This is the modern version of the dilemma first posed by Socrates in Plato's "Euthyphro". Lewis explains why he is unable to accept either horn of the dilemma.
"If the first, if good is to be defined as what God commands, then the goodness of God Himself is emptied of meaning and the commands of an omnipotent fiend would have the same claim on us as those of the "righteous Lord." If the second, then we seem to be admitting a cosmic dyarchy, or even making God himself the mere executor of a law somehow external and antecedent to His own being. Both views are intolerable."
Be that as it may, rejecting the dilemma is not a valid option here. In essence, the question posed is whether God's morality is subjective or objective and there is no middle ground between these two alternatives. [2] However, Lewis' concern with the "objective" horn of the dilemma turns out to be unfounded. An objective law need not be external and antecedent to the being that follows it.
To see this, consider the economic law of supply and demand. The truth of this law depends on the actual interactions between people. The law did not precede the existence of people since it depends on what people do. But neither did anyone create the law. Instead it is a discovered generalization of people's behavior. That is, the law of supply and demand describes what people do or, to phrase it differently, people act according to the law of supply and demand. [3] The law is objective rather than subjective because it exists independently of anyone's opinions about it albeit, in this case, not independently of people's behavior.
Similarly the existence of the moral law for God is conditional on God's nature and therefore not antecedent or external to it. Given God's nature, it prescribes what God should and should not do. That is, God is subject to the moral law which he did not create but which nonetheless depends on his existence. Adding the premise that "God is (always) good", the moral law also describes what God does and does not do. As Lewis says elsewhere, "... the Divine Will is the obedient servant to the Divine Reason." [4]
Note: The solution to the dilemma involves other philosophical issues which I haven't explored here but which I take a generally Aristotelian approach to. These include the problem of universals (what does it mean for abstractions, such as the moral law, to exist?), the is-ought problem (how does the moral law derive from a being's nature?) and the argument from morality (does the moral law require God?).
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[1] "The Poison of Subjectivism" from "Christian Reflections" by C.S.Lewis.
[2] Lewis is aware that he doesn't have a satisfactory solution to the dilemma. He says, "But it is probably just here that our categories betray us. It would be idle, with our merely mortal resources, to attempt a positive correction of our categories - ambulavi in mirabilibus supra me." (Translation: I do exercise myself in great matters, in things too high for me.) However, despite his rejection of the dilemma in this instance, Lewis' general tenor in this essay and other writings is toward the "objective" horn.
[3] The law of supply and command is usually thought of as being true all else being equal. So, for example, when demand increases for a fixed supply of oil, government regulation could prevent the price from rising.
[4] Letter from C.S.Lewis to John Beversluis a few months before his death in 1963. From "C.S.Lewis and the Search for Rational Religion", p295, John Beversluis.
