 Intensionalism is controversial, so the fact that it conflicts with our generalizations linking naturalness to definability is not by itself a big problem for the generalizations. But once intensionalism is denied, there is an urgent need for a systematic theory of properties to replace it. For example: given that  (D)  is false for some \(F\) and \(G\), it would be good to know whether it is false for all of them. If so, are there more general principles which explain this? If not, what marks the difference between the \(F\), \(G\) pairs for which  (D)  is true and those for which it is false? Without a background theory to provide some guidance with these questions, generalizations linking partial definability to naturalness will be of little use: we can still know in many cases that \(p\) is partially definable from \(q\), by knowing of some particular operations that can be applied to \(q\) to get \(p\), but it will be unclear how to argue for any claim of strict partial definability. The task of coming up with a consistent non-intensionalist theory strong enough to answer answer such questions is by no means straightforward. The most popular fine-grained picture holds that properties (and propositions) are “structured”, where this is taken to imply that, for example, bearing \(R\) to \(a\) is identical to bearing \(S\) to \(b\) only if \(R\) and \(S\) are identical, and \(a\) and \(b\) are identical. Unfortunately, when principles of this sort are spelled out in the higher-order logic mentioned in §3.4 above, and generalized in the obvious way, they lead to an inconsistency: the “Russell-Myhill paradox”, first identified in Russell  1903  (appendix B), and subsequently rediscovered by Myhill  (1958):  see Goodman  2017  and Dorr  2018  (§6). But this certainly is not the end of the story, even within the higher order framework. For example, Dorr  (2016)  develops a consistent non-intensionalist theory that entails that  (D)  is false for every \(F\) and  \(G\).[63] 