Here's one way of bringing out the problem.
Assuming there are universals, then:
- 1. Either (a) universals exist extramentally or (b) they do not exist extramentally.
- 2. If (b), then nominalism or conceptualism, QED.
- 3. If (a), then either (i) they only exist in the objects that have them, or (ii) they sometimes exist outside of the objects that have them.
- 4. If (i), then that is Armstrong's view.
- 5. If (ii) then that is Platonism.
- 6. So if (a) is true, then either Armstrong's view or Platonism is true, and so moderate realism either reduces to Armstrong's view or Platonism.
- 7. If Armstrong's view holds, then universals depend on the objects that have them, and therefore cease to exist if the objects do.
- 8. But if the universals cease to exist, then statements about non-existent things, like "dinosaurs are big creatures", do not require universals to be true, and so universals are superfluous.
- 9. So if Armstrong's view holds, then universals are superfluous.
That only leaves Platonism, and there are huge problems with Platonism.
To be fair, I am leaving much unsaid here, and I am not making any distinctions within "Platonism." But ultimately I think this is basically correct; the way Platonism is construed in modern times basically just is the view that there are non-mental, objective, necessarily existent, universals.