The Question is:
God is commonly defined as an omniscient (infinite knowledge), omnipotent (unlimited power), omnipresent (present everywhere) entity.
Is there any logical inconsistency in this definition?
The basic answer depends on the definitions of the "omni" terms:
1. If the base omni terms themselves are logically inconsistent then the definition itself is not logical and cannot be evaluated logically
2. If the original terms are defined in a logically consistent way, then there is no logical inconsistency in the definition.
I believe #1 is self-evident and therefore the remainder of this answer comprises developing #2.
I'm not sure that the omni-everything being is really a "common definition" of God, but I will assent to it for my answer. I tweak the "omni" definitions as follows to make sure they are logically consistent, and add the term "omnibenevolent":
- Omnipotent - all-powerful - can do anything that can be done
- Omniscient - knows everything there is to know about the universe (all matter that exists, all states of that matter, the preceding at all points in time, etc.)
- Omnipresent - The most important aspect of this is that it entails time. God is present everywhere and at all times - simultaneously. This is similar to Omniscience in practice (perhaps the means of his omniscience). The reason for this is that we live in (at least) a 4 dimensional universe, not 3-D.
- Omnibenevolent - I don't like this term, because "all-good" is not quite correct. Better to say all his actions/desires are always good. Of course, this begs the question of what "good" means.
Why do I modify the "omni" terms? Because the terms themselves are often defined in an illogical manner. And when that is the case, of course any definition using them will be logically inconsistent as well. Let me explain with "omnipotent":
If you say that an omnipotent being can do "anything", then you are claiming that an omnipotent being can do both logical and illogical things. This definition of omnipotent exists both partly in and partly outside the realm of logic, and thus logic cannot only speak to part of it, and cannot even understand the rest of it, nor forward any arguments about that part. Logic itself only works within the framework of logical things.
One solution to this is to claim that, indeed, God can do both logical and illogical things - that he truly has all power. This is expounded on in the most popular answer here. However, in this case, we cannot say anything about the part of the definition of God which is outside of logic. Note that unlike what that answer said, this does not mean we cannot know anything about such a God - it only means we cannot know about the part of God which exists outside of logic. The part that operates within logic we can still know and understand.
However, if we define "omnipotent" as "all-powerful", in the sense that an omnipotent being has all power that can or does actually exist at his disposal - anything that can be done, he can do - then we remain within the realm of logic and we can apply logic to determine whether the definition is logically consistent.
I know I am belaboring this point, but it is the crux of the problem with the arguments that are raised against an "omni-x" being.
So, why does the above matter?
The vast majority of attacks against the "omni-everything" being takes the form of asking the being to do something that is logically impossible. In other words, the power to do it cannot exist, based on our understanding of how the universe operates. But why should an all-powerful being be asked to do something that cannot be done? Why should they be asked to create something that cannot be created? If an omnipotent being can do anything that actually can be done, and can create anything that it is possible for some force or being to create - if it can create anything that is comprehensible to us or can be real to us - that is enough!
Can God create a square-circle? Well, if a square-circle is an abstract concept which is illogical (another word for illogical is incomprehensible), then why should you ask him to create something which means nothing to you and has no connection to our universe or reality as we know it? It is a meaningless request!
Granted, it may be possible to create a square-circle, because our knowledge of the universe is limited. We experience it only in 4 dimensions. Various theories imply there may be as many as 10 or 11 dimensions, and if we find that to be true then once we understand what that means, then a square-circle might end up being a logical and possible construct. If so - then an omnipotent being is able to create a square-circle, but as long as a square-circle is a meaningless, incomprehensible construction, then an omnipotent being would not need to create one to have "all-power" that does exist.
The other way to look at omni-potent, is to understand that no other being exists which has a power which he does not, and no other being has a power which he cannot override. This is the same thing as stating an omnipotent being has "all-power which does exist", and is a logically consistent definition.
Once this core principle of making sure that the original definition is logical in the first place, is applied to all the "omni" terms, so that it is even possible to evaluate the resulting definition of an omni-everything being, then the majority of arguments against it simply dissolve away as not meeting the criteria of the definition. I will leave it at that with regard to the other "omni" terms, as I've already gone too long here.
Now, there are some arguments which still stand after we get the logically inconsistent ones out of the way. The main one is the "Problem of Evil/Suffering". The first major philosophical formulation of an answer to it can be found on the Alvin Plantinga Free Will Defense page on Wikipedia. The short answer to that one is that it is better to have a world with real love/relationships (which requires free will), then to have a world without suffering/evil. This, of course, is somewhat subjective and hangs on the definition of what "good" means, but it is a logically consistent solution to the problem, whether one agrees with the conclusion or not.
I give a more detailed answer to the Problem of Evil/Suffering here.