Simon Blackburn claims that moral realists cannot explain "the ban on mixed worlds." Let's say that moral properties supervene on natural properties (or predicates if you like) if there can be no difference in the former without a difference in the latter. Then, we can formulate the following three claims:
(S) □((∃x)(Ax & B∗x) → (∀x)(B*x → Ax))
(N) □(∀x)(Bx → Ax)
(P) ♢(∃x)(B*x & ¬Ax)
According to (S[upervenience]), if having moral property B results in having moral property A ('*' when coupled to B, B* denotes a complete base description of everything that could be relevant to A), then anything which has B has A. (N[eccessity]) tells us that in every possible world, if something has B, then it has A. And, (P[ossibility]) says there is a possible world in which something has B, B results in a moral property, but it is not-A.
Given that (N) is simply not true and (S) and (P) are, then we have worlds where:
- objects have B/A combinations
- objects B/not-A combinations
However, we have no worlds where:
- objects have B/A and objects have B/not-A
Question: Why do realists have a difficulty explaining mixed worlds? Why do projectivists have an easier time explaining them?