Actually, a high order logic program ran the question of if an Abrahamic style god could exist (the omnipotent, omnipresent, etc type of god), and it only found one small problem with the logic. There was an unsupported claim in the logic. I will search for it and post it here as an edit when I find it.
EDIT: Found it
http://mally.stanford.edu/Papers/ontological-computational.pdf
In layman's terms it says:
1.We conceive of God as a being than which no greater can be conceived.
2.This being than which no greater can be conceived either exists in the mind alone or both in the mind and in reality.
3.Assume that this being than which no greater can be conceived exists in the mind alone.
a.Existing both in the mind and in reality is greater than existing solely in the mind.
b.This being, existing in the mind alone, can also be conceived to exist in reality.
c.This being existing in the mind alone is not therefore the being than which no greater can be conceived. (See statement 1 above.)
4.Therefore, this being than which no greater can be conceived exists in reality as well as exists in the mind.
It is short, but hard reading, and requires a LOT of logic.
The full on logic is included in that link, but they expose the only weakness in the logic. It is this:
"if the conceivable thing than which nothing greater is conceivable fails to exist, then something greater than it is conceivable." This claim has no secondary support. It assumes our ability to conceive is unlimited. We do not know if this is true or false, and there is no way to know.