Modern (Modal) forms of the ontological argument go something like this (modal moves in brackets).
- Possibly, God exists (◊G)
- If God exists, necessarily God exists (G → □G)
- Possibly Necessarily God exists (◊□G, From  and )
- Necessarily God exists (□G, from  and S5)
The modal logic works (even if you find S5 unintuitive), and generally one stipulates God such that it has the property of necessarily existing.
The key premise is . Although colloquially it seems obvious God is possible, it is probably best thought of as possible in an epistemic sense: for all we know it is possible that God exists (even if we don’t think it likely). Yet that is distinct from saying it is a metaphysical possibility: that there really are possible worlds in which God exists. And it is this claim which is needed to drive the OA.
So what, though? We can conceive God’s existence without contradiction, seems to be well-formed proposition, etc. etc. So all of these epistemic reasons appear to give warrant to the metaphysical claim via Yablo conceivability or similar measures. What’s the problem?
The same applies to God not existing!
Consider this modal argument
1) Possibly, God does not exist (◊¬G)
2) If God exists, necessarily God exists (G → □G)
3) If possibly, God does not exist, then not necessarily god exists (◊¬G → ¬□G)
4) God does not exist necessarily (¬□G)
5) God does not exist. (1, 4, Modus tollens).
So again, all the modal logic follows (as an added bonus, the argument does not rely on S5, which is controversial), and so once again the argument hinges on whether we should accept the key possibility premise. As far as I can see, we can make exactly the same moves: God’s non-existence seems conceivable without contradiction, well formed proposition, etc.
So the two relevant metaphysical possibilities (◊G, ◊¬G) appear to be in equipoise. Unless there are some good modal concerns suggesting why one is more possible than the other, then the ontological argument serves more as a reminder that our epistemic grasp on metaphysical possibility is not as secure as we think.