Alvin Plantinga, answered from the texts and cited to the page.
The argument I find most compelling begins from a single premise: that maximal greatness is possibly instantiated. Once that is granted, everything follows with logical necessity. Let me lay out the structure. Maximal greatness is defined so that a being has it in a given world only if it has maximal excellence in every world, and maximal excellence in every world requires omniscience, omnipotence, and moral perfection in every world.1
So if there is even one possible world in which maximal greatness is instantiated, then in that world there exists a being that is omnipotent, omniscient, and morally perfect in every possible world.2 But what is logically impossible does not vary from world to world — contingent truths shift across worlds, but impossibility does not.3 So if that being's non-existence would be impossible in some possible world, it is impossible in every possible world, including the actual one.
Hence there actually exists a being who is omnipotent, omniscient, and morally perfect and who exists and has these properties in every world.4 An earlier version of the argument, drawing on Hartshorne and Malcolm, ran into trouble at precisely the point where necessary existence was treated as a great-making property and then predicated of a possible being that might not exist in the actual world.5
The move I make in the version I prefer avoids that problem entirely: instead of speaking of a possible being that has maximal greatness somewhere, I speak of the property of maximal greatness and ask simply whether that property is possibly instantiated.6 No commitment to merely possible beings is required, any more than the proposition that necessarily a unicorn has one horn commits one to the existence of unicorns.
The charge of circularity deserves a direct answer. Circularity in the paradigm case means arguing for A on the basis of A₁, for A₁ on the basis of A₂, and eventually for some Aₙ on the basis of A itself. This argument has no such structure.7 The sole substantive premise is that maximal greatness is possibly instantiated, and I take that premise to be true — more plausible, on reflection, than its denial.
The argument is valid; the conclusion follows from the premise; the only genuinely interesting question is whether the premise is true.8 I think it is, and so I think the argument is sound. What the argument does not do is compel assent from everyone. Someone who finds the conclusion sufficiently counterintuitive may simply take the argument as a reductio of its possibility premise.
That is a coherent response. What the argument does establish is that anyone who accepts the possibility of maximal greatness must accept the existence of a necessarily omnipotent, omniscient, and morally perfect being — and that the possibility premise is more plausible than its denial.
A being has maximal greatness in a given world only if it has maximal excellence in every world. A being has maximal excellence in a given world only if it has omniscience, omnipotence, and moral perfection in that world.The Analytic Theist, pp. 113–114
if W' had been actual, there would have existed a being who was omniscient and omnipotent and morally perfect and who would have had these properties in every possible world.The Analytic Theist, pp. 113–114
while contingent truths vary from world to world, what is logically impossible does not.The Analytic Theist, pp. 113–114
hence there actually exists a being who is omnipotent, omniscient, and morally perfect and who exists and has these properties in every world.The Nature of Necessity, pp. 230–231
If we take it as (21''), on the other hand, then indeed it is useful in the argument, but we have no reason whatever to think it true. So this version of the argument fails.The Analytic Theist, pp. 109–110
Instead of speaking of the possible being that has, in some world or other, a maximal degree of greatness, we may speak of the property of being maximally great or maximal greatness.God, Freedom, and Evil, pp. 119–120
What is it for an argument to be circular? In the paradigm case, one argues for a proposition A, on the basis of A₁, for A₁ on the basis of A₂... Argument A is clearly not an example of it.The Nature of Necessity, pp. 230–231
It is certainly valid; given its premise, the conclusion follows. The only question of interest, it seems to me, is whether its main premise—that maximal greatness is possibly instantiated—is true. I think it is true; hence I think this version of the ontological argument is sound.God, Freedom, and Evil, pp. 119–120