From the point of view of argument, it seems like a big mistake to take on the task of disproving the existence of God. After all, if someone is making the claim that there is a God, then the burden of proof is on them. Also, I don't think it's possible to disprove the existence of God any more than it is possible to prove it.
Still, there are very many people who believe in God, and so we atheists are often in the position of explaining why we don't think that a god of some particular description is at all likely or possible. The inevitable response from the theist is that my argument doesn't apply, because God is some sort of special exception.
Take, for instance, the claim that God is a person. Most Christians, at least, are committed to this claim. (Worse, they are committed to the claim that God is three persons, though only one being, something that seems incoherent on the face of it, and even sillier when they say in addition that God is "simple". But let's not go there....) What is a person, though? We usually think of a person as a being possessed of intelligence, capable of thinking and reacting in complex ways. But God is supposed to be somehow outside of time, so she can't react to anything. Nor can she have thoughts in the sense we do: a succession of mental states that relate to a person's situation at a particular time.
And what is intelligence but the ability to process information in complex ways? But God, being omniscient, can never receive any new information, and, being outside of time, cannot process anything: a process is something that takes place over time.
So how does the theist respond to these challenges? Ed Feser has helpfully given examples in some recent posts. He agrees that God canot have the same sort of consciousness as we do:
There’s nothing it’s “like” to be God if we mean by that a certain kind of stream of thoughts and conscious experiences, like ours but (say) more vivid and encompassing a perceptual awareness of every part of the world at once.But, he says, God's intelligence is "analogous" to ours.
OK, sure - if by "analogous" you mean "utterly and completely different from." God doesn't have perceptions, or mental images, or thoughts, or emotions, God can't gain new knowledge of anything, or change Her mind. God is, in fact, incapable of pretty much everything we associate with personhood.
So is God less than a person? Of course not!
Similarly, to say that it is a mistake to try to grasp the divine intellect by modeling it on our thought processes does not entail that God is less than “personal” in the sense that we are personal (as contrasted with impersonal objects and forces like stones and gravity). Rather, God is more than personal, in that everyday sense of “personal.” His intellect is not inferior to our conscious thought processes (as a stone, gravity, or even the unconscious informational states of a computer are to that extent inferior to our conscious states) but on the contrary beyond and higher than them, just as divine power is beyond and higher than the relatively trivial capacities in created things that we characterize as “powers.”Oh, I see: our poor human personhood is but a pale shadow of Her perfect Personhood.
The only thing that can be said about arguments like these is that they are so clearly born of desperation, so clearly instances of equivocation and special pleading, that they hardly require a response. Any more than the Purple Plutonian Pachyderm.