I’m quoting an admittedly simplified version of this argument by the Christian Apologetics and Research Ministry (CARM). Read the original here.
“Logical absolutes exist. Logical absolutes are conceptual by nature, are not dependent on the space, time, physical properties, or human nature. They are not the product of the physical universe (space, time, matter) because 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 is authoring them. This mind is called God.” Follow the link for CARM’s own list of possible objections and responses to each.
The Transcendental Argument for the existence of God (TAG for short) demands a certain sardonic respect due to its sheer ambition. In its full form, it claims that logic (and by extension rationality, sense, morality and any argumentation at all) can only exist if the Christian God does.
In simplified form it’s not Christian-specific, but it can be used at any point in an argument to override the whole thing and declare that the argument is only possible (or evidence is only understandable as a concept, or our senses are only reliable) if there’s a god, so one must exist.
This approach does not convince many atheists as far as we know. It seems like the equivalent of winning at chess by knocking over all the pieces. Nevertheless it’s difficult to find a clear hole in it which Christians in particular haven’t already closed with an addendum (see how much longer CARM’s list of defenses is than the argument itself).
One good way to make it a lot less convincing, strangely, is to temporarily presuppose the existence of God. If God exists, He still isn’t guaranteed to be the source of logic, because how could we check? We couldn’t go to a universe without God to see whether logic fails there, either because we’re stuck in this universe or because God’s omnipresence extends beyond it. In other words, if God’s there we can’t remove Him to see whether logic is independent of Him.
Therefore even if God existed and we all knew it, that logic is dependent on Him could only ever be an assertion and the Transcendental Argument is still not self-evident. If He doesn’t exist, of course, then the TAG is not only moot but flat out false.
I have two other major objections which CARM’s pre-emptive defenses don’t fully cover. Firstly, logical absolutes, rather than being conventions, eternal or anything else CARM mentions, may not really exist at all but instead may only be apparent.
Secondly, if logical absolutes do exist, saying that they must be the product of an absolute transcendent mind is an argument from ignorance. (Likewise is the assertion that if the physical universe were to disappear they would still be true. Again, how would you check?) Even if they’re not the product of the physical universe or human minds, there may be any number of unknown alternatives besides a transcendent mind, or any mind at all.
I realise that most objections to the TAG are simply alternative hypotheses and doubts as to its basic assertions, but that’s really all you need. If there is any possible alternative, an argument presented as the only possible state of affairs cannot be a proof until it clearly dismisses all competition. Once the possibility of an a priori proof is gone, the TAG loses its power and becomes just another thing theists say.