Disagreement and Relativism

One standard argument for moral relativism goes something like this: No-one agrees on what's morally true. If there were a truth, then they'd agree. Therefore, there is no universal moral truth.

I'm inclined to think that this is unsound. Still, if it were sound, what would follow?

The problem is that there's also no agreement on whether or not there is objective moral truth. There's wide disagreement on whether morality is relative or universal. But, so the argument goes, if there were a universal truth of the matter, we'd agree. Therefore, there is no universal fact about whether or not moral relativism is true or false.

So we also ought to be second-order relativists about whether or not first-order moral relativism is true. Therefore, the argument shows that first-order relativism cannot possibly be true.

(I've pursued the argument as one in favour of moral relativism. It's sometimes used to support an error theory instead. My remarks would apply analogously.)