Let us borrow the template of the argument from Internet Encyclopedia of Philosophy for sake of formality:
St. Anselm's argument runs as follows:
1. It is a conceptual truth (or, so to speak, true by definition) that God is a
being than which none greater can be imagined (that is, the greatest possible
being that can be imagined).
2. God exists as an idea in the mind.
3. A being that exists as an idea in the mind and in reality is, other things
being equal, greater than a being that exists only as an idea in the mind.
4. Thus, if God exists only as an idea in the mind, then we can imagine
something that is greater than Ω (that is, a greatest possible being that
does exist).
5. But we cannot imagine something that is greater than God (for it is a
contradiction to suppose that we can imagine a being greater than the
greatest possible being that can be imagined.)
6. Therefore, God exists.
Let us now look at the framework of the argument and for notation we shall use Ω in
placee of God.
1. It is a conceptual truth (or, so to speak, true by definition) that Ω is a
being than which none greater can be imagined (that is, the greatest possible
being that can be imagined).
2. Ω exists as an idea in the mind.
3. A being that exists as an idea in the mind and in reality is, other things
being equal, greater than a being that exists only as an idea in the mind.
4. Thus, if Ω exists only as an idea in the mind, then we can imagine
something that is greater than Ω (that is, a greatest possible being that
does exist).
5. But we cannot imagine something that is greater than Ω (for it is a
contradiction to suppose that we can imagine a being greater than the
greatest possible being that can be imagined.)
6. Therefore, Ω exists.
Then a natural question follows, what is Ω? Part of the problem that lies is the semantics in which it is cloaked. Gaunilo the monk argued by same line of
reasoning that Lost Island exists substituting that term for Ω:
Gaunilo's argument runs along the same lines:
1. The Lost Island is that than which no greater can be conceived.
2. It is greater to exist in reality than merely as an idea.
3. If the Lost Island does not exist, one can conceive of an even greater
island, *id est* one that does exist.
4. Therefore, the Lost Island exists in reality.
The faulty reasoning becomes apparent when we generalize the argument, substituting either a concept of Non-Being, Non-God, Nothing or Anti-God (if you will). We shall use the notation !Ω. Does the argument still remain valid if we proceed in the opposite direction?
1. !Ω is that than which no greater can be conceived.
2. It is greater to exist in reality than merely as an idea.
3. If !Ω does not exist, one can conceive of an even greater !Ω,
one that does exist.
4. Therefore, !Ω exists in reality.
But by definition of "Non-Being" !Ω does not exist, yet we are asserting it does hence giving us a contradiction. Non-Being ipso facto does not exist if we take the premise that "existence is part of essence". One could also plug in the abstract concept of Nothing but that runs the danger of solipsism from linguistic abuse (ie bearing in mind when we state "Nothing" is that than which no greater can be conceived we agree that we cannot head in the opposite direction to conceive of, say, Ultra-Nothing or Meta-Nothing). Note: we do not interpret Nothing to be zero,] as one can always consider 0 plus an integer to counter the argument.
We can conclude that the fault lies in the not being able to manufacture a well-defined definition for the term greater or stems from taking Deity of Judeo-Christian background, and, St Anselm being an Archbishop, possibly, did not consider the dual of God (as in the argument above where loosely Zen philosophy was utilized of an idea of Non-Being).
As a side-note, I mention that we discussed it in college and our professor argued against the idea of being able to plug anything for Ω because of the meaning of greater. If we do so we need to come to an agreement of the meaning of greater than and thus the problem does not go away for the notions of Non-Being or Non-Existence.
Roughly, It is akin to visualizing God at top of peak of mountain or "upward light cone" and Nothing can be greater than Him. However, we can always conceive of a valley or "downward light cone" where at the very bottom resides Devil or Anti-Christ or the concept of Non-Existence. If we acknowledge St. Anselm's argument than there is nothing that prevents us from imagining something that is infinitesimal or dual to infinite nature of God.
