(premise 1:) God is that, nothing greater (or better) than which can be conceived.
(This is a definition of the term "God.")
(premise 2:) God is (or can be) conceived.
(Or, God exists in the mind.)
(premise 3:) To exist is greater (or better) than not to exist.
(Or, to exist in reality is better than to exist only in the mind.)
________
(conclusion:) God exists.
(logic:) Imagine a God0 that had every attribute
that a God1
has (omnipotence, omniscience, omnibenevolence, etc.) except
existence.
But by premise 3 having existence is better than not having it. So only God1, not God0, could be God according to Anselm's definition (premise 1).
Kant's criticism: existence is not a predicate (i.e., not an
attribute).
Compare:
"Griffins have the body of an eagle, the head of a lion"
and
"Griffins exist."
A slightly different argument:
(premise 1:) God is that, nothing greater (or better) than which can be conceived.
(premise 2:) God is (or can be) conceived. (Or, God exists in the mind.)
(premise 3:) To exist necessarily is greater (or better) than to exist contingently.
(conclusion:) God exists necessarily.
(conclusion:) God exists.