[This is my favourite argument for God’s existence. I love the idea that God’s existence should be demonstrable, and even undeniable, from absolutely nothing but logical thinking.]
Consider in your mind, the idea of That-Than-Which-Nothing-Greater-Can-Be-Thought. Whatever That may be, it is impossible to think anything greater.
You have conceived this concept, and so That-Than-Which-Nothing-Greater-Can-Be-Thought may be said to exist in your thoughts.
Now, either That-Than-Which-Nothing-Greater-Can-Be-Thought exists outside of your thoughts also, or it doesn’t.
But if it doesn’t exist beyond your thoughts, it would be greater (in thought), if it did exist outside of your mind. But then there could be a greater thought than that which, by definition, is That-Than-Which-Nothing-Greater-Can-Be-Thought, which is a contradiction. To think of That-Than-Which-Nothing-Greater-Can-Be-Thought existing outside of our thoughts is greater than to think of That-Than-Which-Nothing-Greater-Can-Be-Thought, being confined to mere thought.
Hence That-Than-Which-Nothing-Greater-Can-Be-Thought must exist also beyond our thoughts, in reality.
What about islands?
How about That-Island-Than-Which-No-Greater-Island-Can-Be-Thought? Must that exist too?
No. As a creation, Islands are necessarily limited, and so we must consider them as at most times and in most places, not existing. Now, an island would be greater if it was, beyond mere thoughts, in every place and time (it’s a lot more convenient to me, at least). However, then it would no longer be an island at all (it would be upon its own coast!).
And so, That-Island-Than-Which-No-Greater-Island-Can-Be-Thought, is a contradiction, and cannot exist, either in or beyond thought. While the thought of the verbal formula [That-Island-Than-Which-No-Greater-Island-Can-Be-Thought] does exist in the mind, the actual thing, even as a conception of logic, cannot exist in the mind, because it contradicts the definition of an island.
And the same goes for all finite things.
To help illustrate, consider that there does not, and cannot, exist a greatest finite number. By virtue of being finite, it has a boundary, an end (fin), and hence, it is always possible to exceed that boundary.
In fact, all definitions impose boundaries, and so, the argument does not work for anything that is defined. And so we arrive at the conclusion, that That-Than-Which-Nothing-Greater-Can-Be-Thought, cannot be defined. And how extraordinarily great It must be, to be too great for our definitions!
St. Anselm of Canterbury, pray for us.
God bless you!