There is an enormous literature on the material in Proslogion II-III. Some commentators deny that St. Anselm tried to put forward any proofs of the existence of God. Even among commentators who agree that St. Anselm intended to prove the existence of God, there is disagreement about where the proof is located. Some commentators claim that the main proof is in Proslogion II, and that the rest of the work draws out corollaries of that proof (see, e.g., Charlesworth 1965). Other commentators claim that the main proof is in Prologion III, and that the proof in Proslogion II is merely an inferior first attempt (see, e.g., Malcolm 1960). Yet other commentators claim that there is a single proof which spans at least Proslogion II-III—see, e.g., Campbell 1976 and, perhaps, the entire work—see, e.g., La Croix 1972. I shall ignore this aspect of the controversy about the Proslogion. Instead, I shall just focus on the question of the analysis of the material in Proslogion II on the assumption that there is an independent argument for the existence of God which is given therein.
Here is one translation of the crucial part of Proslogion II (due to William Mann (1972, 260–1); alternative translations can be found in Barnes 1972, Campbell 1976, Charlesworth 1965, and elsewhere):

اقتباس :

Thus even the fool is convinced that something than which nothing greater can be conceived is in the understanding, since when he hears this, he understands it; and whatever is understood is in the understanding. And certainly that than which a greater cannot be conceived cannot be in the understanding alone. For if it is even in the understanding alone, it can be conceived to exist in reality also, which is greater. Thus if that than which a greater cannot be conceived is in the understanding alone, then that than which a greater cannot be conceived is itself that than which a greater can be conceived. But surely this cannot be. Thus without doubt something than which a greater cannot be conceived exists, both in the understanding and in reality.

There have been many ingenious attempts to find an argument which can be expressed in modern logical formalism, which is logically valid, and which might plausibly be claimed to be theargument which is expressed in this passage. To take a few prime examples, Adams 1971, Barnes 1972 and Oppenheimer and Zalta 1991 have all produced formally valid analyses of the argument in this passage. We begin with a brief presentation of each of these analyses, preceded by a presentation of the formulation of the argument given by Plantinga 1967, and including a presentation of some of the formulations of Lewis 1970. (Chambers 2000 works with the analysis of Adams 1971.)

8.1 Formulation 1

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[*]God exists in the understanding but not in reality. (Assumption for reductio)

[*]Existence in reality is greater than existence in the understanding alone. (Premise)

[*]A being having all of God's properties plus existence in reality can be conceived. (Premise)

[*]A being having all of God's properties plus existence in reality is greater than God. (From (1) and (2).)

[*]A being greater than God can be conceived. (From (3) and (4).)

[*]It is false that a being greater than God can be conceived. (From definition of “God”.)

[*]Hence, it is false that God exists in the understanding but not in reality. (From (1), (5), (6).)

[*]God exists in the understanding. (Premise, to which even the Fool agrees.)

[*]Hence God exists in reality. (From (7), (8).)

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See Plantinga 1967.

8.2 Formulation 2

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[*]The Fool understands the expression “the being than which no greater can be conceived”. (Premise)

[*]If a person understands an expression “b”, then b is in that person's understanding. (Premise)

[*]If a thing is in a person's understanding, then the person can conceive of that thing's existing in reality. (Premise)

[*]Each thing which exists in reality is greater than any thing which exists only in the understanding. (Premise)

[*]If a person can conceive of something, and that thing entails something else, then the person can also conceive of that other thing. (Premise)

[*]If a person can conceive that a specified object has a given property, then that person can conceive that something or other has that property. (Premise)

[*]Hence the being than which no greater can be conceived exists in reality. (From (1)-(6), by a complex series of steps here omitted.)

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See Barnes 1972.

8.3 Formulation 3

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[*]There is a thing x, and a magnitude m, such that x exists in the understanding, m is the magnitude of x, and it it not possible that there is a thing y and a magnitude n such that n is the magnitude of y and n>m. (Premise)

[*]For any thing x and magnitude m, if x exists in the understanding, m is the magnitude of x, and it is not possible that there is a thing y and magnitude n such that n is the magnitude of yand n>m, then it is possible that x exists in reality. (Premise)

[*]For any thing x and magnitude m, if m is the magnitude of x, and it it not possible that there is a thing y and a magnitude n such that n is the magnitude of y and n>m, and x does not exist in reality, then it is not possible that if x exists in reality then there is a magnitude nsuch that n is greater than m and n is the magnitude of x. (Premise)

[*](Hence) There is a thing x and a magnitude m such that x exist in the understanding, and xexists in reality, and m is the magnitude of x, and it it not possible that there is a thing y and a magnitude n such that n is the magnitude of y and n>m. (From 1, 2, 3)

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See Adams 1971.

