Nielsen's combox discussion confirms that the most common point of contention about AP is Kant's. Thus AP is interpreted as premising that something called "existence" is a "predicate" (or, on some accounts, a "property"). Indeed, if that claim be taken to mean that it is a perfection of anything that there is such a thing as it, then AP does rest on a simple logical error. But Anselm didn't appear to be that dense. He meant that (a) it belongs to the concept of God that God is "that-than-which-nothing-greater-can-be- conceived (TTWNG), and (b) a God that existed in his own right, independently of our minds, would be greater than a God that exists in our minds as a merely imagined or "intentional" entity. If so, then God cannot exist only "in the mind" as an imagined or intentional entity. By itself, of course, that does not prove that God really exists; all it proves is that, if God exists at all, he exists really not merely intentionally. As Norman Malcolm once suggested, the question is really whether God, as TTWNG, is relevantly conceivable to begin with, so that it would make sense to say that such an entity really exists.
Given as much, the traditional Thomistic objection to AP is that, since what-God-is (i.e., the "divine essence") is not properly conceivable by us, the premise that God is TTWNG lacks enough content to allow the needed inferences. Now prima facie, that objection seems a bit of a cavil. All AP needs is the claim that, however else the divine essence can or cannot be conceived, it must be conceived as TTWNG and that something which is TTWNG cannot be a merely intentional object. And that claim seems true enough. So the main question really boils down to whether anything which is TTWNG is conceptually possible. And this is where the Thomistic objection gains its real force. In order to show that a TTWNG is conceptually possible, we must possess a fully perspicuous account of what the concept of it entails. But on most classical-theist accounts, we possess no such account. If there is a God, we can say many true things about him; but we don't have a clear enough idea of what they mean to enable us to demonstrate, from incontestable premises, that such an entity is conceptually possible.
And so I take the value of Anselm's argument to be how it shows that, if God's existence is possible at all, then it is necessary. It is not logically necessary, but really so. 'God exists' does not have the force and clarity of a logical theorem; rather, it entails that there is something existing eternally, uncausably, and unpreventably and is God. The value of AP, which isn't really an "ontological" argument like Descartes' and criticized as such by Kant, is that it tells us something about what sort of existence, if any, to ascribe to God.
