Atheist philosophers often try to disprove God's existence by saying that the idea of God is incoherent; they say that some of God's attributes contradict each other. Rather than respond to particular arguments, I will give a general explanation of why we cannot completely understand everything about God, which means that an atheist using an argument from incoherence will have, at the very least, a difficult time disproving God's existence.
Finite VS. Infinite
For something to exist, it must have a reason for its existence, which it can either have in and of itself or get from something else. This means that there must be some first cause that has its existence in and of itself; an infinite regression of things passing on their existence doesn't account for how it got there in the first place. God, who created the universe and gave it existence, is this first cause. Now, for God to exist independently of anything else, there must be something in His nature that makes it necessary for Him to exist; His very essence must be existence. A being's essence is what the being is, so God must be existence. Since God is existence, He must be infinite and without limitations, for a limitation is a deficiency in existence, but a being that is existence can have no deficiencies in its existence. We, on the other hand, are finite and have limitations, and this is the primary reason why we can't understand everything about God.
We are finite, and so are our intellects, which means that we cannot totally understand everything about reality. Some things are just above our ability to comprehend, just as an adult's actions may be above the ability of a child to fully comprehend. Consequently, some of God's attributes may seem irrational and contradictory to us, but in reality, they are perfectly logical.
Spirit/Eternity VS. Matter/Time
For something to be made of matter, it must have limitations. In a material being, no one part of it is the whole being, which means that parts can be broken off. If something is dismembered, it loses a part (or parts) of itself, which is a weakness and a limitation. In addition, if something is made of matter and has separate parts that are not the whole, it must occupy space (otherwise each part would be the whole), which means that a material being cannot do everything it is capable of doing at the same time. For example, if I am now clapping my hands, I cannot be also playing the piano and lifting a heavy box at the same time. This is also a limitation, and God cannot have limitations, so we can see that God cannot be composed of matter. Since God is not matter, He must be something else, and we call this something else "spirit." We, however, live in a world totally composed of matter, so we can only experience matter and its limitations, not spirit and its lack of material limitations. Since we have only experienced matter, we have trouble understanding spiritual (I use the word "spiritual" here to mean a being that is spirit rather than matter) beings. Since God is a spiritual being, it makes sense that some of His attributes would seem incomprehensible to us and may exceed our capacity to understand.
Time is the measure of change in the physical universe. If something changes, it can never be in total possession of itself (you are not the same as you were last year, and neither am I), which is a limitation. Since God can have no limitations, He cannot change, so He must be outside of time and in eternity. We humans are in time and experience things only in time, so we have not even the slightest inkling of what eternity may be like. Because of this, we cannot be expected to understand everything about eternity and how God can act outside of time.
An Unfair Advantage?
So, what implications does this have for the theist/atheist debate? Does it mean that it's impossible to disprove God's existence? Here's my take on the issue.
There are some things about God that we cannot understand and that seem to be contradictions, but there are also very good reasons to believe in God, such as the kalaam cosmological argument, the teleological argument, and the argument from contingency (which I outlined above in the section on God's infiniteness). Thus, we seem to be faced with a problem. God exists, but His nature and/or activity seems to violate the laws of logic. In my opinion, because of the reasons given in this essay, God should be given the benefit of the doubt, and we should believe in Him. However, if there were no good arguments for God's existence, then because there would be no reason to believe in Him in the first place, the arguments from incoherence would, in my opinion, demonstrate that God probably does not exist.
Am I just being prejudiced towards my own side? Am I saying that we should give God the benefit of the doubt just because I believe in Him? It could be. We all have our prejudices and biases; nobody is totally, 100% objective. However, I stand by everything I've said in this essay and hold that our lack of understanding does not disprove God's existence.