I've been reading an interesting thread on JREF about skepticism and the nature of provability.
A poster using the handle "Piggy" has been claiming that it is possible to prove impossibility of claims using various tools that lie outside a strict logical/scientific framework.
The discussion is quite long, and it I'm far from finishing it, but from what I've read so far, Piggy's biggest problem is the feeling he has that it is necessary to be able to declare impossibility.
For myself, I am happy to assume that something is untrue when there is a lack of evidence and sufficient unlikelihood. Just because something is possible, doesn't mean I should take it into consideration if I estimate the chance of it actually happening as sufficiently small.
For instance, it is entirely possible that someone has broken into my garage, moved all the stored contents out onto the grass and used it to keep their car out of the weather. I cannot say that such a thing is impossible. But I don't need to say it's impossible in order to decide not to rush home when it starts to rain to protect my belongings. It's sufficiently unlikely that I can ignore the possibility.
God, and supernatural entities in general, I find far more unlikely than unauthorised garage usage. Absent credible evidence, I feel comfortable assuming non-existence. This is not a leap of faith, it is a rational assumption.