Let us define God as the Supreme Truth.
Prove A: Supreme Truth exists.
Assume ~A: Supreme Truth does not exist.
~A --> B: If Supreme Truth does not exist, then degrees of truth cannot be known.
~B: Degrees of truth are known.
Hence, ~~A: by modus tollens.
Therefore, A: Supreme Truth exists.
I think (~B) is fairly obvious. We know, for example, that "John Lennon was a musician" is less true (that is, it expresses less truth) than "John Lennon was a musician and founding member of the Beatles." This lends greater weight to our key premise - (~A --> B), since something is only known to be more or less true if we have some ultimate standard of truth - the Supreme Truth.
One possible objection might be that we could have a very high standard of truth, but not a Supreme Truth. I think this objection fails, however, since this "very high standard of truth" is itself known to be imperfect to some extent. By what standard do we know that? Without a Supreme Truth, then, one is left with a logically fallacious infinite regress.
I also often hear Dawkins' counter-argument to this, which is a sort of reductio ad absurdum. He says that if this argument were true, then we could also posit degrees of smelliness and conclude that there must be some Great Stinker. Since this is obviously not true, then allegedly the Fourth Way is also unsound. Once again, I don't find this objection compelling. First of all, truth is intangible; we don't reach out and touch truth. Yet, smelliness is something tangible (i.e. it can be physically sensed). Moreover, the Fourth Way is dealing specifically with great-making properties, of which truth is included, and smelliness is not. On both counts, therefore, I believe the objection misconstrues the original argument, and is therefore an unsuccessful refutation.