8.4 Formulation 4

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[*]For any understandable being x, there is a world w such that x exists in w. (Premise)

[*]For any understandable being x, and for any worlds w and v, if x exists in w, but x does not exist in v, then the greatness of x in w exceeds the greatness of x in v. (Premise)

[*]There is an understandable being x such that for no world w and being y does the greatness of y in w exceed the greatness of x in the actual world. (Premise)

[*](Hence) There is a being x existing in the actual world such that for no world w and being ydoes the greatness of y in w exceed the greatness of x in the actual world. (From (1)-(3).)

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See Lewis 1970.
Lewis also suggests an alternative to (3) which yields a valid argument:

اقتباس :

(3′) There is an understandable being x such that for no worlds v and w and being y does the greatness of y in w exceed the greatness of x in v.

and two alternatives to (3)—not presented here—which yield invalid arguments. (Of course, there further two alternatives are crucial to Lewis' overall analysis of the passage: essentially, Lewis suggests that Anselm equivocates between an invalid argument with plausible premises and a valid argument with question-begging premises. In this respect, Lewis' analysis is quite different from the other analyses currently under discussion.)

8.5 Formulation 5

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[*]There is (in the understanding) something than which there is no greater. (Premise)

[*](Hence) There is (in the understanding) a unique thing than which there is no greater. (From (1), assuming that the “greater-than” relation is connected.)

[*](Hence) There is (in the understanding) something which is the thing than which there is no greater. (From (2), by a theorem about descriptions.)

[*](Hence) There is (in the understanding) nothing which is greater than the thing than which there is no greater. (From (3), by another theorem about descriptions.)

[*]If that thing than which there is no greater does not exist (in reality), then there is (in the understanding) something which is greater than that thing than which there is no greater. (Premise)

[*](Hence) That thing than which there is no greater exists (in reality). (From (4) and (5).)

[*](Hence) God exists. (From (6).)

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See Oppenheimer and Zalta 1991.
Oppenheimer and Zalta 2011 provides a “simplified” version of this argument, in which the number of controversial assumptions is reduced. Since they also provide a clear reason for thinking that this new version of the argument is not persuasive, I shall not consider it further here.

