Lewis' Account of Predication
What, in general, does it mean that something A satisfies a predicate 'F'? Traditionally, there are three candidates:
1) 'A is F' means that A is F. That' all. Simple predications can't be analysed.
2) 'A is F' means that A instantiates the property F. Except in some special cases, in particular the case where 'F' is 'instantiates'.
3) 'A is F' always means that A instantiates the property F.
It is not entirely obvious how to locate Lewis here. In some places, when discussing Armstrong's request for analyses (or truthmakers) for predication, he sounds like he favours (1): "the statement that A has F is true because A has F. It's so because it's so. It just is." ("A world of truthmakers", p.219 in Papers)
But note the strange copula "has". I think what Lewis really means is (2): "To have a property is to be a member of the class" ("New work", p.10 in Papers), but this cannot be further analysed. Something A just is a member of the class F. That's all.
In "Tensing the Copula", he gives a reason why (besides Bradley's Regress): There is no such thing as the membership relation. So 'A is a member of F' can't be analysed as '(A,F) is a member of membership'. There is no membership relation because some things are members of proper classes. Hence the membership relation would have to contain (sets of sets containing) proper classes, but that's impossible.
Lewis doesn't mention that, for the same reason, there is also no such thing as parthood, no identity, no such property as being a class, no property of having parts. All these classes would have to contain proper classes. (Moreover, mixed fusions of classes and individuals, like proper classes, don't have any properties at all, not even the property of being a mixed fusion, because they fail to be members of anything, see Parts of Classes, p.8.) So how do we analyse 'A is a class' or 'A is a mixed fusion' or 'A is part of B'? It seems that we need quite a lot of special cases.
Things get even more complicated because in fact, Lewis does provide an analysis of 'A is a member of F'. Or rather, two analyses. In Parts of Classes, before he turned structuralist, the analysis goes 'the singleton of A is part of F'. After the structuralist turn, the analysis gets much more complicated, something like: 'for all things XX satisfying conditions R1...Rn and S1...Sn, there is is an X such that X is one of those XX and its 1-part is A and its 2-part is a part of the XX-F', where R1...Rn are conditions for things to mereologically model ordered pairs (whose elements are their '1-part' and '2-part'), S1...Sn are Lewis' conditions for singleton relations, the XX-F is the thing that is determined as the F-class by the singleton relation (modeled by) XX, and the main quantifier is plural (cf. "Mathematics is Megethology"). As it happens, all predicates in this analysis are logical (including plurals) and mereological, so here is another reason why those predicates can't be analysed in terms of membership: This would be circular.
Actually, as far as I can see, all the special cases now turn out to be logical or mereological.
This reduction of ideology of course burdens us with ontology. For every non-special predicate we need a class of its instances. Lewis thinks that we need this ontology anyway, because it is the ontology of good old mathematics. I'm not sure if that is true: To my knowledge, ZFC is sufficient for most of mathematics, and ZFC contains neither proper classes nor classes of books or dragons. Anyway, there are other reasons why we need that ontology, for example to analyse quantifications over properties.
In this respect, it is unfortunate to have so many special cases: "Parthood is transitive, set membership isn't". Since parthood and set membership don't exist, Lewis has to find a paraphrase for all sentences like these. And, as he himself notes (in an only slightly different context), "even if such paraphrases sometimes exist -- even if they always exist, which seems unlikely -- they work piecemeal and frustrate any systematic approach to semantics" ("New Work", p.16).
Back to the analysis of predication. In "Tensing the Copula" (p.9), Lewis describes how his structuralism affects talk about properties: "we can simulate quantification over properties by embedding a genuine quantifier over classes within a simulated [plural] quantifier over [models of relations determining] hierarchies. Further, it turns out that something is a member of a given hierarchy's property purple just in case it is one of the purple things." Well yes, this turns out. It turns out because it had been put in before: 'the F-class determined by the singleton relation XX' means 'the thing which in the XX-hierarchy has all Fs as members'.
In the next few paragraphs, Lewis discusses some consequences of taking this as his analysis of predication: 'A is F' means that A is one of the Fs. Unfortunately, it is not clear to me from the text whether Lewis really does endorse the analysis (I mean as an analysis, not just as a true fact that somehow "turns out"). If he does, why this long roundabout in structuralist set theory? We don't need an elaborate set theory to tell us that something is purple iff it is one of the purple things. We knew this all along, and if we hadn't known it, set theory couldn't have told us.
What's nice about the plural analysis is that it can even handle some of the special cases: Something is a class iff it is one of the classes. What's less nice is that the analysis can't handle relations. We might try the Burgess-Hazen-Lewis technique: Something A is bigger than something B iff the mereological model of the pair (A,B) is one of those models that --- well, that are models of all those pairs the first of which which is bigger than the second. This doesn't sound like a good analysis. Lewis should better stick with the set theoretical one.
In conclusion, I think Lewis' theory of predication is a very unusual variant of candidate (2): 'A is F' means that for all things XX that satisfy R1...Rn and S1...Sn, there is is some X among those XX whose 1-part is A and whose 2-part is a part of the XX-F', except if 'F' is 'part', in which case 'A is F' means that A is F. Quite a mouthful, even in the abridged form!
There are a couple of reasons why I'm not satisfied with this interpretation. One is of course that such an enormously complicated analysis for simple predications just doesn't feel right. Another is that Lewis nowhere clearly says that this is his account. On the contrary, he spends pages stressing that not much can be said about how things instantiate properties, that they just do, that this is a "non-relational tie" etc. If by 'property' he means classes -- and what else could he mean? -- this amounts to saying that not much can be said about set membership. But this flatly contradicts the fact that he himself wrote an entire book and a long paper about its analysis. Puzzling.