Using Set Theory to Define Atheism

2 X = [P ~[B[G E]]]. The definition argued for by most of the outspoken contemporary self proclaimed atheists I have encountered.
3 X = ~[P[B[G E]]]. The definition Hermes and [me] are arguing for.

Actually no, that would be a category error. But, as far as a label goes, 2&3 would convey the same information when a person declares themselves to be an atheist since they are already part of set P. If they are in set P, but they are ~[P[B[G E]]] then it follows that they are an atheist. So by saying I am ~[P[B[G E]]], I'm also saying that I am [P ~[B[G E]]]. Within the category of entities that are able to be theists, I think that ~T or ~[B[G E]] is the best descriptor for the label atheist.

However, I wouldn't necessarily narrow it to set P. I wouldn't be surprised if other species are able to form beliefs, maybe some concerning the nature of god, but I wouldn't exclude them on the basis of being not being human. But that's besides the point.