First, thank you for all that. You're a lot better at formulating a formal argument than I am. I still feel that the other premises help to build to p4 which helps to understand the thing as a whole, even if they aren't technically necessary. Especially since the conclusion points right back to p1.
Perhaps p1 does come into it--since you didn't reference it in your explanation I didn't think you were using it. But moving from the antecedent to the consequent of p1 isn't necessary if you are trying to disprove omnipotence or omniscience, since the conclusion that God doesn't know something is sufficient. If God doesn't know something then it's true that he doesn't know he doesn't know something, but the simple fact that he doesn't know something disproves omniscience.
That quoted finished product is what I have been trying to get at in my own clumsy way, but your objection doesn't make sense to me. ap4 doesn't require the second existence mentioned to be anything but hypothetical, and it is still true.
If they were distinguishable from each other, to God, then that distinction would make God aware of the thing that He didn't know. Since He would be incapable of making that distinction, the two scenarios are, in fact, from His perspective, indistinguishable.
The only objection I have raised is that you have not provided any justification for ap4. The second existence in ap4 need not be actual for ap4 to be true, but I still don't see any reason to believe ap4 is true.
Now I'll address your last post in more detail:
The only extra clarity I think p4 needs is that I need to make special note that it is from God's perspective. Two different existences would be distinguishable if you had the ability to distinguish, and I did imply that they would be indistinguishable to anyone, and that would be incorrect. I could rewrite the whole thing to be a generic "observer" as well, but it's fine the way it is too. So here is the slightly modified p4:
To any given observer, an existence in which they are aware of everything is indistinguishable from an existence in which they are aware of everything they are capable of being aware of.
Okay, I will stick to your original formulation but alter it in the way noted
here:
ap4: An existence in which God is aware of everything is indistinguishable from an existence in which God is not aware of everything but is aware of everything He is capable of being aware of.
p5: If ap4, then God doesn't know that there isn't something that He doesn't know.
Conclusion: Therefore God doesn't know that there isn't something that He doesn't know.
I deny ap4. There are at least two reasons.
First and most importantly, because you give no reason to believe ap4. You've provided no argumentation in favor of it, and it is far from self-evident.
Secondly, we would say that one is able to distinguish two possibilities if they know that one is true and the other false. But God knows that the first possibility is true and the second false, therefore he is able to distinguish them. How does he know this? God, knowing himself perfectly, knows that he is the sole source of existence, and thus knows that anything which exists outside of him exists only because it is created and sustained in being by him. Thus there is nothing that God is not aware of, for God is not unaware of that which he created, and he created all that exists.
It seems that you have committed the error of misunderstanding what God is. Your argument implicitly denies that God is the creator of everything that exists, which runs contrary to basic theism. You may have proved that some god is not omnipotent, but it certainly isn't the God of Christianity, Judaism, Islam, and the philosophers.
As an aside, I think it is worth pointing out that when you attempt to give an argument disproving God's omniscience you are attempting to show that the common conception of God is incompatible with omniscience. (In this case you are trying to show that omniscience is impossible.) My duty is not a philosophical demonstration that God exists as classically conceived. Rather, I have the duty of showing that the classical conception of God is compatible with omniscience, namely by finding some fault in your argument. It is true that I have provided some philosophical arguments for certain attributes of God, but it is not necessary for me to
demonstrate the classical conception rather than defend its coherence.
For example, your argument implicitly asserts that God is not the creator of everything, or at least that he doesn't know he is. Thus if the premises of such an argument have no rational justification in their favor, the argument fails insofar as it is blatantly contrary to the general understanding of God as creator. Without justification, the argument is just an assertion against God as creator rather than a reason to believe that God either didn't create everything or doesn't know that he created everything. And an assertion has no force against the commonly accepted definition of God.
It all comes down to the question, "How does God know that He knows everything?" You're saying that's how he's defined, but how does He know that definition is correct? He can't confirm it (p2, p3).
God
knows what he is, and he is being/existence itself,
ipsum esse subsistens. He is the sole cause/source of being through which everything else exists. He is the creator.
The reason there cannot be two Gods is because there would be no way to differentiate two perfect beings. For example, they are not composed of matter, they have no constitutive parts, they have infinite power, they have perfect knowledge, etc. The only way they could be differentiated is if one had an imperfection, but then that one would not be God. Furthermore, to be
ipsum esse subsistens--one that can create
ex nihilo as a true source of existence--is to be perfect ("fully actualized") and therefore to be God. Nothing but God can create
ex nihilo and sustain entities in existence.
See, now this is why I'm expecting a circular answer to my argument. You seem to be conflating a contradiction of a person's nature with a logical contradiction. I'll give it to you that if God can't do the logically impossible (square circles and all that) then He can still be omnipotent.
The reason that God cannot create a square circle is because of the nature of squares and circles. The reason God cannot create a non-rational man is because of the nature of man. The reason God cannot create a creature that does not depend on him for being is because of the nature of creatures.
Here's a metaphor. I have a regular old 2001 Ford Taurus. No modifications. It is a contradiction of how my car works to drive on three wheels. It simply isn't the nature of my car. It is not logically impossible to have a three wheeled car. I could even install hydraulics in my car to enable it to drive on three wheels.
It would be a contradiction to say that your car could at the same time be a nuclear submarine, for cars travel on ground and run on gasoline or diesel fuel, and nuclear submarines travel underwater and run on nuclear energy. The natures of cars and nuclear submarines are incompatible. There is no reason to believe three wheels contradicts the nature of a car any more than to believe that three legs contradicts the nature of a dog.
(Admittedly, cars and submarines are artifacts and thus do not have an intrinsic nature, but the basic argument still holds)
The nature of God and existence, as you've seemed to describe it, is that existing things rely on God to continue to exist, and God must continue to maintain their existence. That's fine, if that's how you think God works. But it is not logically impossible for a god to exist that does not work that way.
It is logically impossible. Creatures are not sufficient unto themselves for their existence. Their essence is distinct from their existence. There is nothing in the nature of creatures that causes their own existence. If there were, they would neither come into existence nor cease to exist. Contingency belongs to the nature of creatures.
(St. Thomas
writes on the question, giving the standard answer.)