There seem to be quite a few atheists who act like there is only one proposition, "gods exist", in play. They also separate questions of belief and knowledge, and then ask two binary questions. One, regarding belief, and the second, regarding knowledge.
Do you believe "gods exist"?
"Yes" = theist
"No" = atheist
Do you "know" "gods exist"?
"Yes" = gnostic
"No" = agnostic
The first problem is that this will only give you 3 possible answers:
(YY) = gnostic theist
(YN) = agnostic theist
(NN) = agnostic atheist
Using just those 2 questions, the 4th option is an impossibility:
(NY) = gnostic atheist
It's impossible to not believe "gods exist" but "know" "gods exist". Even if it were possible, it wouldn't be representing what we mean by "gnostic atheist". The "gnostic atheist" must have been asked an entirely different 3rd question:
Do you "know" "gods don't exist"?
"Yes" = gnostic (apparently)
"No" = agnostic (I assume)
The second problem is that we should have the "gnostic atheist"'s answer to the 2nd question, and everyone else should have been asked the same 3rd question, and their answers given. That gives us the 4 positions:
(YYN) = gnostic agnostic theist
(YNN) = agnostic agnostic theist
(NNN) = agnostic agnostic atheist
(NNY) = agnostic gnostic atheist
The third problem is that a belief question, regarding the "gods don't exist" proposition, should have been asked as well as the knowledge question, and it too should have been asked of everyone.
Do you believe "gods don't exist"?
"Yes" = ?
"No" = ?
^Fourth problem...there aren't really any binary belief labels for this question. I'll use the "negative" and "positive" terms, even though those words don't really describe beliefs in answer to the question. Now we've got:
(YYNN) = gnostic agnostic negative theist
(YNNN) = agnostic agnostic negative theist
(NNNN) = agnostic agnostic negative atheist
(NNNY) = agnostic agnostic positive atheist
(NNYY) = agnostic gnostic positive atheist
We arrive at this convoluted mess by attempting to assign labels to all the specific answers, rather than labelling the 5 different positions, at the end.
We'll clean that mess up a bit by saying all theists are "negative", so that's a redundancy. We'll also say that it's redundant to note that a gnostic is agnostic about the opposing question, and that the remaining "agnostic agnostic" positions are agnostic to both, so we'll clean that up too, with only one "agnostic". Now the broad definition "atheism" and narrow definition "agnosticism" looks like this:
(YYNN) = gnostic theist
(YNNN) = agnostic theist
(NNNN) = agnostic negative atheist
(NNNY) = agnostic positive atheist
(NNYY) = gnostic positive atheist
Labelling, the final positions with broad definition "agnosticism" and narrow definition "atheism":
(YYNN) = theognostic
(YNNN) = theist
(NNNN) = agnostic
(NNNY) = atheist
(NNYY) = atheognostic