You assume that God’s character could have been anything whatsoever. This is not how theists understand God. We understand God to be a necessary being. His character is what it is necessarily. It could not have been otherwise.Thank you. That is a reasonable summary of the dilemma.
This is in fact a fairly well-known apologetic argument that has been made a number of times before. For example, What is the Euthyphro dilemma? Is it a challenge for Christians? | carm.org and What is the solution to Euthyphro's Dilemma?
The response to this apologetic tactic is known as well. Quite simply, you have not resolved Euthyphro's Dilemma, you have simply restated it in a slightly different form. The skeptic can still ask the same question:
If "God is goodness itself", and morality springs from his intrinsic nature, what does that mean?
Can God's character be pronounced to be good because it can be measured by some external standard? If so, we do not need God, we can simply consult this standard.
But if not, then goodness simply means "what God is." Whatever God was would be good. If God was cruel and unjust, cruelty and injustice would be good. If God was capricious and vindictive, then caprice and vindictiveness would be good.
As you correctly put it above, this "suggests that goodness is arbitrary. God says that murder is wrong. But he could have just as well said that murder is good. This makes morality fairly meaningless."
This is still the case if you say that God's character is the foundation of goodness; just a small change to say "God's character makes murder wrong. But his character could just as easily have made murder right." The obvious question is, how do you know that God's character is indeed good? By what yardstick do you measure it? Itself? That would be circular reasoning.
And so Euthyphro's Dilemma continues...
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