EdenOne:
I agree this is all a matter of probability. I think the issue here is that you can accomplish what you set out to obtain by simply applying the standard tools of scientific inference. Suppose you have the following hypothesis:
H : God exists.
Now you gather evidence and that affects your belief* in H. Suppose (say) H becomes very unlikely on the evidence, then one way to quantify that is to say the probability H is true is very low. In facts, on this view, claims of certainty about non-analytical statements are merely approximating probabilities very near 0 or 1 with 0 or 1.
The point I am making is you should distinguish between two things: 1) the hypothesis (god exists, unicorns exist, seashells exist, the Higgs bosone exists) and 2) the degree of belief in these statements which is principle never 1 (total acceptance) or 0 (total acceptance of the negation).So you don't need to alter your original hypothesis H to
H' : God is absent
to allow for the remote possibility god may exist -- it is build into the framework of inference that such a remote chance might exist.
* Belief has various meanings and i am not using the religious one.