I was going through the list of algebras that might work either for a formal proof or, if this was too complex, one that could had an intuitive structure for a pseudo-formulation, and I started sketching out some possibilities. A central issue is that even though the study of modality (including epistemic) in linguistics started with logic and the philosophy of logic, systems were designed not to compare beliefs but to express them. That is, possible world semantics and modal logics tend to address beliefs as possibilities like "it is possible that god exists" vs. "necessarily god exists". I realized, though, that while this isn't so helpful for what I was after, it's even better.
The logics described above typically do not simply add the two modal operators ( the symbol □ indicating and ◊ possibility). They will use these in conjunction with frames, domains, relations, worlds, etc. However, what is useful is how to express certainty and possibility in such systems.
The view expressed here is that atheists neither believe nor disbelieve in god(s). Moreover, an atheist is defined entirely by not being a theist.
Presumably, if a theist is anything, a theist must believe that some concept of god has a reality or existence outside of that theist's mind.
A fairly intuitive way to extend logics to include beliefs is to think of an infinite set of world s.t. (such that) for any situation that is possible there is a world in which it is true. By possible, I refer to limits such as a world in which impossibilities can exist that impossible the way that something being both true & false at the same time is impossible. Symbolically, ~◊( A & ~ A), must be true. I do not mean that ambiguous concepts or things that seem impossible can't exist. Logical impossibilities, like "both A & ~A can be true", we can interpret precisely and thus understand the impossibility precisely.
What doxastic logics, possible world semantics, modal logics, etc., and the combinations thereof have in common is that beliefs are taken to relate to something (a truthmaker, a token, it doesn't really matter here) that is what makes the belief true or false. So, for example, a theist would say the proposition "god exists" is necessarily true: it is true in all possible worlds. That's because for any proposition, there is that relation (that truthmaker thing) which makes the proposition true in some possible world (with exception noted above). By saying god actually exists, the theist means there is no possibility that god doesn't exist, and therefore no possible world for which the proposition "god exists" is false.
I'm not saying, of course, that this means god exists. That the proposition "god exists" is true in all possible worlds is not true for every set of all possible worlds. It is simply the way that a theist would express the existence of god(s). An agnostic, on the other hand, describes the proposition "god exists" differently. The agnostic describes a different set of all possible worlds for which the proposition "god exists" is made false in some possible worlds and true in others.
Now for the interesting part. For the atheist (strong, positive, etc.) who believes god doesn't exist, we just use the opposite of what the theist expresses: it is necessarily true that for all possible worlds the proposition "god exists" is false. No problem.
But now there is nothing left. Given the proposition "god exists", we've exhausted all the ways in which the truth of this statement could be evaluated. Possible worlds are indeed designed to consist even of those things we can't conceive of, are hard to define, etc. Yet we have nothing that enables us to speak of an atheist who neither believes nor disbelieves (except in the way an agnostic does).
Nor do we escape this through other ways of extending logic to cover predicates like "believe". Belief ascriptions, for example, suffer from a similar problem here as in classical logic. For they are ways in which beliefs are described, a function or relation B with arguments B(x, p, m, t) that tell us x believes the proposition p as presented in mode m at time t (not all beliefs require the fourth argument). However, such a function is, as in classical logic, binary. There is no possible difference between "not believe A" and "believe not A". Models, worlds, frames, etc., may differ in semantics and the same even one of them, like worlds, may differ in one system compared to another. We can even go further and incorporate fuzzy logic, but even if we allow our membership function to assign a truth value to the proposition "god exists" to the unaccountably infinite set [0,1], 0 would be a certainty that god doesn't exist and anything else would represent a possibility.
Surprisingly, then, the issue at the moment isn't which system to use or how, in that system, to represent proofs that relate the definition of theists to what that entails for atheists, it is that despite the diversity of systems and the diversity of ways belief can be expressed about a diverse set (including things like "lsihler"), the only way evaluate the proposition "god exists" is truth, some form of possibly true, or false. There are literally, in this case, infinitely many "third ways" that aren't possible in classical logic, yet still not believing god exists entails believing that god does not exist or that it is possible god exists.
