Evangelical Outpost points us to a blog called Prosthesis where the question whether God can do the logically impossible is entertained. The answer, apparently, is yes and no.
All this is based upon the assumption that God created the laws of logic which are therefore contingent upon Him and which He can therefore transcend. However, were He to do so, we could not discern what was happening since we cannot conceptualize outside the framework provided by those laws and would be unaware of any phenomena occurring extra-logically. So God can transcend logic, but we couldn't know He was doing it.
What if the laws of logic are not discretionary, however, but rather are ingrained, like Love and Goodness, into the very nature of God? What if logical laws, like the moral law, are infused into the created order such that any world God creates would be governed by the same logical principles? If God suffuses the creation with his own nature then the laws of logic could not be other than what they are, nor can God violate them without somehow transcending His own nature. It might, then, be quite impossible for God to create a square circle or to create a world in which it would be true to say that God did not create it.
Theologians who maintain that logic is a contingent set of rules created by God generally wish to hold on to a very high view of omnipotence. They wish to be able to say that there are no limits of any kind constraining what God can do. This may be the case, but if God is the sort of Being who can do anything at all theology would seem to be pointless. After all, how can one talk about whether God exists when one of His attributes would be that He can both be and not be at the same time?