But we can tell him he did something wrong by the standard against randomly killing people. And we can act consistently with that standard by sanctioning him. In fact, it would be inconsistent for us to give him a pass.

I understand your point is that we can tell the killer that he has acted wrongly according to our standard (that one ought not randomly to kill people). But if people in general are bound only by their own standards, why should that matter to him? It seems to me I cannot provide him compelling grounds as to why he ought not to have done what he did, and that to punish him would be arbitrary. Sorry if I'm not getting it.

I'm not sure I'm following the argument here. I'm saying that all normativity is hypothetical. It sounds like you're arguing there is a categorical 'ought' for believing mathematical truths because it would be very strange to say we only 'ought' to believe 2 + 2 = 4 in reference to some goal. So if there are some categorical 'oughts,' there might be others.

Is it something like that?

This states the thought very clearly -thanks.

If so, then I would offer the goal of "in order to be logically consistent."

I acknowledge the business about the nature of the compulsion behind mathematical judgement is pretty opaque. What I had in mind is illustrated by this dialogue. As it shows, the problem gets right back to the compulsion to be logically consistent. It's possible this doesn't really engage your thoughts, though.

There are some who think moral oughts reduce to logical consistency, so we ought act in a certain way in order to be logically consistent. I don't have a good counter-argument to that, other than asking to examine such a theory and wondering how being able to point out a logical consistency is going to rein in people with desires that run counter to it any better than relativism can.

If the view is correct, then you can at least convince rational people that it is not rational to kill people. Isn't that an important result?