But before I offer the argument, I think I ought to first clarify what my position on religion and theological propositions are. I am an igtheist, which is a person who does not believe that theological propositions (e.g. "God caused the world to spring forth from non-existence") express meaningful ideas which can be used in a coherent and logical manner. So to an igtheist, trying to argue against God's existence is just as illogical as trying to argue for his existence. If igtheism is true, then all theological propositions are pseudo-propositions which do not meet the basic requirements necessary to form a complete idea. Some are igtheists because they observe that the word "God" has an inconsistent meaning among individuals; indeed, if you line up a hundred theists and question them on what the characteristics and properties of God are, you will get very contradictory answers between them. However, I am not an igtheist for this reason (although I also think this reasoning is also a very good one to follow).
My argument for igtheism is an argument against the meaningfulness of the word "God", and it is somewhat similar to A.J. Ayer's argument for igtheism, but with significant differences. For those not familiar with Ayer's phenomenalism, here is a summary I wrote:
Any word which refers to an empirical object (e.g. Table, Apple, Hydrogen Atom, Human, Universe) is really just a noun which refers to a collection of observations about the object (sense-data). So, for example, the word "Apple" is equivalent to "approximately 0.100kg, the colour red, roundness, the taste of apple, etc." These things are what the word Apple actually mean. There is no, so to speak, an "essence of an apple" which exists independently of these sense-data. So if, for some reason, the human body was not equipped with the sense organs necessary to experience these kinds of sensations, then the world Apple would literally become meaningless. We might as well use the word "smidditybop", because it contains just as many sense-data (exactly zero).
So if we accept that an empirical object is no more extensive than its perceived properties, then words which do not refer to even a single sense-data or possible observation fail to be meaningful.
However, this leaves us with the problem of objects such as "Apple the size of the universe" being meaningless as such (it is impossible to ever observe an apple the size of the universe, because no such thing exists). This is where I differ from A.J. Ayer's phenomenalism. For me, an empirical object may consist of either sense-data and/or modified sense-data. Modified sense-data are variations of sense-data which are generated by applying the human mind's imagination to already-existing forms of sense-data. So if we take the sense-data "size of an apple" and apply our imagination to it, we can get the modified sense-data of "size of the universe". So, the word "apple the size of the universe" is still meaningful when we apply my "modified sense-data theory of meaning" to it.
So now I will make my argument for igtheism.
Premises:  God is a transcendent thing, completely unlike any normal things we see in the natural world such as matter and energy.  Because God transcends reality, he is invisible, inaudible, odourless, tasteless... completely undetectable to the senses.  God is an empirical object, and theological propositions involving him are empirical (a posteriori) ones.  The modified sense-data theory of meaning is true.
Conclusion  Because the word "God" is an empirical noun that does not refer to any amount of sense-data and/or modified sense-data, it does not mean anything in the empirical (a posteriori) sense.  It then follows that all theological propositions fail to be a posteriori. And because they are not a priori either, they are meaningless.  Igtheism is true.
I feel like I probably could have written this post better, and with less unnecessary information. Really, this is just using the phenomenalistic aspect of logical positivism to render the word "God" meaningless rather than the whole proposition "God exists". I am not a logical positivist, but I do think that Ayer was still right to consider all theological propositions meaningless.
So, any thoughts? Counter-arguments? Should I try to clarify anything in my argument?
As an igtheistic atheist, I am particularly interested in seeing what an agnostic atheist may have to say about this.