Divine Command Theory
Divine command theory states that morality is determined by the will or command of God.
** If God wills x or y, then it must be right, for there is no rule over God to which he is himself a subject. **
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The Euthyphro dilemma is put in modern form by Leibniz:
"It is generally agreed that whatever God wills is good and just. But there remains the question whether it is good and just because God wills it or whether God wills it because it is good and just; in other words, whether justice and goodness are arbitrary or whether they belong to the necessary and eternal truths about the nature of things"
As Leibniz explains, the command view of morality is that it is arbitrary - perhaps it is identical to the will or commands of God - whatever those commands might be.
However, pace the DCT, we have no reason to believe that God is a moral authority except on the basis of the system of morality he has commanded. There is no reason to consider commands to be superior in any valuable sense except for the tautological fact that they have commanded that their commands are right. Note that if we say that God is good, therefore we do have reason to follow his commands, we are arguing against divine command theory. In divine command theory, God is good because he gets to define and command what is and is not good.
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What does the Euthyphro dilemma argument show about the relationship between morality and religion?
Any time someone believes in the command theory or in the alternative of non-arbitrary morality, there is always some human that is claiming authority of a kind, that is, saying "I know that, for this or that reason, we can judge, with good authority, that such and such a theory is the right one." It would be mere tautology to say that the authority by which he knows his theory correct is itself some moral authority or God. And if one says "God reveals truth to us in revelation", then that only begs the question who gets to judge that an experience was a revelation from God - who is the real authority on Earth that is actually making moral judgements and deciding what the moral reality is?
The dilemma makes clear the way in which we as humans are confused and circular about our uses of the concept of authority, trying to pin one type of authority onto another, until we have gone full circle without ever having to explain it. It shows that religion and morality are both as human as ever, for it is the humans that are trying to work out where authority is, whether it is in the divine, or elsewhere, and our judgements are always going to be limited and human, and ultimately depend upon humanity for the source of authority of our judgements.
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Is there any other important way in which morality depends on religion?
The way we have been informed about moral values is connected to the history of the development of moral values in history, and this has been a process that is closely connected to religious culture, as well as class and other social phenomenon. However, this history and learning is arguably arbitrary and contingent too - we believe that something is moral because a system of values has been handed to us by our contemporary cultural and religious situation.
Value is always judged by beings with reasons for making particular judgements or commands (for anything else would reasonably be ignored by them, and would lack authority). I do not say that it is created by such beings. Therefore, morality depends upon religion insofar as religion can show itself to be reasonable in its views that command x or y or z is a command of a God that knows better than us.