Anselm's Monologion Arguments:

(I) Proof that there is some one being that is Supremely Good (Monologion, c.1):

(1) Whenever various things are said to be F in greater, lesser, or equal degree, either (a) different Fs are F through different things, or (b) there is some one thing through which they are all F.

(2) (1a) is clearly false.

(3) Therefore, whenever various things are said to be F in greater, lesser, or equal degree, there is some one thing through which they are all F. (One-over-Many Principle) (1,2)

(4) There are innumerable goods, which can be experienced by the senses and discerned by reason.

(5) Therefore, there is some one thing through which all goods are good. (3,4)

(6) That through which all goods are good is a great good.

(7) Therefore, that through which all goods are good is good through itself (per se). (6)

(8) What is F through another (per aliud) is not greater than or equal to what is F through itself (per se). (Independence Principle)

(9) Therefore, there is some one thing which is Supremely Good. (7,8)(Parallel argument that there is some one thing that is Supremely Great, Monologion, c.2.)

Objection to (2): Apparent counter-examples--some horses are good because they are swift and others because they are strong; a horse is good because it is swift, but a swift robber is bad.

Anselm's Reply: Whatever is good is good either because it is useful (= instrumentally good) or honorable (= intrinsically valuable), and the instrumentally good is good ultimately because of the intrinsically good.

Query/objection to (6): Anselm's argument is reminiscent of Plato's for the Forms. If what lies behind (6) is assent to the Self-Predication principle, then it combines with the One-over-Many and the Transcendence of Perfection Principles to generate the "Third Man" argument: for any batch of Fs, One-over-Many posits some one thing F-ness through which they are F; Self-Predication would identify F-ness as a great F, thereby producing a new collection of Fs (viz., the old collection + F-ness) to which One-over-Many can be reapplied, etc.

Reply: (a) Anselm doesn't explicitly endorse a Self-Predication Principle, and if he did it would get him into more trouble later. (6) may ride on the independent intuition of the explanatory priority of Goodness and/or the general causal intuition that you can't produce what/better than what you actually have.

(b) Anselm doesn't understand the Transcendence of Perfection to imply that paradigm F transcends every collection of Fs but only collections of sensory and changing Fs, that can be more or less F or be equaled in F-ness by something else.

Proof that there is some one being that is maximally and is the highest of all (Monologion, c.3):

(1) Everything which is, either (a) is through something or (b) through nothing.

(2) Nothing exists through nothing. [= not-1b]

(3) Therefore, whatever exists exists through something. [=1a]

(4) Either (a) there is one being or (b) more than one being through which all extant things exist.

(5) If (4b) there are more than on being through which all extant things exist, then (a) either these are related to some one thing through which they all exist, or (b) these exist each through itself, or (c) these exist through one another.

(6) If (5a), then not-4b, because all things exist through that one being through which these exist.

(7) If (5b) then not-4b, because there is some one thing through which all these exist each through itself. [One over Many Principle}

(8) (5c) is absurd.

(9) Therefore, not-4b. (5,6,7,8)

(10) Therefore, 4a.

(11) That through which all being exists exists through itself (per se).

(12) Whatever is F through another (per aliud) is less than what is F through itself(per se). [Independence Principle]

(13) Therefore, there is some one thing though which all beings exist which is the greatest and highest of all beings.

Note on (2): (2) amounts to the assumption that existence requires an explanation.

Note on (4)-(10): Compared to Monologion, ch.1-2, this argument seems to provide a defense for the claim that all beings have their being through some one being, instead of appealing immediately to the One-over-Many Principle. But this is no real advance over chs. 1-2, because the One-over-Many Principle is invoked at (7).