Supervenience-based arguments for moral naturalism have tended to apply only to moral properties, not to relations. One might have thought that they could easily be generalised so as to apply to relations as well. However, as I'll argue here, this may not be so easy.
In the case of properties, it can be shown that the following is a valid argument.
(1) The A-properties (strongly) supervene on the B-properties; i.e. for any possible objects x and y, if x and y instantiate the same B-properties, then x and y instantiate the same A-properties.
(2) The B-properties are closed under complementation and arbitrary intersections and unions.
(3) Every A-property is coextensive with a B-property.
(Of course, this doesn't yet establish that the A-properties are B-properties, but it's a step in that direction.)
Suppose, then, we try something similar with relations. The natural way to revise (1), (2), and (3) so that they apply to relations instead of properties seems to be as follows.
(1*) The A-relations (strongly) supervene on the B-relations; i.e. for any possible objects x and y, if x and y stand in the same B-relations to the same things, then x and y stand in the same A-relations to the same things.
(2*) The B-relations are closed under complementation and arbitrary intersections and unions.
(3*) Every A-relation is coextensive with a B-relation.
In this case, however, the argument is invalid. The following is a counter-instance. Let B contain only the four relations "is identical to", "is not identical to", "is either identical or not identical to", and "is both identical and not identical to". And let A contain the relation "is taller than". B is closed under Boolean operations, so (2*) is true. And (1*) is also true: the antecedent is false whenever x ≠ y (because then, e.g., x is identical to x but y is not identical to x), and the consequent is true whenever x = y. But (3*) is false: "taller than" is not coextensive with "is identical to", because nothing is taller than itself; nor is it coextensive with "is not identical to", because some things are not taller than some other things; and so on.Like