Nevermind. We might as well discuss the trinity.

Ok. I've been giving this some thought.Correct my mistake in the following.
1). God is immortal, by scriptural assertion.
2). What is immortal, by definition, cannot die.
3). The Word is God, by scriptural assertion.
4). Jesus is the Word, by scriptural assertion.
5). (From 3 & 4) Jesus is (and always has been) God.
6). (From 1 & 5) Jesus is immortal.
Have I made any mistakes yet?
First of all, let me correct my statement about death. Though it may be unorthodox, I am not comfortable with saying that only "the human side" of Jesus died on the cross. I think that Jesus was a whole person and cannot be separated into two halves. (I am pretty sure that the writers would have seen death the way you do as well). He is the unique God-man. A single individual.
Ok. Onto the question. I don't think that this is too off topic. Let me explain.
Since I am stateing that God cannot break the rules of logic, it seems you are saying that I need to relinquish my belief that He can't do this, because I believe in the doctrine of the trinity, which you assert is illogical.
You have conceded that the statement "One God in Three Persons" is not itself illogical, but that it leads to contradictory beliefs. These are what I am looking to discuss.
I think that your reasoning above is almost correct. But you are leaving out some very important things. I am assuming you are heading in the direction of showing that I believe "God can be alive and not alive at the same time", so I'll reason from there.
Agreed.1). God is immortal, by scriptural assertion.
Agreed.2). What is immortal, by definition, cannot die.
Agreed.3). The Word is God, by scriptural assertion.
Agreed.4). Jesus is the Word, by scriptural assertion.
Agreed.5). (From 3 & 4) Jesus is (and always has been) God.
Disagree. It would follow from 1 & 5 alone. But you are leaving out some important information. Namely, that Jesus is a man (since the Word became flesh), and always will be fully man.6). (From 1 & 5) Jesus is immortal.
Though He became a man, He never ceased being God. However, He:
Phi 2:7 ...emptied Himself, taking the form of a bond-servant, and being made in the likeness of men.
Phi 2:8 Being found in appearance as a man, He humbled Himself by becoming obedient to the point of death, even death on a cross.
Jesus emptied Himself of His divine attributes, including immortality (IMO).
So I do not believe that God, (meaning the Godhead) was "alive and dead at the same time". I do believe that a member of the Godhead died while the others were alive. This is made possible by the incarnation.
I even believe that Jesus, though He is God, was able to be "separated from the Father", by dying a real death (by whatever definition you want).
This is indeed a mysterious thing. But is it illogical in light of the trinity and the incarnation? I don't see how.
You cannot simply say "God cannot die. Jesus died. Therefore He is not God" (or in your case the trinity is illogical). It does not take into account the trinity, and incarnation/kenosis doctrines. Remeber, trinitarians don't think that any one member of the trinity makes up the totality of God, but all three do. In order for God to be "alive and dead at the same time" the Father, Son, and Holy Spirit would all have to be "alive and dead at the same time", which is of course impossible.
If (and I realize that it may be a big if in your case), the trinity is correct, then my position is not breaking any rule of logic that I can see.
Though I am not articulating it as well as I should, I think that after a bit of discussion, I can hash out more clearly why it's not illogical. Again, it is possible that it's unbiblical. But not illogical (which is what we're discussing).
God bless,