So the Ontological Argument was either first or most famously formulated by St. Anselm of Canterbury way back in like the 13th century or something, and his formulation goes something like this:
God is that than which no greater can be thought.
Let us suppose that God does not exist.
Then we have a contradiction by definition, since we can think something greater than our idea of God, i.e. that+existence.
Therefore, God must exist.
It's sort of the Xeno's paradoxes of arguments for God, in that it is charmingly, obviously wrong, but deciphering exactly how is kind of difficult. The first attempts at refuting this argument got all mixed up with questions like, did it imply that the best island possible must exist in reality and shit like that, but most people confronted with it sort of just went, "Huh?" and moved on with their lives. Kant said some stuff about it which as near as I can figure was about how you just can't fucking do that, deriving actual existence from purely theoretical exercises. I didn't accept this argument even when I was a christian, because it's a form of argument that could only possibly be valid in this particular case, so the question of whether or not it is a valid form of argument is logically equivalent to the question of God's existence and we're right back where we started.
Anyway, Anselm's formulation is not the one Craig references here; rather, he's using Alvin Plantinga's formulation, which goes like this:
1) It is possible that a maximally great being exists.
2) If it is possible that a maximally great being exists, then a maximally great being exists in some possible world.
3) If a maximally great being exists in some possible world, then it exists in every possible world.
4) If a maximally great being exists every possible world, then it exists in the actual world.
5) If a maximally great being exists in the actual world, then a maximally great being exists.
6) Therefore, a maximally great being exists.
First off, let me just note that it is not at all clear to me that this whole possible world business is all that different from the world ensemble Dawkins postulates to answer the teleological argument, for which Craig ridicules him, but that's not the main point here. The point is that Craig insists that, "In order for the ontological argument to fail, the concept of a maximally great being must be incoherent, like the concept of a married bachelor." Setting aside the issue of whether that's actually true, what is demonstrably false is his further insistence that "the concept of a maximally great being doesn't seem even remotely incoherent." Observe.
Let's assume that the concept of a maximally great being is coherent. Let us represent this entity by n. Now let us posit an entity n + 1, which has all of the characteristics of n, and the additional characteristic, "can take n in a fight" (or "is greater than n", but I like my formulation best). It should be obvious that n + 1 is just as coherent as n - if you're not super comfortable with the idea of something being "more omnipotent", then you clearly haven't been watching enough anime. (I realize this seems like a flippant point, but it's not. The inability to conceive of something "more omnipotent" is actually a failure of the imagination. Or, if you're still not convinced, replace "can take n in a fight" with "is capable of making a better universe than n" because if you candidly think that this is the best of all possible worlds, well, congratulations, you're a white straight cisgendered christian male in the wealthiest one percent of the United States of America, and fuck you.) Point being, using the same inductive reasoning that the ontological argument (or at least Anselm's formulation of it) uses to derive God, I can derive the existence of a better God. This, of course, contradicts the idea that n is maximally great, therefore we have a proof by contradiction that the concept of a maximally great being is in fact incoherent, therefore the ontological argument holds no water.
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