This argument for the existence of God tries to deduce from the meaning of the term "God" the fact that this term also has a referent. Normally, semiotics teaches that the signified is a different beast than the referent. But is that true for the signifier "God"?
It seems that when God signifies "a being than which no greater can be thought" (TW), this conception includes within itself the fact of God's existence in reality.
So, what we do when trying to think of the greatest possible being is we start enumerating its attributes: the being than which no greater can be thought must be omnipotent, omniscient, 3, 4, 5, actually existing, 7, 8...
Now I have argued that its existence remains a conception, such that "from the idea of a perfect being only an idea of its actual existence follows, not its actual existence." Am I right?
Let me further suggest that there are two distinct criteria for something's existing at work here:
(a) (∃X)[X is a TW] -- normal existential statement; and
(b) (∀X)[X is TW → X exists] -- from definition of the being than which no greater can be thought.
In (b), the universal quantifier ranges over both real and ideal things. If X is an idea, then, being in the mind, it cannot acquire actual existence. God can exist actually, and an idea of God can exist mentally, but an idea of God cannot exist in both ways.
If X is a real thing, then those who do not admit God's existence will insist that (a) is false. Then "X is TW" is false for all X, and the truth value of "X exists" in (b) is undefined.
In two sentences in (ST, I, 2, 1, reply 2), St. Thomas make both points. (1) "Yet, granted that everyone understands that by this word 'God' is signified something than which nothing greater can be thought, nevertheless, it does not therefore follow that he understands that what the word signifies exists actually, but only that it exists mentally."
(2) "Nor can it be argued that it actually exists, unless it be admitted that there actually exists something than which nothing greater can be thought; and this precisely is not admitted by those who hold that God does not exist." We can't argue for the consequent of (b) unless we admit (a), and (a) is not self-evident. ◊(∃X)(X is a TW) is true, but we don't know at this stage of our proof whether X exists in the actual world.
(a) and (b) may be clarified as follows:
(a') (∃X)(X's essence is described by the phrase "TW");
(b') (∀X)(X's essence is described by the phrase "TW" → X exists).
Consider a second version of the ontological argument. Let X be a being that is pure actuality. Let also it be possible for X to exist (lest it can be argued that in not existing X has no potency to come to exist, because its existence is impossible). Then if X did not exist or existed but could corrupt and perish, then existence would stand to X's essence as act to potency, and X would no longer be pure act, contrary to the definition. In other words, the meaning of the term "pure actuality" entails existence of pure actuality.
There are two interpretations of this argument. (1) If the concept of pure actuality is not incoherent, and it's not, then X exists. (2) If anything is pure act, then it has existence by its very nature, i.e., X is imperishable. The reasoning is similar.