The dot-matrix test for naturalness
In section 6 of "Redefining 'Intrinsic'" (Philosophy and Phenomenological Research 62, 2001), Lewis introduces an interesting test for comparative naturalness of properties. The test is based on two-dimensional dot-matrix pictures, where distance along the horizontal dimension measures intrinsic dissimilarity, and distance along the vertical dimension extrinsic dissimilarity. Roughly (p.385), a natural property demarcates a regular region in the dot-matrix. Less roughly (p.391), two aspects of the region are important for naturalness: spread and scatter.
If a property region is horizontally spread, some of its instances are intrinsically quite different from others. Similarly, vertical spread indicated that some instances differ widely in extrinsic respect. Hence spread detracts from naturalness.
If a property region is horizontally scattered, many things that lack the property resemble things that have the property intrinsically at least as closely as any other thing that has the property. Similarly for vertical spread with intrinsic resemblance replaced by extrinsic resemblance. Scatter also detracts from naturalness. (This is not quite right. See below.)
Note that, on this account, any region that includes some but not all dots in a column or row is at least slightly 'scattered', namely at its borders. This is why, in general, horizontal scatter models the extent to which a property divides extrinsic duplicates, and vertical scatter the extent to which a proprty divides intrinsic duplicates. A purely extrinsic property like 'lonely' only includes full rows, a purely intrinsic property only full columns.
Lewis indicates (still on p.391) that vertical scatter always outweighs lack of vertical spread. Let intrinsic completion be an operation on properties which for every property F returns the extension F+ of F that includes all intrinsic duplicates of all instances of F. (So the intrinsic completion of a purely extrinsic property is the universal property.) What Lewis indicates is that for any property, its intrinsic completion is at least as natural as the property itself.
In sum, a property F is natural to extent that the following conditions are met, where (2) is far more important than (3) (more generally, I guess Lewis would hold that each of (1) and (2) outweighs each of (3) and (4)):
- all Fs resemble each other intrinsically
- few non-Fs intrinsically resemble some F at least as closely as any other F does
- all Fs resemble each other extrinsically
- few non-Fs extrinsically resemble some F at least as closely as any other F does
I don't know if Lewis intends these conditions to provide a reductive analysis of naturalness. Such an analysis would presuppose intrinsic and extrinsic resemblance. It would be a kind of 'resemblance nominalism', which Lewis usually regards as a minor variant of taking 'natural' as primitive (e.g. "New Work for a Theory of Universals", p.15, Fn.9 and "Against Structural Universals", p.79 -- page references to Papers in Metaphysics and Epistemology). Judging from Lewis' remarks at the end of section 6 (back in "Redefining intrinsic"), he isn't interested in reductive analyses here, but rather in analytic cross-connections between 'intrinsic', 'resemblance' and 'natural'.
In fact, there might be a good reason to think that his conditions for naturalness are conditions for a different kind of naturalness than the one used in most of his other papers. The reason is that it would be odd to classify many properties as more natural than perfectly natural. But this is precisely what his conditions do, because they rely on overall similarity rather than similarity in one particular respect. If, for example, specific masses are perfectly natural, then all the things weighing exactly 207g comprise a perfectly natural class. But those things differ widely in both intrinsic and extrinsic respect: Some of them are chunks of rock, others are lonely spheres, others tiny dragons.
It would be interesting to explore the connections between these kinds of naturalness. Maybe the whole problem is only superficial, due to a misleading name, and would disappear if we called the perfectly natural properties 'basic natural properties'? But maybe not, because then we would not only lack a definition of 'natural' in terms of 'perfectly natural' (the problems I mentioned recently would only get worse), we would also lack a converse definition of 'perfectly natural' in terms of 'natural', so that we couldn't even resort to taking gradual naturalness as primitive.
Another interesting task would be to explore what kind of naturalness does what part of the work Lewis offers for universals. For example, on the above criteria, extrinsic properties would all turn out to be fairly unnatural, making it somewhat puzzling how we manage to have so many words for them.
For now, I would be satisfied if I could properly state Lewis' criteria. The ones I gave above aren't quite his. To see that, we have to look at his applications. For instance, for his definition of 'intrinsic' to come out right, R = 'being rocklike and not seamlessly embedded in any more inclusive rocklike thing' should be no more natural than RL = 'R and lonely'. We have to check whether RL satisfies (1)-(4) at least as well as R.
(1): Given the plausible assumption that every R is an intrinsic duplicate of some RL and vice versa, the RLs resemble each other intrinsically just as closely as the Rs. Tie.
(2): Some of the Non-Rs intrinsically resemble some of the Rs quite closely, namely those rocklike things that are part of bigger rocklicke things. All these Non-Rs are also Non-RLs. Therefore, again by the plausible recombination assumption, these Non-Rs also resemble some the RLs quite closely. So far, tie. But now think of the Rs that are Non-RLs. They too closely resemble the RLs in intrinsic nature. In sum, the Rs fare better than the RLs on condition (2).
(3): The Rs widely differ extrinsically. The RLs don't, they are all extrinsic duplicates. So here the RLs fare much better than the Rs.
(4): Some of the Non-Rs extrinsically resemble some of the Rs quite closely (because R is fairly intrinsic). All these Non-Rs are also Non-RLs and therefore extrinsically resemble some of the Rs just as closely. The remaining Non-RLs, that is, the Rs that are not RLs, can't resemble the RLs perfectly in extrinsic respect, because they aren't lonely. On the other hand, the RLs are all extrinsic duplicates. Therefore none of those remaining Non-RLs extrinsically resembles any RL more closely than any other RL. Tie.
In sum, R does better on (2) and RL on (3). But remember that (2) outweights (3). So overall, R comes out more natural than RL. This contradicts the result Lewis obtains by his dot-matrix picture (P7). And indeed, in (P7), the RLs -- the dots below the dashed line -- aren't scattered at all. Where's the mistake? It seems that (2) is not quite the rule Lewis uses. To determine the horizontal scatter of RL he doesn't compare its instances to all the Non-RLs, but only to the lonely Non-RLs. His rule seems to be this one:
2'. few of the non-Fs that are extrinsic duplicates of some F intrinsically resemble some F at least as closely as any extrinsic duplicates of that (latter) F which is also an F.
But to me this looks like an odd rule. Why this restriction to extrinisc duplicates? Does this, rather than (2), really capture an intuitive concept of naturalness? I can see that in a way a class is unnatural if many of its non-members are exact intrinsic duplicates (or near-duplicates) of its members. But why do these duplicates also have to be exact extrinsic duplicates of some other member of the class? And even if we replace (2) by (2'), don't we then need a further rule for Lewis' claim (p.391) that it is unnatural for a property to divide intrinsic duplicates, which would again create trouble for RL?
I'm pretty sure that these questions can be answered. For example, I think an adjusted (4) might provide the mentioned further rule without any trouble for RL. Maybe I'll think about answers sometime later. For today I've enough stared at dot-matrix pictures.