As the title states this thread is about belief. Not belief in any one thing, but belief itself.
First, lets define belief as: That which a being thinks to be true. So when I think the statement "The pen will drop to the ground when I let go of it" to be true, I believe it. Same goes for thinking a mathematical or similarly logical identity to be true. What this means is that all things that one knows one believes in but not all things that one believes in are known. Which I think should be agreeable, though I don't want to get bogged down in epistemology.

Suppose a statement "I believe X", where X is not necessarily a particular word but rather a concept or even group of concepts. To fulfil all possible states of belief, one merely negates the belief so one can either:

-I believe X (state a)
-I do not believe X (the negation state a' = b)

There is no other possible state that does not belong to the above two as something is either itself or not itself. Note that these can be applied to all individual objects, even those incapable of possessing belief. A rock will not believe X(b), because it lacks the capacity to possess the belief.

The statement: "I believe not X" (state c) is fundamentally different to the above two and its meaning will depend on X since the object is being negated rather than the belief. Thus, not believing X (b) is not the same as believing not X (c). Through contradiction one cannot rationally believe X (a) and believe not X (c), so believing not X is surely a subset of not believing X (b).
To illustrate this, take the rock again. Whilst it does not believe X (b), since it is incapable of possessing belief it cannot be said to believe not X (c) since that requires the capacity of belief, which therefore means that the two statements must be separate.

Hopefully that's clear, so let's move onto more complex cases. For example, what if someone believed X, then lost their memory. What then is their state of belief afterwards?
Well provided (a) is not innate (not that I mean this to imply any beliefs are or are not innate), then they are being reverted back to their default state which must be (b). Furthermore, when they are sleeping or unconscious, they must also be in state (b), being unable to have the capacity to hold belief (a) (unless one has an extremely specific dream).
However, this does create a very transient approach to belief, in that it is implied that every time someone wakes up they have to update their beliefs again since they will all (non-innate anyway) be left in the default state (b). Not only this, but it could be used to state that one only believes X at the moment they think it to be true and then afterwards they relapse into the default state as they are no longer thinking about X anymore.

To resolve this, I will keep belief as this transient process, but what should be important in assessing someone's views is what their state of belief was under the following criteria:
-That they have the capacity to make the belief
-That they actively considered the object of the belief
Thus it is not current belief that is used in practicality, but last held belief when the concept X was considered.

One final area that I think I should cover, is what if one is not sure whether they believe X, ie:
-I am not sure if I believe X (d)
Does this break the idea of the dichotomy between (a) and (b)? Well no, as going back to the original definition of belief if one is not sure that something is true, then they are not thinking it to be true. It is another subset of (b). However, it is a subset that also cannot coexist with state (c) since one cannot be not sure they believe X and believe not X because then they would not be in state (d) to begin with because state (c) implies that one is sure they believe not X. However, states (c) and (d) are not exhaustive of state (b) in that one can be sure that they do not believe X and not believe not X simultaneously.

Would like to hear people's thoughts on this in case there is anything that I have missed or have misunderstood, but I do think that this sort of approach makes the concept of belief more straightforward and easier to analyse.