The ontological argument is the earliest and likely most famous (infamous?) argument for God's existence. It was originally formulated in the 11th century by Anselm of Canterbury (later Archbishop and still later Saint).
Like Leibniz, who we discussed before, Anselm was wicked smart. But he was much less concerned with developing a complete philosophy of the world. Rather, he was mainly interested in developing his love for God (and helping his students to do the same) through the exercise of his reason and intellect. He called this "faith seeking understanding."
In fact, maybe the coolest aspect of the ontological argument is that it appears in the middle of the Proslogion, which Anselm wrote as a prayer.
The term 'ontological' was of course not used by Anselm himself, but was applied to it much later. In philosophical discourse, 'ontology' means basically the study of being as such (the root onto- means 'being'). To explain this, I'll borrow an illustration from one of my professors: A biologist's studies might at some point take him to a volcanic area to study some form of life that thrives in that environment. But the biologist does not study the volcano itself. This, rather, is the field of the volcanologist. The biologist studies the volcano for the organisms that live on it. The volcanologist studies the volcano insofar as it is a volcano.
Similarly, ontology is the study of 'being qua being,' or being insofar as it is being. This name has been applied to various arguments for God's existence, beginning with Anselm's, because these arguments all establish God's (necessary) existence from His very being, or the very concept of God itself. In essence (and far too simply), because we have a concept of God, He must actually exist.
Think I've pulled one over on you? Read on.
(1) God is that than which nothing greater can be conceived. (Definition)
(2) God exists in the understanding but not in reality. (reductio assumption)
(3) Existence in reality is greater than existence in the understanding alone. (Premise)
(4) God's existence in reality is conceivable. (Premise)
(5) If God did exist in reality, then He would be greater than He is. [from (2) and (3)]
(6) It is conceivable that there is a being greater than God is. [from (4) and (5)]
(7) It is conceivable that there be a being greater than the being than which nothing greater can be conceived. [from (1) and (6)]
(8) It is false that God exists in the understanding but not in reality. [from (2) through (7)]
(9) God exists in reality. *
So this argument works by way of what logicians call a reductio ad absurdum, which means that we assume a controversial premise (in this case, premise 2), and then we draw a contradiction from that premise. Since contradictions cannot be true, we then know that the premise in question must be false.
Anselm's controversial premise is that it is possible for God to exist only in our minds but not in reality. But if we understand God to be the greatest conceivable being (which we all intuitively do), then this is impossible, because it would be greater for him to also exist in reality. So, on pain of contradiction, God must exist in reality.
And all this because we have a (relatively) clear idea of God. Like I said, I love this argument.
Confused? That's alright; it isn't easy. I'll let it simmer with you for a while, and next time we'll deal with some of the major objections that have been given and I'll share what I think is the usefulness of the argument.
* This is roughly Alvin Plantinga's formulation.