I have attempted to show elsewhere that the Monotheist Hypothesis and the Atheist Hypothesis may be reduced to the two logical statements
There exists X who is Y (= Monotheist Hypothesis)
There exists no X who is Y (= Atheist Hypothesis),
Expressed in this form, then neither the Monotheist nor the Atheist hypotheses exhibit any logical contradictions. But both statements are also completely devoid of any empirical information. Hence neither the Monotheist, nor the Atheist hypothesis have any consequences. Debating over them would seem superfluous and irrational.
Two irreducible arguments?
However, as unperson aptly points out in a /msg, there are at least two atheist arguments which don’t seem to be immediately reducible to "There exists no X who is Y". Here are the two supposedly irreducible atheist arguments (A and B below):
(A) Judging by the many flagrant contradictions in the monotheist Scriptures, the Monotheist Hypothesis actually reduces to the self-contradictory statement "There exists X who is Y and non-Y". Hence the Monotheist Hypothesis should be dismissed as a logical contradiction.
(B) Contradictions aside, the Monotheist Hypothesis also makes certain empirical predictions. For example, the Monotheist Hypothesis holds that X is Almighty and All-Good. However, actual observations show many instances of Evil in the world. Hence X is not Almighty or not All-Good, or neither (the so-called Theodicy Problem). This would empirically disprove the Monotheist Hypothesis and the Monotheist Hypothesis can be dismissed on empirical grounds.
Demonstrations to the contrary
Below I’d like to demonstrate that the irreducibility of these two arguments is in fact illusory. In reality both the Monotheist Hypothesis and the Atheist Hypothesis are reducible to non-contradictory -- but at the same time empirically non-informative -- statements. Thus neither of them has any practical or moral consequences.
On the illusory nature of argument (A):
There are indeed many contradictions in the monotheist Scriptures. For example, in some passages X strongly condemns killing, while in other passages X equally strongly commands not only killing but even genocide (annihilation of tribes not believing in X). This is just one of many obvious contradictions -– of "Y and non-Y" -- in monotheist Scriptures.
But this argument of scriptural contradictions is relevant for fundamentalist monotheists only. In socially and intellectually advanced societies fundamentalist monotheists comprise a small minority, on par with fanatical Nazis or fanatical Communists. The overwhelming majority regards fundamentalists as cranks. They may of course still pose a serious threat to society, if they gain positions of political significance, like in Tehran, Riyadh and to some worrying extent in Washington, D.C.. Such threats must be eliminated, but this can only be done by political means -- philosophical discussions with cranks are futile.
Most modern monotheists, in contrast, do no longer regard their Scriptures as literal truth or "the true words of X". They understand that the Scriptures were written in another age, for readers living in radically different societies. They use them for general spiritual inspiration, in a similar manner that secularists use non-religious classics as an inspiration. Modern monotheists don’t use the Scriptures as sources of accurate descriptions, commands or recommendations. Furthermore, modern intellectual theology has long since done away with all attempts at logically justifying X’s existence. X’s existence is not accessible to logic, only to Faith -- this is the mainstream position of modern monotheists.
So the modern Monotheist Hypothesis may well be reduced to "There exists X who is Y". Hence the corresponding Atheist Hypothesis "There exists no X who is Y" would still be applicable, because the many "Y and non-Y"-contradictions in the Scriptures are irrelevant, being beyond the concerns of modern intellectual monotheists.
On the illusory nature of argument (B):
Here the argument maintains that the claimed properties (= Y) of X predict Good, but in reality Evil is observed instead.
If a certain hypothesis predicted that something is "circular", but in reality it was observed that it is "rectangular", then the hypothesis would be refuted, unequivocally. However, "evil" and "good" are very far from clearly definable, unequivocal concepts like "rectangular" and "circular".
According to the philosophy of the logical positivists (particularly the meta-ethical theory Emotivism of the American philosopher C. L. Stevenson) such concepts as "evil" and "good" can not be given any concrete meaning at all. They are rather expressions of the speaker’s emotional attitude toward an object or circumstance.
If some people are saying "good" about something, then we can be assured that there are others who will say "evil" about exactly the same thing. If I discover that you are busy burning my Armani suit in a bonfire, then I will probably term your action "evil" at first. But if you were to explain that my suit was infected by the H5N1 (avian flu) virus, then I’d probably be grateful and say "Good!". So "good" and "evil" may sometimes even be used interchangeably. From a logical point of view they are simply meaningless terms.
The statement "X is All-Good" consequently doesn’t make any logically interpretable empirical predictions about X and hence it can not be empirically refuted. So it seems that the atheist position can always be reduced to non-empirical statements like "There exists no X who is Y".
Both the Monotheist Hypothesis and the Atheist Hypothesis are logically non-contradictory, but at the same time totally devoid of empirical consequences. Both are essentially meaningless expressions, hardly deserving the effort of rational discussion.