## Plural Reference

March 16, 2008

Colin has a post over at at inconsistent thoughts which I thought was quite interesting. The idea was to treat plural and singular reference as the same species of reference by simply relaxing the constraint that the reference relation must be functional. In the case of an ordinary singular variable or noun phrase there will be exactly one thing standing in the reference relation to it, but in the case of plurals there may be many.

I like this approach for several reasons. For one thing, it does away with the difference between the object, a, and the singleton ‘plurality’ containing only a. Of course, pluralities aren’t objects in their own right, so the only way to make sense of the difference here is that it is a difference in kind of reference involved. But prima facie, there is no difference in the case of a singular name, and a plural term denoting a “singleton”.

Colin then poses a problem for this kind of view: how does one then account for collective (non-distributive) plural predicates – predicates as found in “the students surrounded the building”. It’s not the case that each student surrounded the building, they did so collectively, so we can’t simply say that “The F’s surrounded the building” iff for each x such that Ref(“the F’s”, x), x surrounded the building. Nor is there any other obvious way to do it along these lines.

Maybe you could take the same line with predicates as with terms (in particular, with collective plural predicates.) So instead of a predicate standing in the reference relation to exactly one set of objects, as per usual in model theory, it could stand in it to many. So we treat the predicate reference relation as non-functional too. We could then say Ref2(“surrounded the building”, P) holds iff P is a set of things that surrounded the building, and generally a plural predicate is true of a term iff the predicate refers to the set of referents of the term. What do people think?

Update: Just an extra detail: nothing relies on the semantic value of a predicate being a set. We could treat predicates as plurally referring to their members. This way Ref2 would take plurality rather than a set as its second argument, making Ref2 an irreducibly plural predicate. I take it that having plural predication in the metalanguage is not problematic though.

1. This is really nice. I was trying to work something out by having the extension of a predicate be some subset of the powerset of the domain. But I couldn’t get it to work out right. You can get collective predication at the expense of distributive predication, and vice versa. But this approach has both. (I take it distributive predicates will be related to the singletons of all the objects that would be in an ordinary extension.)

2. Thanks! By the way, I don’t see what is wrong with your proposal yet. The distributive predicates will just be the downwards closed subsets of the powerset of the domain (i.e. $x \in F \wedge y \subseteq x \Rightarrow y \in F$.)

The only reason I don’t like this proposal is that it understands plural reference in terms of singular reference to sets. In most cases your idea will work, just not when we get to tricky cases, like “the ordinals” (and of course, the other reasons Boolos gave not to understand plurals like this…)

3. Andrew, I think you and I came to a similar solution here. I’m working on a weird framework that actually takes sets for elements in the domain, but as such I’ve just decided that the simplest way to handle the non-distributive predicates issue is to have the satisfaction of an atomic formula ‘Fa’ be defined in terms of the union of everything that ‘a’ denotes, which will help distinguish between cases of the many things being in the extension of ‘F’ and each one of individually being in the extension of ‘F’. Your approach might actually be better, I have to play around with it to see what are the advantages of doing it one way or the other.

4. Just to explain a bit more, here’s where I was running into trouble on your type of approach: I was still thinking in terms of ‘Fa’ being satisfied so long as each thing ‘a’ refers to is an element of the extension… so adding multiple extensions didn’t help get around the non-distributive predicate issue because on my way of thinking about it, so long as the many thing ‘a’ refers to belong to some extension of ‘F’ inevitably they each belong to some extension of ‘F’… your idea is a big improvement, requiring as you put it, that the “predicate refers to the set of referents of the term”.

5. Hi Colin, thanks for your comments. I see – so if you treat the sets as being first order objects in the domain you can explain plural predication in terms of singular predication on sets.

My problem with this approach is that plurals can’t take values from the full powerset of the domain on this account on pain of contradiction. And in some cases this is needed, for example the collective predicate “the F’s have a fusion” should be the entire powerset of the domain, since intuitively, any subset of the domain has a fusion. Of course, once you treat plural predicates as simply subsets of the full powerset of the domain you have Aarons suggestion, which is naturally isomorphic to mine if you assume everything is set sized. (The set of subsets P denotes on Aarons account, will be the set of those subsets of the domain P refers to on mine.)

6. I’m not entirely sure I get it. Why would we need the powerset of the domain to enter the picture in the first place? If I’ve got a domain of sets and I am saying that plural predicates are evaluated in terms of unions of denoted sets being in their extension, then the most we would need is that every collection of sets from the domain has a union in the domain. That’s easy as long as we start with a domain that is actually the powerset of some other non-empty set. Anyway, a lot of these details are quirks of the models I am constructing, so perhaps bizarre in the abstract.

7. Hmm, maybe I’ve misunderstood you. I thought your suggestion was that you started off with a domain, D, of objects – things that our first order variables range over. Included in D are elements that represent sets of objects from D. Then first order predicates and plural predicates receive a uniform treatment: they are subsets of D. The extension of first order predicates will be ordinary objects like you and me, but the extension of a plural predicate would include elements of D that represented sets from D.

Now the elements from D that represent subsets of D can’t exhaust every subset of D by Cantors theorem. So that means there can’t be a plural predicate that is true of any arbitrary subcollection of the domain, there aren’t enough elements in the domain to represent sets for that. But to my mind you want there to be plural predicates like this, e.g. “the F’s have a fusion” or less controversially, “the F’s are some things”. Does that sound right, or did I just misunderstand your proposal?

8. No, sorry I was unclear, I meant that I am working in models where the domain *just is* some sets. These (sets) are the objects over which our variables range. In particular, the domain is the powerset of some other non-empty set. So whatever a term denotes, those things are sets and predicates take sets of sets for their extension. What I was thinking previously was this: let Fa be true just in case the union of everything a denotes is in the extension of F. This works fine as long as every collection of sets in the domain has a union in the domain, which is guaranteed by the fact that it is a powerset. But on reflection I like your approach much better, so thanks 🙂