Anselm’s first ontological argument

1. God is that than which nothing greater can be conceived. (definition)

2. God exists in our minds, i.e., we have the idea of God. (premiss)

3. But it is greater to exist in reality and in mind than just to exist in the mind. (premiss)

4. That than which nothing greater can be conceived exists in our minds. (by (1) and (2))

5. But by (3), that than which nothing greater can be conceived cannot exist just in the mind.

6. Therefore, that than which nothing greater can be conceived exists (by (4) and (5)).

Anselm’s second ontological argument

1. God is that than which nothing greater can be conceived. (definition)

2. We have an idea of God. (premiss)

3. But it is greater to be such that one cannot be conceived not to exist, than to be such that one can be conceived not to exist. (premiss)

4. We could conceive of God as being such that he cannot be conceived not to exist. (premiss: this Anselm omitted)

5. By (1), (3) and (4), if we conceive of God, then we conceive of God as a being that cannot be conceived not to exist.

6. But we do conceive of God. (by (2))

7. Therefore, the God of our conception is a being that cannot be conceived not to exist. (by (5) and (6))

8. Therefore, God exists. (from (7))

Plantinga’s ontological argument

1. God is a maximally great being. (definition)

2. A maximally great being is one which is maximally excellent (including being omnipotent and omniscient) in every possible world. (definition)

3. It is (logically) possible that there is a God. (premiss)

4. Therefore, there is a possible world W in which God exists. (by (3))

5. Therefore, in W, the following is true: God is maximally excellent, and in particular exists, in every possible world. (by (4))

6. Therefore, in W, the following is true: It is logically impossible for there to fail to be a maximally excellent being. (Since what is true in every possible world is that which logically cannot fail to be true.)

7. But logic does not vary between possible worlds—what is contradictory in one world, is contradictory in others, and hence what is impossible in one world, is impossible in others. (premiss: S5)

8. Therefore, (in the actual world) it is logically impossible for there to fail to be a maximally excellent being. (by (6) and (7))

9. Hence, there is a maximally excellent being in every possible world. (by (8))

10. Hence, there is a God (by (1), (2) and (9)).

A Leibnizian ontological argument

1. God is a being that has all perfections. (definition)

2. Necessary existence is a perfection. (definition)

3. It is possible that God exists.  For:

3.1. Perfections are simple ideas. (premiss)

3.1. Simple ideas are all mutually compatible, since a proof of their incompatibility would require further articulation of the simple ideas. (premiss)

3.2. A being all of whose properties are mutually compatible is possible. (premiss)

3.3. Therefore, all of God’s properties are mutually compatible. (By (1), (3.1) and (3.2).)

3.4. Therefore, (3).

4. Therefore, our idea of God is a genuine idea.  Hence premisses (2) in Anselm’s arguments are true, and premiss (3) in Plantinga’s argument is true.  Hence, there is a God.b