First, let me be clear by what is meant by a "transcendental argument for the nonexistence of God." This isn't an argument that defeats the existence of all possible gods, it's only aimed at a certain conception of God. If someone believes in a god that doesn't possess the qualities that this argument attacks, then it obviously doesn't apply.
Many conceptions of God suppose that God is the creator of all things external to God, that God has control over everything external to God, and furthermore that God isn't contingent on any higher transcendental "truths" than God. My argument (which amplifies the first published transcendental argument for the non-existence of God by Michael Martin) aims to show that God is not the creator of all things external to God, that God does not have control over everything external to God, and that God is contingent on higher transcendental "truths" than God, so that therefore any God attributed with those characteristics can't exist.
1) "God is the creator of all things external to God"
The Law of Identity describes a state of affairs in which something is itself, formulated symbolically as A = A. This state of affairs is both self-evident and incorrigible, which is what epistemologists would call "properly basic," meaning that it's necessarily true because even its negation assumes its efficacy.
For example, if I were to even hypothetically say, "The Law of Identity is false," to what am I referring? The Law of Identity. Is the Law of Identity the Law of Identity? Yes! To even attempt to doubt it, I have to assume its truth -- which is ultimately self-refuting. There are volumes on this subject, but let us just keep in mind for now that identity is incorrigibly true.
Is identity part of God? Well, it's true that God = God (identity), and that God must be either God or not-God (excluded middle), and that God can't be both God and not-God at the same time and in the same respect (noncontradiction), so clearly God exemplifies identity. This doesn't mean identity is part of God though, since every last one of us exemplifies identity.
For example, my name is Erin and I exemplify identity. Erin = Erin, but identity is external to me. Why would I say this? Because if I never existed, things would still be what they are (and they wouldn't be what they aren't). So identity isn't a part of me, it's just an attribute I possess.
If God didn't exist, would identity continue to exist? Yes. Let's say that the proposition "God exists" is represented by the character X.
Is ¬X = ¬X true? Yes, we must agree that it is. Clearly, identity would still function just fine in the absence of God just as it functions in the absence of Erin.
Thus, identity is external to God. This is important because it brings us to the next question: can God create identity?
Consider for a second the absurdity of God trying to create identity. How could God be God in the first place to create identity if identity weren't already inherently true?
If identity is external to God, and God can't create identity, then God didn't create all things external to God. This suggests there is a higher transcendental "truth" than God, and therefore God can't be the highest transcendental reality. (Out the window, all you old and dusty ontological arguments for God!)
2. "God has control over everything external to God"
Does God have control over identity? Even some of the most die-hard theologians would argue that God doesn't. Let's ignore for a second the bizarre cart-before-the-horse ramifications of God controling a higher transcendental "truth" than God and ask:
Can God create a square that is a circle at the same time and in the same respect?
Can God exist and not-exist at the same time and in the same respect?
Can God be absolutely benevolent and absolutely malevolent at the same time and in the same respect?
I think we would have some intersting thinkers on our hands, indeed, if they answered yes to either of these questions.
Furthermore, many theists argue that it's "against God's nature" to be malevolent. For these folks in particular: do you see how saying that God is unable to do something "because of his nature" suggests that God conforms to higher transcendental truths which are outside of the control of God?
Since God is unable to change God's own nature, or to actualize logical contradictions, God is not able to controll all things external to God.
3) "God isn't contingent on any higher transcendental truths"
As I've argued from (1) and (2), this must clearly be the case. God is contingent on higher transcendental states of affairs.
Now this really raises some tough ramifications. For one, any kind of ontological argument for the existence of God that relies on God being "that which nothing greater can be conceived" fail immediately and inherently. This is the most serious consequence of this line of argument, and where I rest my case against such arguments.
Secondly, the three statements that I tackled can't be said to be true of any god. This is far less serious, since any rational theist can go right on believing in a god that's contingent on higher transcendental truths.
However, this really raises the most hair-raising question: if God wasn't necessary to cause identity, then we have at least one non-God thing that exists independently of God's creation that even theists must logically admit given the argument is sound. There goes theistic arguments that God is necessary to explain any non-God existence at all! While it doesn't follow from the argument, it does set the stage for the question: why should we assume the existence of the material universe must be explained as "created" or "beginning" in the first place?
You decide.
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EDIT: Oh yeah, PS. I forgot to mention that this argument also undercuts the basis for the Transcendental Argument for the Existence of God. Whoops! Tiny oversight.