(K) Shakespeare and the writer of Hamlet are one and the same person.
(L) Shakespeare wrote Hamlet.
In the referential sense, (J) and (K) state the same fact, since both claims refer to Shakespeare and tell us that Shakespeare has the property of being numerically identical to himself. In the informational sense, however, (J) and (K) state different facts. Unlike (J), (K) refers to Shakespeare in a way that also tells us that Shakespeare wrote Hamlet. In the informational sense, it is (K) and (L) that state the same fact.
Parfit then goes on to consider the moral naturalist's thesis:
(Q) moral rightness is the same as some particular natural property
(R) this natural property is the same as this natural property.
Parfit writes: “To defend their claim that (Q) states an important truth, these naturalists must therefore claim that, in the relevant, informational sense (Q) and (R) state different facts.” This then provides the basis for his Fact Stating Argument.
In rough gist, the argument challenges the naturalist to explain what natural fact is informationally equivalent to (Q) -- or to any other true positive normative claim like "Torturing babies is wrong." If there is no eligible natural fact available, it looks like the naturalist is committed to non-natural facts after all. If, on the other hand, the naturalist proposes a candidate natural fact, it looks like they're giving up the "non-analytic" part of their view: the information provided in their candidate natural fact would pin down an analytical reduction of 'moral rightness'. So, non-analytical naturalism is untenable: under pressure it must transform into either analytical naturalism, or non-naturalism.
Can non-analytical naturalists avoid this conclusion? The key move seems to be in the assumption of a correspondence between information and facts. Informative claims that referentially correspond to trivial facts must also -- informationally -- correspond to some other, less-trivial fact. This is, in effect, the fact that we learn when we acquire the information in question. This seems like an innocent enough assumption. But if we remember the naturalist's commitment to thinking that all facts are natural facts, then we find an immediate conflict between what we might call this information-fact correspondence thesis, on the one hand, and the core commitments of non-analytic naturalism (irreducibly normative claims / concepts / information without any irreducibly normative facts) on the other hand. Is there any principled case to be made for abandoning the former?
It seems to me that the information-fact correspondence thesis is highly plausible, and provides an important method for disciplining our metaphysical practice. (It's too easy to claim to be a naturalist these days!) This principle places serious constraints on what phenomena naturalists can feasibly hope to accommodate -- which seems only appropriate for what is, after all, a very metaphysically constraining thesis! A similar argument could easily be constructed to cast doubt on non-analytical ("type B") materialism in the philosophy of mind, for example.