St. Anselm of Canterbury
Before we walk through Anselm’s argument for the existence of God, first consider a notable point from Anthony Kenny on the subject of “proofs” for God’s existence:
All such proofs start from a phenomenon, or class of phenomena, within the world, which demand explanation. They go on to show a particular type of explanation will not lead to intellectual satisfaction, however frequently it is applied. [ … ] Proofs of the existence of the God, if they are not to be mere appeals to ignorance and incomprehension, must not depend on particular features of the world which yet are unexplained. The appeal to God is not based on particular failures of explanation but upon the provable inability of a particular pattern of explanation to give an intellectually satisfying understanding of phenomena of a certain type. 
Thus, rather than examine the phenomenon of explanation (Leibniz) or movement (Aristotle), Anselm looks to the phenomenon of conception. As he writes in chapter II of his Proslogion, “Now we believe that you are something than which nothing greater can be thought. So can it be that no such nature exists, since, “The fool has said in his heart, ‘There is no God'” (Psalm 14:1; 53:1)? When this same fool hears me say ‘something than which nothing greater can be thought,’ he surely understands what he hears” .
Anselm and Conceivability
Anselm distinguishes between two kinds of “realities”: existence in the understanding and existence in reality. The former simply means to say that God exists “subjectively, as a concept in the mind of man” , whereas the latter simply means that God exists objectively, “as a being external to consciousness” . According to Smith’s expounding on the argument: “The atheist denies what the theist affirms, but both affirm that the idea of God exists subjectively as an object of the understanding. And this subjective idea is sufficient to prove that God exists as an objective being external to consciousness” .
However, a more unraveling discussion regarding the matter of conceivability can be found from Robert Maydole (2012), where he argues  for the given proposition:
- (A3) The concept of whatever a definite description that is understood refers to has existence-in-the-understanding.
First noting from Anselm where he writes:
When the fool hears mentioned a being than which a great is inconceivable, he understands what he hears [ … ] Moreover [ … ] if this being is understood, it is in the understanding [ … ] For as what is conceived, is conceived by conception, and what is conceived by conception, as it is conceived, so is in conception; so what is to understood, is understood by understanding, and what is understood by understanding, as it is understood, so is in the understanding. What can be clearer than this? 
Maydole thus furthers the following analogical argument for (A3):
- (1) Concepts are in that which conceives of concepts.
- (2) Whatever is in that which conceives of concepts has existence-in-the-understanding.
- (3) Therefore, the concept of whatever a definite description that is understood refers to has existence-in-the-understanding.
According to Maydole thence “the analogical argument certainly appears to be inductively strong, with a degree of inductive strength that is proportional to the degree of likeness between conceiving and understanding” . This seems to demonstrate the existence in the understanding portion of the argument, but what about existence in reality?
Anselm and Reality
Anselm first argues, that “that than which a greater cannot be thought [ … ] if it exists only in the understanding, it can be thought only to exist in reality as well, which is greater” . However, just what exactly does Anselm mean when he says “that than which a greater cannot be thought” exists in “reality” as well? To distinguish from our earlier discussion of conceivability, existence in reality can be understood to mean ontological completeness. Maydole defines: “Something is ontologically complete if and only if every property or its negation is a member of the set of all its properties” .
Maydole uses the illustration of his computer being ontologically complete. For instance, the properties of the computer from which you are using to read this possesses every possible property or its negation, including the property or its negation of containing a hydrogen atom that was form a split second after the Big Bang. The set of properties of that computer also happens to be infinite. However, the limited idea of the computer itself is a finite one because it doesn’t include every property or its negation. Thus, we can formulate the following argument :
- (1) Things that have existence-in-reality are ontologically completely. (Premise)
- (2) The property of being ontologically complete has the property of being great-making. (Premise)
- (3) For every property X and Y, X has Y if and only if everything that has X has as a property that has Y. (Premise)
- (4) The property of being ontologically complete has the property of being great-making property that has the property being great-making. (3, UI)
- (5) Everything that has the property of being ontologically complete has a property that has the property of being great-making. (2, 4, Equiv., Simp., MP)
- (6) Hence, everything that has the property of existence-in-reality has a property that has the property of being great-making. (1, 5, UI, HS, UG)
- (7) The property of existence-in-reality has the property of being great-making if and only if everything that has the property of existence-in-reality has a property that has the property of being great-making. (3, UI)
- (8) Hence, the property of existence-in-reality has the property of being great-making. (6, 7, Equiv., Simp., MP)
Thus, for a neo-Platonist as to the likes of Anselm, he would concede to the idea that things existing in the understanding and things existing in reality might share most or the same properties, such as the properties of being “inconceivable” or “something to be greater.” This plausibly seems to be the case.
-  Anthony Kenny, The Unknown God: Agnostic Essays (Continuum Press: 2004) p. 26
-  Anselm, Proslogion, trans. Thomas Williams (Hackett: 1995) pp. 99-100
-  George H. Smith, Why Atheism? (Promotheus: 2001) p. 151 – emphasis his.
-  Ibid.
-  Ibid.
-  Robert Maydole, “The Ontological Argument” in The Blackwell Companion to Natural Theology (Wiley-Blackwell: 2012) p. 556
-  Anselm, Basic Writing, 2nd edn. Trans. S. N. Deane (Open Court: 1962) p 157
-  Maydole, p. 558
-  Anselm (1995), p.100
-  Maydole, p. 559