The ontological argument (OA) is, historically, as follows:
When talking about language, I often use the signifier/signified model; here, I’ll be talking about members of a class as defined by some sign, too. In terms of notation,
- The signifier of x = sr[x]
- The signified of x = sd[x]
- Members of x = m[x]
Now, what exactly is the signified of a sign? I’m not talking about its function linguistically, I’m talking about its ontic being; i.e., where is the signified stored, so to speak. There are two ways one go: materially and ideally. If one takes the materialist route, one would say that the signified is constructed solely by the human mind; if one takes the idealist route, one would say there is some “place” in the ideal “world” where signs or signified are “stored” (in quotes because for many the ideal has no physicality and the words place, world and stored are merely metaphoric).
For materialists, the following argument applies:
And this makes sense; just because we define X as Y doesn’t imply an instance of class X even if Y is that it exists. Furthermore, in every exhaustive signified (i.e. something of which all attributes are accounted for--if x, for instance, is the greatest being) there is the implication of existence--to have anything signified something must also exist (again, at least for a materialist). This is in line with the existentialist mantra of “existence precedes essence”; i.e., something must have existence to have essence, or some class must have membership to be signified.
Logically, this can be expressed of anything as follows: 
(where E(x)= x exists). However, this is not necessarily true in the converse; to say so would be to affirm the consequent, a logical fallacy--that is to say, one cannot from that logic conclude that if existence is a predicate of the signified of x then there is a member of x.
Now, back to the OA. Where g is God, what the conclusion is really saying is that 
. In my notation, therefore, the entire argument is as follows:
In reality, the only conclusion one can draw from premises P1 and P2 is that 
, a conclusion quite different to a non-empty membership. Hence, a hidden premise must exist for the conclusion to follow; in particular, it’s that 
. As earlier noted, this premise cannot be proven via affirming the consequent (assuming the earlier given premise, the converse of this one, is true--I accept that it could be contested); therefore, it must be proven elsehow.
That has dealt with the material idea. The ideal idea is quite simple: it has the hidden premise that there is some ideal world of signs, or Forms in Platonic terminology. This must be proven if one wishes to take this route. Furthermore, it must be proven that there is the Form of God; i.e., that within the world of Forms there is God’s Form. One may be of the opinion that within the world of Forms, there is a form of every possible thing and that God is possible, but this seems to be contrary to popular theology. Consider the following argument:
This argument goes on ad infinitum; there are infinite great beings all of which are in some way lesser than that which is above it. This not only seems absurd, but is also contrary to every faith I’ve ever encountered. While that doesn’t make it untrue, this is a piece directed at people of faith who believe the OA is a sound argument (and those who aren’t but are merely interested).
Hence, we have concluded that without further evidence of these hidden premises the OA is at least unverifiable.
Note: I’m aware that there are more modern and complex forms of the OA; here I’m more exploring the historical argument (which is still used today) and what that uncovers about contemporarily immediate questions (which I’ll leave up to you). This is less a polemic against an argument, despite its superficial appearance; rather, it is a personal exploration of an argument with the purpose of personally furthering understanding of related fields.