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Anselm's ontological argument can be summarized as follows:
P1. It is a conceptual truth (or, so to speak, true by definition) that God is a being than which none greater can be imagined (that is, the greatest possible being that can be imagined).
P2. God exists as an idea in the mind.
A being that exists as an idea in the mind and in reality is, other things being equal, greater than a being that exists only as an idea in the mind.
P3. Thus, if God exists only as an idea in the mind, then we can imagine something that is greater than God (that is, a greatest possible being that does exist).
P4. But we cannot imagine something that is greater than God (for it is a contradiction to suppose that we can imagine a being greater than the greatest possible being that can be imagined.)
C. Therefore, God exists.
Anselm presented a further argument in the same vein:
1. By definition, God is a being than which none greater can be imagined.
2. A being that necessarily exists in reality is greater than a being that does not necessarily exist.
3. Thus, by definition, if God exists as an idea in the mind but does not necessarily exist in reality, then we can imagine something that is greater than God.
4. But we cannot imagine something that is greater than God.
Thus, if God exists in the mind as an idea, then God necessarily exists in reality.
5. God exists in the mind as an idea.
Conclusion. Therefore, God necessarily exists in reality.
This contains the notion of a being that cannot be conceived not to exist. He argued that if something can be conceived not to exist, then something greater can be conceived. Consequently, a thing than which nothing greater can be conceived cannot be conceived not to exist and so it must exist. This can be read as a restatement of the first argument.
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