In the nutshell version TAG goes like this:
- Logical absolutes (principle of non-contradiction, law of excluded middle, etc.) exist.
- Logical absolutes are conceptual by nature.
- They are not dependent on space, time, physical properties, or human nature.
- They are not the product of the physical universe (space, time, matter). If the physical universe were to disappear, logical absolutes would still be true.
- Logical Absolutes are not the product of human minds, because human minds are different, not absolute.
- But, since logical absolutes are always true everywhere, and not dependent upon human minds, it must be an absolute transcendent mind that is authoring them.
- This mind is called God.
But the train comes off the rails right at the beginning.The first problematic notion is the idea that logical absolutes exist in such a way as to require them to be accounted for. They do exist in the same sense that numbers and triangles and other abstract objects exist. But their existence isn't really anything mysterious. When we say that a triangle (in plane geometry) is necessarily a closed three sided figure with internal angles that sum to 180 degrees, we don't think of this fact as something that requires an explanation or a cause or a ground for its existence. At least, not since the demise of Platonism we don't. It isn't necessary to invoke the mind of God (or any other mind) as the storehouse in which the perfect form of Triangle is held. All that's required for us to claim that a triangle necessarily has three sides is simply to assert that in all possible worlds any idea or object that can be called a "triangle" by our definition of the term will have three sides. Mystery dissolved.
Something similar is going on with logical absolutes. To say that the principle of non-contradiction is a logical absolute simply means that in any possible world no proposition and its contradictory can be true in the same way at the same time. A figure can't simultaneously be a triangle and not a triangle if we mean the same thing by "triangle" in both instances. But this fact doesn't need the mind of God (or any other mind) to obtain. Even in the absence of all minds it would be the case. No one would know it, but it would be the case. The expression of the principle of non-contradiction is conceptual or propositional, but it doesn't follow from this that there is some kind of mysterious essential nature of logical principles that requires them to be lodged in a mind in order to exist.
But even if we accept Slick's characterization of logical absolutes as existents that have to be caused by a mind somehow, the rest of his argument doesn't follow. If they are mind dependent like he claims and if we don't know from the outset whether there is an eternal mind, then he has no ground for claiming that they are universal and independent of the physical world. If all minds are physical and logical principles are mind dependent, then they are not universal. Assuming jointly that they are mind dependent and absolute essentially begs the question. So Slick would need to provide either evidence or an argument for their absoluteness; he tries to do this in the tortuously convoluted full argument, but his defense of the absoluteness of logical absolutes is merely a series of assertions.
He argues that they can't be the product of human minds, but the only reason he gives for this is that "human minds are different, not absolute" but surely this is beside the point. The variation in human minds doesn't prevent the collaborative construction of other conceptual systems: history, science, and language are all mind dependent conceptual systems that arise as the result of many different minds. Why should logic be any different if indeed it is mind-dependent in the way Slick claims? His response would likely be along the lines of, "But there are disagreements in all these fields. No one disagrees with the laws of logic." But there are plenty of disputes and unresolved issues in formal logic, and even forms of logic that reject bivalence and other logical principles classical logic would call absolutes.
And finally, his conclusion that a transcendental mind called God must be "authoring" logical absolutes is incoherent. If he means that God invented them by fiat, then God could have given us other logical absolutes, in which case the actual logical absolutes end up being contingent, and not absolute at all. But this eliminates the need to invoke an eternal mind in the first place, which Slick only had to do to explain logic as absolute. But if God didn't invent them by fiat and if they are to be truly universal and absolute then even God is bound by them; not even God can contradict the principle of non-contradiction. Which means that they don't depend on God in any relevant sense.
Ultimately, TAG suffers from all the same problems as the other a priori arguments for the existence of God. Logic can only be twisted so far; you can't conjure beings out of concepts, which is what all of these arguments try to do. They're the philosophical equivalent of magic tricks, designed to fool the credulous and impress the superstitious.