say of her that she believes that it might be snowing.