Less than perfectly natural properties
My logfiles indicate that people are more interested in silly logic puzzles than in pointless remarks on footnotes in the metaphysical writings of David Lewis. Let's see if I can get my readership down to zero with this one.
Besides perfectly natural properties, Lewis also needs somewhat less natural properties in his philosophy of language and elsewhere. What determines how natural a property is? Lewis gives three different answers, in four different places, none of them longer than two sentences.
The answer that appears twice (On the Plurality of Worlds, p.61, and "Putnam's Paradox", p.66 in Papers in Metaphysics and Epistemology) is that naturalness is measured by complexity or length of definition in terms of perfectly natural properties:
Some few properties are perfectly natural. Others [...] are at least somewhat natural in a derivative way, to the extent that they can be reached by not-too-complicated chains of definability from the perfectly natural properties. (Plurality: 61)
I don't see how this is supposed to work. I'm not even sure that every less-than-perfectly natural property is definable in terms of perfectly natural ones. The problem is that some intuitively not very unnatural properties involve particular individuals, like 'inhabitant of Berlin' and, presumably, 'green' (since presumably a definition of 'green' would have to mention 'ordinary people' and 'ordinary conditions', which are people like us and conditions like those common hereabout).
Even granted that these properties are definable in terms of fundamental properties, such a definition of 'green' or 'breakfast' will be extremely complex and complicated. Much more so than the definition of 'has a mass of less than 207.336 or more than 10^13 - 2.58 grams. But the latter property should come out less natural than 'blue'. Similarly, 'blue or haevier than 107.336 grams' should come out less natural than 'blue or green', but length and complexity of definition don't deliver that.
The two other answers appear in "New Work for a Theory of Universals" (page references again to Papers in Metaphysics and Epistemology).
Once we are away from the perfectly natural properties, one thing that makes for naturalness of a property is that it is a property belonging exclusively to well-demarcated things. (49)
I think "one thing that makes for" is meant to indicate that this criterion is only one of several factors that determine degrees of naturalness. Without any hint of what the other factors might be this is hard to evaluate, but just by itself it doesn't work very well. For instance, it would classify 'cloud' as unnatural, and 'table or spoon or clock weighing less than 2.47 kg' as fairly natural.
The final, best, and chronologically first, answer is this one:
But also we would have less-than-perfectly natural properties, made so by families of suitably related universals. (13)
Lewis adds a footnote explaining that he has in mind a distinction that is based on something like Armstrongs account of resemblance among universals. This is the best answer just because there is a need of such an account anyway (to explain why Orange resembles Red more closely than Blue), and it is plausible that degrees of naturalness and resemblance of properties are closely related: A property is natural to the extent that its instances instantiate properties that resemble each other; properties resemble each other to the degree that their instances comprise a natural property (or class).
The problem this time is that in another paper ("Against Structural Universals"), Lewis has demolished Armstrong's account of resemblance between universals, which relied on structural universals having their constituents as parts (Universals and Scientific Realism, Vol. II, §16.2 and §22). Hence Armstrong later (Universals: An Opinionated Introduction, p.137): "Where I do see trouble for a Universals theory is the question of resemblance of universals", which, as he explains, might after all be taken as primitive.
Conclusion: Lewis' three answers all fail.