logic is not a natural law. logic is beyond nature. p & ~p is false no matter what the circumstances are, and no god can change that.
I disagree. There's no reason to believe that it be necessary that a being must follow the logic that we would suppose it created. It doesn't make sense, but then why should it? However, we have good reason to believe that in this case (Judeo-Christian God) He must follow the logical rules of His own universe. Why?
If God does not have to follow logic, then there is absolutely no reason, no excuse, that suffering should ever exist. Current religious text argues that God loves us all, wants us to be happy, prosper. So why isn't every single person happy? This is what is generally known as the "Problem of Evil" as an argument against God.
If God must follow the logical rules of his own universe, then reasons can be imagined why suffering must exist. Perhaps suffering is a necessity of existence in order to experience joy. Perhaps suffering is a way to learn to appreciate the happiness. Maybe a lot of suffering comes from free will (although this would only partially explain away suffering) There are several arguments that can be made that suffering is sometimes a logical and necessary consequence of existence.
However, if God does not need to follow the rules of His own universe, then there is absolutely no excuse for the Problem of Evil. There is absolutely no reason why anyone should ever suffer because any value or good or cause or anything that would make evil a part of the world could be removed, even if such an action would contradict basic notions we have about logic. Any part of God's plan that involved suffering of any part should be nonexistent if He can change the universe to His will even at the expense of logical requirements.
However, we know suffering exists. this means that either:
a) God does not exist (as an all-powerful, all-knowing, all-good creator of the universe)
b) God exists but He is not an all-good being, because he allows suffering despite when He could change it even at the expense of logical necessity but chooses not to.
or
c) God is subjected to laws of logic, so some evils are either necessary consequences of reality or required for a greater good in His plan. He has the power to do anything that is physically possible/impossible but not logically impossible.
Generally this means that if you believe in God you choose option C because it allows solutions to the Problem of Evil while maintaining a general notion of the meaning of the idea "God." Sure you can argue that God can act against logical possibility, but you'll be left with no options of allowing anything bad to happen. Ever.
I hope this was helpful.