Divine command, natural law and arbitrariness

People often levy an arbitrariness objection against divine command theory:

1. If God simply chooses what we ought, why did he choose to command kindness rather than cruelty?

It occurs to me that an advocate of theistic natural law probably cannot levy the arbitrariness object. For there is a structurally very similar question about theistic natural law:

2. If God simply chooses which natures to create, why did he choose to create beings with our basic physical structure and a nature that requires kindness rather than beings with our basic physical structure but a nature that requires cruelty?

It might be retorted that logical space does not contain a nature that specifies cruelty and yet the same basic physical structure as ours. This is plausible to me, but the main reason to doubt that there could be such a nature is some theistic story such as that all natures are ways of imitating God, and it is incompatible with divine goodness that he be imitable in such a cruel way. And this, in turn, is quite parallel to the standard divine command response to (1), that it is incompatible with divine goodness that he command cruelty to beings like us.

I think theistic natural law does have advantages over divine command theory. But a better resolution to the arbitrariness objection does not seem to be one of these advantages.

Arbitrariness and contingency

I’ve come to be impressed by the idea that where there is apparent arbitrariness, there is probably contingency in the vicinity.

The earth and the moon on average are 384400 km apart. This looks arbitrary. And here the fact itself is contingent.

Humans have two arms and two legs. This looks arbitrary. But it is actually a necessary truth. However there is contingency in the vicinity: it is a contingent fact that humans, rather than eight-armed intelligent animals, exist on earth.

Ethical obligations have apparent arbitrariness, too. For instance, we should prefer mercy to retribution. Here, there are two possibilities. First, perhaps it is contingent that we should prefer mercy to just retribution. The best story I know which makes that work out is Divine Command Theory: God commands us to prefer mercy to just retribution but could have commanded the opposite. Second, perhaps it is necessary that we should prefer mercy to retribution, because our nature requires it, but it is contingent that we rather than beings whose nature carries the opposite obligation exist.

Now here is where I start to get uncomfortable: mathematics. When I think about the vast number of possible combinations of axioms of set theory, far beyond where any intuitions apply, axioms that cannot be proved from the standard ZFC axioms (unless these are inconsistent), it’s all starting to look very arbitrary. This pushes me to one of three uncomfortable positions:

• anti-realism about set theory

• Hamkins’ set-theoretic multiverse

• contingent mathematical truth.