The a-not-a trap; Frequently employed by atheists; sometimes by presuppositionalists; is any argument along the lines of;
or
or for presuppositionalists,
First, it is important to note that you cannot reach these people. They aren't interested in the truth.
In every other case, familiarize yourself with the Repudiation of Philosophy then see following.
One answer is that just like a square circle (or a transcedental number) is only an impossibility in your mind, because it can exist in reality, any proof or disproof of God cannot be shown from an examination of the physical universe. The reason WHY is twofold; one, it is explicit because of creation ex nihilo, such that God does not exist or is not represented by any created thing, and two it is implicit because the God that does exist will exhibit a fundamental quality of perfection that cannot exist in the physical universe (because of it's granularity), thus is something only able to be conceptualized by a mind.
The danger then is that if we apply or overlay OUR perception over top of God's, we are limiting God by our limited perspective. Case in point, that square circles or married bachelors can in fact exist, in reality – physical objects which appear to be squares from one angle or circles from another – and there is no way to know from our perspective if the object is a square, a circle, or a compound object, from our limited perspective.
Similar, a married bachelor is a logical contradiction only if you are bound by time; for any being for which time is not a bound, such a being would observe a being as both married at one point in time and a bachelor before that point; for such a being both cases would exist simultaneously in a sort of hypostasis.
The atheist is essentially setting up a straw-man god which he then knocks down. The atheist will freely admit this by stating he does not “assume God”. So that when he does attack the notion of God he invariably attacks a different God than the one being proposed by the T.
The essential problem then is that by creating his own version of God, it is by default limited by his own perception – because he does not acknowledge that he is NOT an authority on perception and logic.
Such an A. constructs his worldview out of his perception and logic, such that it is fundamental to his worldview. In contract, the T. states that his logic and perception is a received quality. There is no proof for this other than the word of the T. based on ancestral hearsay – ex. apostolic succession, and the national revelation of Israel. They will never accept this because they have no reason to accept it – nothing in their perception or logic tells them not to trust their perception or logic. You cannot reach these people unless they witness a miracle.
Other types of atheist are those that openly state (such as M. Dillahunte) that even if God were real, he would not worship him. Usually the reason is the rejection of God's morality.
A great parallel of this would be the a culture where everyone was a bachelor – such a culture would have no need for an institution of marriage. As such, the A. who rejects God or his morality will tend to reject adjacent morality in general and would end up allowing almost every form of hedonistic impulse. This is why throughout history (and even today) every sort of sin is found to be promoted in secular cultures. Such cultures end up allowing any form of degredation simply because it is prevelant in other cultures and then, eventually, because morality itself is realized as subjective in and of itself. Thus, to the Atheist, there is no ground of morality – either because he rejects morality or above, because they have no tradition of logic or morality (and then fall back onto a subjective morality, which will eventually be found to be illusory) – which will eventually be seen to have no practical value in the A. society.
The only way out of this A not A dichotomy is to point out that the A. is not an authority on logic or perception – in fact, other human beings are equally capable of realizing logic or achieving perceptions.
If the A. then claims that he is in fact an authority, or challenges the idea that he is not an authority, the claim that you for example are an authority may be given. If this is denied, then the A. has essentially capitulated his argument; he's right because he's right, and you're wrong because you're wrong. You cannot reach these people.