How to Define Theoretical Predicates: The Solution
Long ago, I worried about how the Ramsey-Carnap-Lewis account of theoretical terms could be applied to predicates. I noticed two reasons why Lewis's proposal to just turn the predicates into singular terms ("Instead of [...] 'F ---', for instance, we can use '--- has F-hood'", HTDTT p.80) is no good: first, it entails that completely false theories, say about witches or gods, leave their theoretical predicates undefined, whereas in fact those predicates are clearly empty (and thus defined); second, the proposal can turn consistent theories into inconsistent theories. This second problem can be generalized: For many predicates, there is no corresponding property that could be denoted by a singular term. Exactly which predicates these are depends on one's theory of properties, but "having parts", "being self-identical", "being a set" and "being a property" are generally good candidates, besides of course "not instantiating oneself".
The solution is what I mentioned at the end of my old post but didn't believe at the time because I wasn't yet familiar enough with Plural Quantification: it is to lift the entire account up to second-order level. As follows.
Suppose we're interested in truths involving predicate F. We assemble the relevant theory:
all F are G; many F are made of H; some are located on this planet; etc.
Then, to express the theoretical role, we do not turn the predicate into a name and ramsify that out, but simply use second-order, plural quantification. The Ramsey sentence becomes:
for some xx, all xx are G, many of the xx are made of H; some of the xx are located on this planet; etc.
This doesn't give us a singular condition for an entity -- a property, as it were --, but a plural condition for some entities. Hence at the second step, we check if any entities (plural!) in our more fundamental ontology collectively satisfy the condition. If so, we can identify those entities with the Fs. (If, as in the case of witches and Gods, nothing satisfies the plural condition, we'll have to conclude that the Fs don't exist.)
Lewis makes use of this pluralized version of the Canberra Plan all the time. Like here:
We have the word "property", introduced by way of a varied repertory of ordinary and philosophical uses. The word has thereby become associated with a role in our commonsensical thought and in a variety of philosophical theories. To deserve the name of "property" is to be suited to play the right theoretical role. [...] But it is wrong to speak of the role associated with the word 'property', as if it were fully and uncontroversially settled. [...] The question worth asking is: which entities, if any, among those we should believe in, can occupy which versions of the property role? My answer is, in part, that sets of possibilia are entities we should believe in which are just right for one version of the property role. (Plurality, pp.55f.)
And here:
I am not opposed to states of affairs, ways things might be, possibilities, propositions, or structures. I believe in all those things. That is to say, I believe in entities that deserve the names because they are well suited to play the roles. The entities I put forward are the same in every case: sets of worlds. (Plurality, p.185)
Lewis here doesn't identify property-hood, or proposition-hood with anything. Both of these entities don't even exist in his ontology. What he does is identify the properties (plural) and the propositions with sets (plural) of possibilia.