Reading the introductory anthology that I'm teaching from, I noticed an article by Philip Quinn in which Quinn argues for the divine command theory, that morality depends on God and specifically God's commands. On this theory an action is right if and only if God commands that we do it, wrong if and only if God forbids us from doing it, and neutral if and only if God neither commands nor forbids it. The classic problem with this is presented as the Euthyphro dilemma. Does God command the good because it is good, or is it good because God commands it? Most people would think that God chooses those commands because they are the morally right commands; they do not become morally right in virtue of God commanding them. Leibniz argued that if the divine command theory were true, then God's commands would be essentially arbitrary, that God could have no reason for the commands he gives, and there would be no sense in which God is praiseworthy for the commands he gives since any commands would be as good as any others.
Quinn responds to this argument by claiming that if God is goodness itself (as in, this is the "is" of identity), then his actions cannot arbitrary, or non-praiseworthy, while morality would still depend on God (since everything depends on itself). I'm not sure this solves the problem, but even if it did, it seems to be a cure that's worse than the disease. What would it mean to say that God is identical to goodness itself? Goodness is an abstract entity, but God is a concrete particular. Goodness cannot causally interact with anything, but God does (if God created the universe). According to Christianity, one is supposed to worship God, and God is supposed to do things for his followers--reward their devotion by giving them eternal bliss in heaven--how is it possible for the abstraction of goodness itself to do any of these things? And what sense would it make to worship goodness itself? No matter how perfect the number 1 is, it wouldn't make any sense to worship it.
Now, does this assertion solve the problem? If God is identical to morality, then it's impossible for God to be anything other than morality, but it doesn't follow that the moral commands God gives would necessarily be the ones God has given. This is a case of scope ambiguity. God necessarily is morality, but it's not necessarily the case that the particular moral rules would necessarily be as they actually are if they depend on God. Given the dependence of morality on God, God's commands are still arbitrary because there would be no moral facts for God to consult by means of which to evaluate his own choices. God could only compare his choices to themselves or to his preferences (even though it makes no sense to say that goodness itself makes choices or has preferences). But, given the lack of an independent measure, God's choices must be based on no reason. Similarly, they must not be praiseworthy since they would still be good no matter what they were.
I think the only way to avoid these problems with the divine command theory is, literally, to label moral goodness "God". But in reality this is a kind of atheism. If I claim that God exists because I have named by dog "God", I have not established the existence of God. And if I say that there is no other God besides my dog, then I am an atheist. I would be denying the existence of the being with all the characteristics relevant to our conception of God.
It's equally clear that this "God = goodness" claim is just an argumentative dodge that even Quinn does not really accept. Almost immediately after making the nonsensical claim that God is goodness, he discusses the problem of God's order to Abraham to kill Isaac. But if God really were goodness, he could not order Abraham to do anything. The story would qualify as one giant category mistake. The idea that God just is goodness is so transparently nonsensical that no one could actually believe it; it's an argumentative ploy used and then discarded once it has effected the all-important response to problems with the divine command theory.