Three Ontological Arguments

I have been trying to provide some clear and plausible versions of the ontological argument for one of my classes. This is a lot harder than it sounds. Below is what I have so far. The first argument is an attempt to capture what Anselm was arguing in his first version in Proslogion. The second tries to do the same thing for Descartes’s version in Meditations V. The third tries to capture the essence of the Malcolm/Hartshorne version, which, they say, is based on Anselm’s second version. Any comments or suggestions would be appreciated.
Anselm’s First Argument:
1) “God” = df “the greatest conceivable being (GCB).”
Premise: Definition of “God.”
2) The GCB exists in the understanding.
Premise: Even “the fool” knows what “GCB” means.
3) If the GCB exists in the understanding, then either (a) it exists only in the understanding, or (b) it exists both in the understanding and in reality.
Premise: Anselm’s assumption that existence can have two modes, existence in the understanding and existence in reality.
4) The GCB either (a) exists only in the understanding, or (b) exits both in the understanding and in reality.
Conclusion: From 2 and 3 by Modus Ponens.
5) It is greater to exist both in the understanding and in reality than only in the understanding.
Premise: Intuitive principle.
6) The GCB exists only in the understanding.
Premise: Assumption for reductio ad absurdum.
7) It is possible to conceive of a being greater than the GCB, namely one that exists both in the understanding and reality and not only in the understanding.
Conclusion: From 5 and 6; it is possible to conceive of a being greater than one that only exists in the understanding.
8) But this is absurd.
Premise: By definition, it is impossible to conceive of a being greater than the GCB.
9) Therefore, it is not the case that the GCB exists only in the understanding.
Conclusion: Rejection of 6 by reductio ad absurdum.
10) Therefore, the GCB exists both in the understanding and in reality.
Conclusion: From 4 and 9 by disjunctive syllogism.
11) Therefore, the GCB exists in reality.
Conclusion: From 9 by simplification.
12) Therefore, God exists in reality (i.e. God exists).
Conclusion: From 1 and 11; definition of “God” as “GCB.”
 
A Quasi-Cartesian Version:
1) “God” =df “the supremely perfect being.”
Definition of “God.”
2) “The supremely perfect being” =df “the being that must possesses every perfection.”
Definition of “supremely perfect being.”
3) Existence is a perfection.
Premise. Intuitive; it is better to exist than not.
4) If we conceive of the supremely perfect being, then we conceive of that being as possessing the perfection of existence.
Conclusion: From 2 and 3; definition of “supremely perfect being.”
5) We have the concept of a supremely perfect being
Premise: Introspection reveals that we have such a concept.
6) We conceive of the supremely perfect being as possessing the perfection of existence.
Conclusion: From 4 and 5 by Modus Ponens.
7) We conceive of the supremely perfect being as existing.
Conclusion: From 6; to possess the perfection of existence is just, in other words, to exist.
8) We conceive of God as existing.
Conclusion: From 1 and 7; God is defined as “the supremely perfect being.”
9) God exists.
Conclusion: From 1-8; if reason requires us to conceive of God as existing, then we must conclude that he exists.
 
Possible Worlds Version:
1) “God” =df “a perfect being.”
Premise: Definition of “God.”
2) If a perfect being exists, then that being exists in every possible world.
Premise: Perfection cannot exist contingently, i.e. only in some possible worlds but not in others.
3) If a perfect being exists, then that being necessarily exists.
Conclusion: From 2; definition of “necessarily exists.” A necessary being is one that exists in every possible world.
4) If it is possible that a perfect being exists, then that being possibly necessarily exists.
Conclusion: From 3; principle of modal logic. If p ⊨ q, then ◊p → ◊q.
5) If it is possible that a perfect being exists, then that being necessarily exists.
Conclusion: From 4; principle of modal logic: ◊□p ⊨ □p (If something is possibly necessary then it is necessary.).
6) It is possible that a perfect being exists.
Premise: Assumption.
7) A perfect being necessarily exists.
Conclusion: From 5 and 6 by Modus Ponens.
8) God necessarily exists.
Conclusion: From 1 and 7; definition of “God.”
9) God exists.
Conclusion: From 8; principle of modal logic. □p ⊨ p (necessarily p entails p.)

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