Anselm gives two reasons why God/ that-than which etc. must exist. The first:
If that than which a greater cannot be thought can be thought not to exist, then that than which a greater can be thought is not that than which a greater can be thought.
Quare si id quo maius nequit cogitari, potest cogitari non esse: id ipsum quo maius cogitari nequit, non est id quo maius cogitari nequit.
Notice that Anselm’s argument here stays entirely on the cognitive level. If you can’t think something greater than X, then you must think of X as existent. Anselm gives no other premise, but he might have been assuming one of these:
1.) If we must think something, then it is so. The normal response to “we cannot conceive other than P” is that P is so. If, for example, you notice that you cannot but think 2+2=4, you normally take this as evidence that it really is so. Even if it were logically necessary to assume that the moon was green cheese, then we could never establish it wasn’t. In fact, the possibility of disconfirming the antecedent (say, by taking soil samples from the moon) is a proof that such a claim can’t be logically necessary.
If this is what Anselm is thinking, then the attempt to refute his argument has to argue that P being logically necessary does not entail P expressing an existent fact. Perhaps an appeal to fictional characters might do this. It’s logically necessary to say that Sancho was the Squire of Quixote, but not that Sancho was. That said, most modern logics seem to deny just this claim about fictional beings for the same reason that they make contingent past events possible.
2.) The thesis contradicting Anselm is incoherent, and where one contradictory is incoherent, the other is true. Presumably the first thing an an atheist would have to do, on Anselm’s account, is give an account of God that could either exist or not. This is either Anselm’s account or it is not. If not, then he is speaking of someone other than Anselm, and so speaking past him; if it is Anselm’s account, no coherent account of it can be formed as non-existent.
Anselm’s second reason is more explicit, and is usually overlooked:
If some mind could think something better than you, a creature would ascend above the creator, and it might judge the creator, which is completely absurd.
Si enim aliqua mens posset cogitare aliquid melius te, ascenderet creatura super creatorem, et iudicaret de creatore; quod valde est absurdum.
The “creature/creator” language is shorthand. Anselm simply doesn’t want to fatigue the reader with “that than which…etc.” To avoid the same problem, let’s say
A = that than which something greater can be thought
B= that than which something greater cannot be thought.
Anselm then appeals to the claim that judgment requires the transcendence of the one making the judgment over the things he is judging, and so if A could judge B, then it would have to transcend B. It’s one thing to say it though – what could it mean?
1.) Judgment is made over the logically contingent. Perhaps Anselm is taking “judgment” in the sense of what requires an indifference to the think judged, i.e. an attempt to consider it one way or another. Our ability to judge B in the relevant sense means our ability to relate to it as existing or not, which is exactly what Anselm is denying. But this would not explain the transcendence.
2.) Judgment is made over things one transcends. While contraries and contradictories drive each other out in the real world, they do not do so in the mind, and in this sense mind must transcend all contraries. What judges (mind) therefore transcends all contraries and contradictories (what either exists or does not). But if mind itself is an A, then the things mind judges: mind :: A, the mind : B. Three terms of the proportion exist, therefore the fourth does also.
3.) Any judgment presupposes the reality of the principle of judgment. This is basically some version of the Platonic forms. Given that A’s have an indifference of being, they can only be in relation to something else, i.e. a B.