8.6 Critical Appraisal

Considered as interpretations of the argument presented in the Proslogion, these formulations are subject to various kinds of criticisms.
First, the modal interpretations of Lewis 1970 and Adams 1971 don't square very well with the rest of the Proslogion: the claim that “being than which no greater can be conceived” should be read as “being than which no greater is possible” would have us render the claim of Proslogion 15 to be that God is a being greater than any which is possible. And that is surely a bad result.
Second, the Meinongian interpretations of Barnes 1972, Adams 1971 and Oppenheimer and Zalta 1991 produce arguments which, given the principles involved, could easily be much simplified, and which are obviously vulnerable to Gaunilo-type objections.
Consider, for example, the case of Oppenheimer and Zalta. They have Anselm committed to the claim that if anyone can understand the phrase “that than which F”, then there is something in the understanding such that F (see their footnote 25); and they also have him committed to the claim that if there is something which is the F-thing, then it—i.e., the F-thing—has the property F (see page 7). Plainly though, if Anselm is really committed to these principles, then he could hardly fail to be committed to the more general principles: (1) if anyone can understand the phrase “anF”, then there is at least one F-thing in the understanding; and (2) if there are some things which are the F-things, then they—i.e., the F-things—must have the property F. (It would surely be absurd to claim that Anselm is only committed to the less general principles: what could possibly have justified the restrictions to the special cases?)
But, then, mark the consequences. We all understand the expression “an existent perfect being”. So, by the first claim, there is at least one existent perfect being in the understanding. And, by the second claim, any existent perfect being is existent. So, from these two claims combined, there is—in reality—at least one existent perfect being.
This argument gives Anselm everything that he wants, and very much more briefly. (TheProslogion goes on and on, trying to establish the properties of that than which no greater can be conceived. How much easier if we can just explicitly build all of the properties which want to “derive” into the initial description.) So, if Anselm really were committed to the principles which Oppenheimer and Zalta appear to attribute to him, it is hard to understand why he didn't give the simpler argument. And, of course, it is also hard to understand why he didn't take Gaunilo's criticism. After all, when it is set out in this way, it is obvious that the argument proves far too much.
Third, some of the arguments have Anselm committed to claims about greatness which do not seem to correspond with what he actually says. The natural reading of the text is that, if two beings are identical save that one exists only in the understanding and the other exists in reality as well, then the latter is greater than the former. But Barnes 1971, for example, has Anselm committed to the much stronger claim that any existing thing is greater than every non-existent thing.
Given these kinds of considerations, it is natural to wonder whether there are betterinterpretations of Proslogion II according to which the argument in question turns out NOT to be logically valid. Here is a modest attempt to provide such an analysis:
We start with the claim that the Fool understands the expression “being than which no greater can be conceived”, i.e., even the Fool can entertain the idea or possess the concept of a being than which no greater can be conceived. Now, entertaining this idea or possessing this concept requires the entertainer or possessor to recognise certain relationships which hold between given properties and the idea or concept in question. For example, given that you possess the concept of, or entertain the idea of, a smallest really existent Martian, it follows that you must recognise some kind of connection between the properties of being a Martian, really existing, and being smaller than other really existing Martians, and the concept or idea in question.
Following Anselm, we might say that, since you understand the expression “smallest really existent Martian”, there is, in your understanding, at least one smallest really existent Martian. (Or, apparently following Descartes, one might say that real existence is “part of”—or “contained in”—the idea of a smallest really existent Martian.) However, in saying this, it must be understood that we are not actually predicating properties of anything: we aren't supposing that there is something which possesses the properties of being a Martian, really existing, and being no larger than any other Martian. (After all, we can safely suppose, we don't think that any Martians really exist.) In other words, we must be able to have the concept of, or entertain the idea of, a smallest really existing Martian without believing that there really are any smallest Martians. Indeed, more strongly, we must be able to entertain the concept of a smallest really existent Martian—and to recognise that the property of “really existing” is part of this concept—while nonetheless maintaining that there are no smallest existent Martians.
It will be useful to introduce vocabulary to mark the point which is being made here. We could, for instance, distinguish between the properties which are encoded in an idea or concept, and the properties which are attributed in positive atomic beliefs which have that idea or concept as an ingredient. The idea “really existent Santa Claus” encodes the property of real existence; but it is perfectly possible to entertain this idea without attributing real existence to Santa Claus, i.e., without believing that Santa Claus really exists.
We can then apply this distinction to Anselm's argument. On the one hand, the idea “being than which no greater can be conceived” encodes the property of real existence—this is what thereductio argument establishes (if it establishes anything at all). On the other hand, it is perfectly possible to entertain the idea of a being than which no greater can be conceived—and to recognise that this idea encodes the property of real existence—without attributing real existence to a being than which no greater can be conceived, i.e., without believing that a being than which no greater can be conceived really exists.
Of course, the argument which Anselm actually presents pays no attention to this distinction between encoding and attributing—i.e., between entertaining an idea and holding a belief—and nor does it pay attention to various other niceties. We begin from the point that the Fool entertains the idea of that than which no greater can be conceived (because the Fool understands the words “that than which no greater can be conceived”). From this, we move quickly to the claim that even the Fool is “convinced”—i.e., believes—that that than which no greater can be conceived possesses the property of existing in the understanding. And then the reductioargument is produced to establish that that than which no greater can be conceived cannot exist only in the understanding but must also possess the property of existing in reality as well (and all mention of the Fool, and what it is that the Fool believes, disappears).
As it stands, this is deeply problematic. How are we supposed to regiment the references to the Fool in the argument? Is the reductio argument supposed to tell us something about what even the Fool believes, or ought to believe? Are the earlier references to the Fool supposed to be inessential and eliminable? How are we so much as to understand the claim that even the Fool believes that that than which no greater can be conceived exists in the understanding? And how do we get from the Fool's understanding the words “that than which no greater can be conceived” to his believing that that than which no greater can be conceived possesses the property of existing in the understanding?
Following the earlier line of thought, it seems that the argument might go something like this:
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[*](Even) the Fool has the concept of that than which no greater can be conceived.

[*](Hence) (Even) the Fool believes that that than which no greater can be conceived exists in the understanding.

[*]No one who believes that that than which no greater can be conceived exists in the understanding can reasonably believe that that than which no greater can be conceived exists only in the understanding.

[*](Hence) (Even) the Fool cannot reasonably deny that that than which no greater can be conceived exists in reality

[*](Hence) That than which no greater can be conceived exists in reality.

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While this argument does not look very compelling, it is plausible to claim that it would have seemed compelling to someone who failed to attend to the distinction which we have drawn between entertaining ideas and holding beliefs, and who was also a bit hazy on the distinction between the vehicles of belief and their contents. When the Fool entertains the concept of that than which no greater can be conceived he recognises that he is entertaining this concept (i.e., he believes that he is entertaining the concept of that than which no greater can be conceived—or, as we might say, that the concept is in his understanding). Conflating the concept with its object, this gives us the belief that than which no greater can be conceived possesses the property of existing in the understanding. Now, suppose as hypothesis for reductio, that we can reasonably believe that that than which no greater can be conceived possesses the property of existing only in the understanding. Ignoring the distinction between entertaining ideas and holding beliefs, this means that we when we entertain the idea of that than which no greater can be conceived, we entertain the idea of a being which exists only in the understanding. But that is absurd: when we entertain the idea of that than which no greater can be conceived, our idea encodes the property of existing in reality. So there is a contradiction, and we can conclude that, in order to be reasonable, we must believe that that than which no greater can be conceived exists in reality. But if any reasonable person must believe that that than which no greater can be conceived exists in reality, then surely it is the case that that than which no greater can be conceived exists in reality. And so we are done.