In explaining away non-naturalists' persisting disagreement with him, Streumer suggests that they tend to "conflate properties with the meanings of the predicates that ascribe these properties" (28). But I think that non-naturalists are instead appreciating a deep and important connection that holds between properties and possible subject matters. Even Kripke/Putnam cases of the necessary a posteriori involve claims that are informative because they (indirectly) relate distinct properties:
[W]henever identity claims are informative, this is because they tell us that two distinct properties are co-instantiated by a single object. The cognitive significance of ‘water’, say, may be given by a certain complex functional property: roughly, being the clear drinkable liquid found in lakes and rivers around here. This differs from the cognitive significance of ‘H2O’, which is instead given by a certain chemical property. The claim ‘water is H2O’ is informative rather than trivial because it relates these two distinct properties. This is possible because the concept water is “gappy”: it refers to whatever actually fills the associated functional role. This functional role could, for all we know a priori, be filled by all manner of chemical substances. Hence it is informative to learn that H2O is the particular chemical property of the watery stuff.
The fundamental problem with normative naturalism is thus that it doesn't have enough properties to ground any new claims (or cognitive significance) beyond what could already be expressed in non-normative terms. But Open Question and Knowledge arguments establish that normative claims cannot be reduced to non-normative claims. When we call an act "right", we are saying something other than that it is "D1 or D2 or ..." But then what are we saying -- what new information are we conveying -- if not that the act possesses a distinctive further property (i.e., that of being right)?
Informative claims need to bottom-out in informative property ascriptions somehow, and it just doesn't seem that the naturalist has the resources to do this in the case of normative claims. (See also: 'Non-Physical Questions', for a straightforward application of this principle to refute physicalism in the philosophy of mind.)