The Euthyphro dilemma seems to undercut moral realism

A version of the Euthyphro dilemma:

1. If morality is not grounded in a particular being’s nature, then one either must ground it externally in another being, or not ground it at all (i.e. be ultimately arbitrary and capricious).

2. It is not probable that morality is grounded in God’s nature.

3. Therefore it is probable that morality is grounded in a being external to God, or that morality is not grounded at all.

But:

4. It is more probable that morality is grounded in a necessary being than in a contingent being.

I can agree with with premise #1, but not premise #2. But if #1, #2 and #4 are all true, then:

5. Therefore it is probable that morality is not grounded in any being and is ultimately arbitrary and capricious.

#5 contradicts moral realism. Therefore if an atheist wishes to be a moral realist, he must reject at least premise #1 or #2 (refuting the Euthyphro dilemma, at least that version of it) or premise #4.

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