Anselm begins by defining the most central term in his argument - God. Without asserting that God exists, Anselm asks what is it that we mean when we refer to the idea of "God." When we speak of a God, Anselm implies, we are speaking of the most supreme being. That is, let "god" = "something than which nothing greater can be thought."

  Anselm's definition of God might sound confusing upon first hearing it, but he is simply restating our intuitive understanding of what is meant by the concept "God." Thus, for the purpose of this argument let "God" = "a being than which nothing greater can be conceived."...

Within your understanding, then, you possess the concept of God. As a non-believer, you might argue that you have a concept of unicorn (after all, it is the shared concept that allows us to discuss such a thing) but the concept is simply an idea of a thing. After all, we understand what a unicorn is but we do not believe that they exist. Anselm would agree.

Two key points have been made thus far:

1. When we speak of God (whether we are asserting God is or God is not), we are contemplating an entity whom can be defined as "a being which nothing greater can be conceived.";

2. When we speak of God (either as believer or non-believer), we have an intra-mental understanding of that concept, i.e. the idea is within our understanding.

Anselm continues by examing the difference between that which exists in the mind and that which exists both in the mind and outside of the mind as well. What is being asked here is: Is it greater to exist in the mind alone or in the mind and in reality (or outside of the mind)? Anselm asks you to consider the painter, e.g. define which is greater: the reality of a painting as it exists in the mind of an artist, or that same painting existing in the mind of that same artist and as a physical piece of art. Anselm contends that the painting, existing both within the mind of the artist and as a real piece of art, is greater than the mere intra-mental conception of the work. Let me offer a real-world type example: If someone were to offer you a dollar, but you had to choose between the dollar that exists within their mind or the dollar that exists both in their mind and in reality, which dollar would you choose? Are you sure...

At this point, we have a third key point established:

3. It is greater to exist in the mind and in reality, then to exist in the mind alone.
Have you figured out where Anselm is going with this argument?

A. If God is that than greater which cannot be conceived (established in #1 above);
B. And since it is greater to exist in the mind and in reality than in the mind alone (established in #3 above);
C. Then God must exist both in the mind (established in #2 above) and in reality;
D. In short, God must be. God is not merely an intra-mental concept but an extra-mental reality as well.

But why? Because if God is truly that than greater which cannot be conceived, it follows that God must exist both in the mind and in reality. If God did not exist in reality as well as our understanding, then we could conceive of a greater being. But there can be no greater being. Thus, there must be a corresponding extra-mental reality to our intra-mental conception of God. God's existence outside of our understanding is logically necessary.

Criticisms:   The very first proposition "God is greater than" is a tremendous presumption that is without validation.  It is a presupposition that is not proven, resulting in a circular argument.  Likewise the whole notion that something must have being merely because we have semantically forced it to exist is outside of most rationale today.

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