For the purposes of this discussion, I will be using the term "presupposition" and "axiom" effectively interchangeably and prefer the latter term because it strikes harder to my point that I think God is an abstraction, not an entity. An axiom is a formal starting point of a reasoning process, and though they are not strictly statements that are self-evidently true, they are statements often used as if that is the case, at least for the purposes of that particular reasoning process.
I wish to define a "God Axiom," then, as a presupposition that "there exists a Deity defined or described in some specified way." A particular God Axiom will, of course, specify in what way God is intended to be understood.
There are people who accept some God Axiom or another, collectively known as "theists," though I do not like that word and tend not to use it, and there are people who do not accept any God Axiom, collectively known as "atheists," another word I do not like, but for a different reason. There are also people who deem that it is too difficult or impossible to effectively judge all of the possible God Axioms to take a side, and they are agnostics, in the usual parlance.
The primary reason that I see this presuppositional attitude as axiomatic is a statement I read from John W. Loftus in Why I Became an Atheist. This statement can be found online here and reads,
As a former student of James D. Strauss at Lincoln Christian Seminary in Lincoln, Illinois, I credit much of my approach to Christianity to three things that Strauss drilled into us as students, but in reverse. When doing apologetics, he said, "if you don't start with God, you'll never get to God." Strauss is not a Van Tillian presuppositionalist because he doesn't start with the Bible as God's revelation, but he does start "from above" by presupposing that God exists and then argues that God's existence makes better sense of the Bible and the world than the alternatives. Again, "if you don't start with God, you'll never get to God." Since this is such an important, central issue, I'll focus on why we should not start "from above" with a belief in God, but rather "from below" beginning with the world in which we find ourselves. If successful, my argument should lead us to reject the existence of the sort of God thought to confirm the biblical revelation. (emphasis mine)This is consistent with the acceptance of an axiom--a starting place to a formal process of reasoning.
I will refer to a "God Hypothesis" as a hypothesis that "there exists a Deity defined or described in some specified way." It looks like this is saying the same thing, but the difference is that a hypothesis is a falsifiable article and an axiom isn't. In other words, we can test a hypothesis, where we merely must accept or reject an axiom.
Among those who accept the God Axiom, it is not possible to consider the existence of God as a hypothesis because the existence is taken as a given by the God Axiom. They have started with God. The question lying behind a God Hypothesis is rendered "Does this thing that exists exist?" becoming a tautology. This should serve to explain why theists typically reject the phrase "God hypothesis" and scornful of those who use it.
There is an interesting caveat here, though: there is not a unique God Axiom. Indeed, each believer is likely to possess her own distinct one. This caveat is superficially dodged for philosophical purposes by monotheists who claim to accept a Monotheistic God Axiom: "There exists a unique Deity that we define or describe in this specific way." It sounds good on paper, but we all see how that works out in practice. The most charitable thing that can be said for this is that if they are right, none of them can know it. (And they show it readily.)
This caveat is interesting because it shows that those people who accept some God Axiom typically reject at least some (often all) other God Axioms. That means that theists have to be treated, as some have noted, simultaneously as theists and as quasi-atheists--people who reject most, but not all, possible God Axioms. They accept their God axiom and reject the others. In the following discussion of the atheistic possibilities, this should be borne in mind and applied as fits the case.
There are two cases within those who reject God Axioms (usually on the grounds that they are not sufficiently evident or parsimonious to be philosophically justified axioms about the fundamental nature of reality). We could divide these in terms of people that hold various ideas, but I think it's more helpful to do so by the kind of God Axiom considered prototypical.
If the prototypical God Axiom being rejected is taken to be something unfalsifiable, like the Deist God Axiom or any that asserts God is purely beyond the physical world and totally supernatural, then it still makes no sense to talk about a God Hypothesis. Hypotheses, really, need to be falsifiable to make sense as hypotheses.
If the prototypical God Axiom being rejected is taken to be a Theistic God Axiom, one that posits that God somehow has or does interact with the world, then the relevant God Hypothesis is a completely valid way to consider the matter. That is to say, if someone sees God as being described in a way that involves interactions with the world but does not accept an axiom of that God's existence, then that person is right to view the matter of that God's existence as a hypothesis.
Of course, the problem here is that no God Hypothesis has ever withstood scrutiny by those who do not accept the relevant God Axiom(s). Thus, when Richard Dawkins (or I) talk about "the probability that God exists" or "the plausibility of the God hypothesis"--which I assert is almost surely zero--we are talking about the degree of confidence that we can put in claiming that these statements, viewed as hypotheses, can be taken to be true.
Despite theistic protests, this isn't a crafty way of redefining things in a way that is not fair to them. Offering to treat God's existence as a hypothesis, instead of just dismissing it as an unlikely axiom, is a major concession to their position which only deserves anything like serious inquiry on an ad populum. It only seems unfair because it doesn't work out for them--and if it did, we'd hear it from the hilltops and everywhere else.
The ongoing argument
This leaves theists in an unfortunate and perhaps dishonest position of having to shift between God Axioms at need. They wish to defend their acceptance of a particular Theistic God Axiom, but those who do not accept it are justified in seeing it as a God Hypothesis that has a very low plausibility or no plausibility at all. At best, the two groups are talking past one another, then.
To make their cases, though, apologists need to prevent people from seeing the fact that they are accepting an axiom that, if rejected, can be viewed as a hypothesis with no support for it. Only by switching as seamlessly as possible to another God Axiom--like the Deistic one (via cosmological arguments including the Kalam)--can they slide past analysis by rendering it temporarily inapplicable. (And how convenient for them that people have called all of these notions "God.")
Then, once the conversation or its observers are sufficiently caught up in treating an axiom like a hypothesis, the apologist will slide back to the Theistic God Axiom that they want to promulgate. This is apologia. This is the art called theology. To call this kind of behavior obscurantism is a kindness.
Theists are caught between two rocks and a hard place. Ahead of them, they have to subject their beliefs about God to the onslaught with which we evaluate hypotheses, and that they apparently cannot win. On the one side, their God may be real but is too remote to interact with the world. And on the other side, they have to take their God's existence axiomatically, which is a concession that what they believe in--so far as we can know--is an abstraction by which they interpret the world.