Anselm of Canterbury
St. Anselm (1033-1109), considered by many to be the father of Scholasticism, was among the earliest medieval thinkers to apply Aristotelian logic to the Christian faith. Following mainstream medieval Christian theology, Anselm believed that reason led to the same truths as those revealed by God (e.g., in scripture). In the passages of Proslogium reproduced below, Anselm demonstrates the existence of the "highest of all existing beings," God, through rational argument. This argument is the famous "ontological argument," which is based on the notion that God must have the trait of existence in order to be God.
From the Proslogium
Even the fool is convinced that something exists in the understanding, at least, than which nothing greater can be conceived. For, when he hears of this, he understands it. And whatever is understood, exists in the understanding. And assuredly that, than which nothing greater can be conceived, cannot exist in the understanding alone. For, suppose it exists in the understanding alone: then it can be conceived to exist in reality; which is greater.
Therefore, if that, than which nothing greater can be conceived, exists in the understanding alone, the very being, than which nothing greater can be conceived, is one, than which a greater can be conceived. But obviously this is impossible. Hence, there is no doubt that there exists a being, than which nothing greater can be conceived, and it exists in the understanding and in reality.
And it assuredly exists so truly, that it cannot be conceived not to exist. For, it is possible to conceive of a being which cannot be conceived not to exist; and this is greater than one which can be conceived not to exist. Hence, if that, than which nothing greater can be conceived, can be conceived not to exist, it is not that, than which nothing greater can be conceived. But this is an irreconcilable contradiction. There is, then, so truly a being than which nothing greater can be conceived to exist, that it cannot even be conceived not to exist; and this being thou art, O Lord, our God.
A Bonus: Excerpts from Gaunilo's In Behalf of the Fool
The fool might make his reply:
This being is said to be in my understanding already, only because I understand what is said. Now could it not with equal justice be said that I have in my understanding all manner of unreal objects, having absolutely no existence in themselves, because I understand these things if one speaks of them, whatever they may be?
Unless indeed it is shown that this being is of such a character that it cannot be held in concept like all unreal objects, or objects whose existence is uncertain . . . .
Moreover, the following assertion can hardly be accepted: that this being, when it is spoken of and heard of, cannot be conceived not to exist in the way in which even God can be conceived not to exist. For if this is impossible, what was the object of this argument against one which doubts or denies the existence of such a being?