1. It is proposed that a being has maximal excellence in a given possible world W if and only if it is omnipotent, omniscient and wholly good in W; and
2. It is proposed that a being has maximal greatness if it has maximal excellence in every possible world.
3. Maximal greatness is possibly exemplified. That is, it is possible that there be a being that has maximal greatness. (Premise)
4. Therefore, possibly it is necessarily true that an omniscient, omnipotent and perfectly good being exists.
5. Therefore, it is necessarily true that an omniscient, omnipotent and perfectly good being exists. (By S5)
6. Therefore, an omniscient, omnipotent and perfectly good being exists.
From what I read in the Wikipedia article, the two objections to this argument seem to be whether to buy into Axiom S5 (which I don't understand quite well enough to comment on, but since it is apparently broadly accepted I will grant it for now), and whether to accept the possibility premise (3).
As it turns out, I am pretty sure I don't accept the possibility premise, but that's not even what I want to talk about. I think that (2) is not even a valid statement, i.e. we could not even differ over whether it is possible because it is ill-formed.
The problem is that (2) refers to a single being inhabiting every possible world. This is nonsensical. An entity that exists in world W cannot be the same entity that exists in world X, at least by the definition of "world" that we mean when using logic. If we attempt to paraphrase Plantinga's assertion with more specificity, it breaks down:
2. It is proposed that a being which exists in world W has maximal greatness if, in any possible world X it exists and is maximally excellent.
In other words, in order to be maximally great, a being -- one single hypothetical being, not two identical hypothetical beings -- has to exist in more than one possible world. This is nonsensical. You can't even invoke some sort of transcendence-of-God argument here, because by "possible worlds" we aren't talking about parallel universes, we are talking about thought experiments. The "possible worlds" are entirely separate.
If we paraphrase (2) to be less objectionable, then Plantinga's argument falls apart:
2. It is proposed that a being which exists in world W has maximal greatness if, for any possible world X in which an identical being exists, that other being would be maximally excellent.
3. Maximal greatness is possibly exemplified. That is, it is possible that there be a being that has maximal greatness. (Premise)
4. Therefore, possibly it is necessarily true in any world X in which a maximally excellent being exists that an omniscient, omnipotent and perfectly good being exists.
5. Therefore, it is necessarily true in any world X in which an omniscient, omnipotent and perfectly good being exists that an omniscient, omnipotent and perfectly good being exists. (By S5)
6. Therefore, tautologies are fun.
Perhaps a more succinct way of doing it is that Axiom S5, if I understand it, cannot be distributed across multiple possible worlds. Otherwise, I could make this absurd argument:
1. It is proposed that Plantinga's argument is incorrect in world W if and only if there is an error in Plantinga's argument in world W.
2. It is proposed that Plantinga's argument would be logically invalid if it is incorrect in all possible worlds W.
3. Plantinga's argument might be logically contradictory. (Premise. Since people are still arguing over it, I think this is a fair premise.)
4. Therefore, it is possible that it is necessary that there is an error in Plantinga's argument.
5. Therefore, it is necessary that there is an error in Plantinga's argument.
6. Therefore, there is an error in Plantinga's argument
In order for the "possibly-necessary implies possibly" aspect of Axiom S5 to hold water, the proposition it modifies must only say something about one possible world, not a generalization about all possible worlds. Otherwise, it could be used to prove anything you want.
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