Perhaps the most intriguing of the sundry logical proofs of God’s existence is the unum argumentum of Anselm of Canterbury, commonly known by Kant’s term, the “ontological argument” (although Kant applied the term to a different argument).
I shall let Anselm present his case, from his Proslogion:
“But surely that than which a greater cannot be thought cannot be only in the understanding. For if it were only in the understanding, it could be thought to exist also in reality–which is greater. Therefore, if that than which a greater cannot be though existed only in the understanding, then that than which a greater cannot be thought existed only in the understanding, then that than which a greater cannot be thought would be that than which a greater can be thought! But surely this conclusion is impossible. Hence, without doubt, something than which a greater cannot be thought exists both in the understanding and in reality.”
Crucially, this argument does not deal with reality, as do the famous Five Ways of Thomas Aquinas, but rather with the thought. Fundamentally, Anselm is not arguing for the existence of God; he is arguing that to doubt the existence of God involves a mental contradiction.
Also important to this argument is Anselm’s definition of “greater”. While I shall not hear attempt to analyze Anselm’s rather complex usage of the term, the important point is contained within the aforequoted passage: to exist in reality is greater than to exist only in the understanding. Thus, “greatest” is equivalent to “greatest in all respects except existence, and existing in reality”. Furthermore, “that than which a greater cannot be thought” is equivalent to “the greatest that can be thought”. Thus, we may restate Anselm’s argument as “That which is thought to be greatest in all respects except existence and is thought to exist cannot be thought to be only in the understanding.” Phrased this way, it becomes nearly trivial, for it is manifestly absurd to think that something defined as existing does not exist. Thus, the argument is a reductio ad absurdum: to deny the existence of God is self-contradictory.
But at the same time, so is to deny the existence of “a unicorn that exists”, for if one asserts that such a unicorn does not exist, clearly one is not referring to the same unicorn. Thus becomes manifest the invalidity of the argument: if we permit such a reductio, anything may be proven to exist. The problem is that thoughts themselves may be false, having no correspondence to reality. To think of a unicorn that exists is to think of a unicorn and to think that it exists; clearly, the second thought is false. Existence cannot signify the unicorn; it can only make a statement about the unicorn, which statement may be true or false. Similarly, to think of the greatest in all respects except existence that exists is to think of the greatest in all respects except existence and to think that it exists; the second statement does not signify the subject, but makes an additional claim about it which may be true or false. Anselm’s inclusion of existence as an element of goodness, however, serves to hide this invalid operation.