Say we agree a.) logical possibility is whatever does not involve contradiction, and b.) God is able to do anything that does not involve contradiction. It seems to follow that logical possibility has a real basis in the power of God, and since the power of God has a real existence outside the human mind, therefore logical possibility is a real thing existing outside of the human mind. I disagree with the conclusion, and claim that logical possibility has no basis in reality apart from the presence of things the human mind, and that any attribution of logical possibility to things is by an extrinsic denomination, the way you could point to something and say “That is the major term of the syllogism” or “She is the subject of the sentence” (before objecting, be sure to read to the last paragraph).
Some basic axioms:
1.) The possible and the impossible are opposed as contradictories, since “impossible” simply means “not possibile”.
2.) What necessarily is (or is not) is divided from what possibly is (or is not).
Note that 2.) is the division between the necessary and contingent – and so “possible” in this sense is synonymous with “contingent”. But in 1.) possible is not synonymous with contingent - If it were, it would follow that whatever was not contingent would be impossible. Such a claim requires saying that God is impossible – but it has more basic problems than that. First of all, it’s not what we mean to say when we use the term “possible”, and second even a necessary proposition would be an impossible proposition, though this claim that would fall victim to the crudest liar paradox, i.e. “no proposition is necessary.”
It follows that there is a real distinction between the possible as opposed to the impossible (1) and the possible as opposed to the necessary (2). When we speak of “the absence of formal contradiction” we are speaking of (1), since if what lacks contradiction is possible in sense (2) then the necessary would be contradictory. Therefore logical possibility is possibility in sense (1). Now since anything that actually exists is not impossible (1), it follows that God can be called a “possible being” if one is speaking of logical possibility. But here again, this sort of possibility is by extrinsic denomination, the way one might say a real thing is “the subject of a sentence”, even though to be a subject only belongs to something so far as it is known and signified by a mind.
But I also agree that God is able to do all that does not involve formal contradiction, and that the object of God’s power is real. So what then? First note that even if one appealed to such a reason to argue that logical possibility was in some way real, the argument could only have force after one had proven that God existed. We absolutely can not appeal to some notion of “possibility” (absence of formal contradiction) in an ontological proof for the existence of God. But even after one has proven the existence of God, logical possibility is not a real possibility, even though God can do all that is logically possible. The reason is that the whole question of omnipotence clearly takes place under a consideration of what is able to be done or made (namely, by God) and so this restricts the notion of the logically possible to what is able to be done. We come to a consideration of the omnipotence of God with an understanding that it is a consideration of what can be done or made, but the logically possible as such has no such limitation, simply because what is not impossible is a broader category than what can be made or done.
There is therefore a third sort of possibility which must be distinguished from the possible as opposed to the impossible and the possible as opposed to the necessary, namely (3) possibility as opposed to what cannot be made. The impossible is not contained in (3), and so everything in 3. is possible in the sense of (1), but not vice versa. This opposition is at the heart of people confounding possibility in (1) with possibility in (3). Further, both the possible and the necessary are found in (3), though every possible being in (2) is in (3) though not every necessary being is.
This last sort of possibility is, I’d argue, the sense of “possible” and “impossible” as real predicates. We can also call it possibility that is in the power of an agent. With God, the extent of this possibility is absolute and unqualified, but it arises from other agents too. Real possibilty (as opposed to impossibility) thus always reduces to the power of some agent, as Aristotle and St. Thomas both argued.