Euthyphro is one of Plato's dialogues which presents us with a meta-ethical dilemma that has been addressed throughout philosophical and theological history (meta-ethics being the study of the ground or foundation of ethics). In this debate, Socrates asks Euthyphro why God assigned the particular moral laws he did, such as to not commit murder or adultery. The problem this creates is that if God assigned these laws because they are good in and of themselves, then there is a "higher" reality than God, and God commands them because he must align himself with this reality just as much as we do. God, in other words, is not absolute; neither the ground of morality nor of reality. But if we say that these laws are not good in and of themselves, then these laws are simply arbitrary, and God could have made them differently. The "good" would have been to commit murder and adultery if God said so. In this case, God is not intrinsically good because the appellation of "good" is entirely arbitrary (this is the position that Euthyphro takes in the debate).
Traditionally, Christianity has split the horns of this dilemma. Moral laws are intrinsically good, not arbitrary. But their goodness is not derived from something outside of God; rather, they are derived from God's own intrinsically good nature. The ground of morality, in other words, is identical to the ground of reality. The error of the Euthyphro dilemma is that it tries to put the two concepts -- the goodness of certain acts and God's command of them -- into a cause-and-effect relationship with each other. If the goodness of these acts is what causes God to command them, then they are higher than he. But if his command of them is what makes them good, they are arbitrary. Neither, however, is the case: these two concepts are both effects from a common cause, namely, God's own nature.
Now, as far as I can tell, this option would be available to any general theistic position. But in my (admittedly limited) knowledge of Islam, Muslim theologians have not availed themselves of this resolution. A core doctrine of Islam is that God is completely transcendent; that is, he transcends even our moral and rational categories. God may give moral commandments, but ultimately, they are not expressions of his nature -- if they were, then he would not transcend them. Since they have their origin in his command of them, but not in his nature, they could have been different, and are therefore arbitrary, as Euthyphro thought.
Thus, in the Qur'an, God is represented as capricious. For example in the battle of Badr, God had told Muhammad (indirectly -- since God is completely transcendent there is no direct communication between him and humanity in Islam) that he would outnumber his enemies. When Muhammad's army got there, they found to the contrary that the enemy outnumbered them; but there was no way to avoid the battle at that point, and the Muslims ended up winning anyway. Later, when Muhammad asked why God told him that they would outnumber the enemy at Badr when they didn't, the response in the Sura of the Spoils of War is essentially, "If God had told you the truth, you wouldn't have gone". Thus, God lied to Muhammad in order to accomplish his goals (which makes me wonder what else he lied to Muhammad about).
Or take the Qur'an's explanation of Jesus' crucifixion, which Muslims deny: God made it seem that Jesus was crucified, but he really wasn't. Islamic tradition explains this by claiming that God put the image of Jesus on someone else (sometimes thought to be Judas Iscariot), and this person was crucified instead of Jesus. In any case, God made things appear differently than they really are in order to accomplish his objectives. He tricked people so he could get what he wanted.
In contrast to this, the God of the Bible cannot lie; not that he merely does not or will not, but he cannot. Unlike Islam, in Christianity morality and rationality are two things that put us in touch with God, because of their origin in his nature. God does not transcend morality and rationality, he is their very ground. That's part of what it means to say that we are created in his image -- there is a connection between humanity and God, even after the fall. We are created in his image because we have the capacity for morality and rationality. There's more to it than that of course, but that's at least some of it.
So it seems that Islam has pitched its tent with Euthyphro, by accepting that the moral laws are good because God commands them, and that they are thus arbitrary. Now -- to get even more speculative -- when I think about this, I wonder whether it has any connection to the bloody nature of Islamic history, and with Islamic terrorism today. Of course, other religions have had their share of violence as well, but Islam seems to stand out in this regard, despite what the popular media says. Committing an evil act in the name of Christianity can only be done by essentially contradicting the central commandment of Christianity to love God and to love other people. But if morality is not directly linked to the ground of reality, it can be reasonably ignored as long as one is doing so in the name of the ground of reality. If murder is not intrinsically bad, then if you can serve God by committing murder, there's really no reason why you shouldn't.
(cross-posted at Quodlibeta)