Semantics For Superplural Reference

In this section, I want to lay out Rayo’s (Rayo, 2006) semantics for a language contain- ing super and higher plural reference and quantification. The reason for this is that I will eventually borrow some ideas from it, turning it into a conjunctivist categorematic natural language semantics. But it will be nice to see the basic ideas in a slightly simpler format.

Rayo begins by noting the advantages of doing semantics without sets, such as, as we’ve seen above, the avoidance of set theoretical paradoxes, and proceeds to show how we give a semantics for a language containing plural and superplural expessions without having recourse to them. He begins with plural terms, like ’the elephants’. Rather than denoting the set of elephants, it refers to the elephants themselves. More formally, we have the following reference condition:

• ∃xx.(∀y « xx iff Elephant(y) & Refers(’the elephants’, xx)) Which he glosses:

there are some things–the xxs-such that: (a) for every y, y is one of the xxs iff y is an elephant, and (b) ’the elephants’ refers to the xxs

He also holds, interestingly, that predicates function in the same way. For example, the predicate ’is an elephant’ itself just stands for the elephants like the plural definite description, and not, as in standard theories, for a function or a set.

He then moves on to consider non-distributive predicates like ’are scattered’ or ’surround the castle’. He suggests we can think of them as super-plurally referring, with the following reference condition:

• ∃xxx.(∀yy(yy « xxx iff Scattered(yy)) & Refers(’are scattered’, xxx))

Superplural variables are notated withthreeletters, while plural variables are notated with two, and singular with one. To understand what this means, Rayo says something worth quoting at length:

The reference of " . . are scattered on the table", for example, is the super- plurality to which all and only pluralities scattered on the table belong. But it is important to be clear that apparently singular quantification over "super-pluralities" is a syntactic abbreviation for super-plural quantification over individuals. Super-plural quantification is not singular (first-order) quantification over a new kind of "item" ("super-plurality"), nor is it plural quantification over a new kind of "item"("plurality").Super-plural quantifica-

counterparts, it is quantification over individuals, which are the only kind of "item" there is. I would like to insist that thinking of super-plural quan- tification as an iterated form of plural quantification–plural quantification over pluralitie–would be a serious mistake. Plural quantification over plu- ralities can only make sense if pluralities are taken to be "items" of some kind or other. And a plurality is not an "item": apparently singular quan- tification over pluralities is a syntactic abbreviation for plural quantification over individuals. (Rayo, 2006, emphasis added)

I’d like to emphasise something about this. Rayo uses superplural reference to give an account of a somewhat tricky semantic phenomenon, namely non-distributive predi- cation. There are other ways, of course, one can go. To see this, consider a case where we have a predicate that typically, but not always, gets a collective reading:

(152) The men lifted a car, then pulled it to the finish line (in a bid to be crowned world’s strongest man).

Here the predicate ’lifted a car’ is functioning distributively, although typically it is non-distributive. And we can capture this fact in several ways. For one, we can say that the predicate is ambiguous. It has, semantically, two distinct meanings, one mapping an object to a truth value, and the other mapping some objects to a truth value. Or, we can say that the sentence contains a covert distributivity marker: a hidden ’each’ (or ’together’, as the case may be) which takes one from a collective to a distributive predicate, or vice versa. Or, finally, one can say that the crucial difference is that there’s two different types of reference which ’lifted three goodyear tires’ can exhibit.

The point I want to make is that we should think of the different types of refer- ence as another direction of freedom available to us, alongside the positing of more (and more complex) semantic values, or of hidden syntactic complexity, to account for various tricky semantic phenomenon. Another way to put this is that there are three places we can look when faced with a problematic sentence: to the sentence itself; to the world, or rather to the semantic values the world makes available; or to the speaker or thinker and their acts of reference. The first two options are well understood across a range of domains, such as, for example, quantifiers in object position, or donkey pro- nouns, or domain restriction. The latter is less so, and is the approach I (following Rayo) propose to take.

That said, let’s return to our discussion of Rayo. Now he thinks there are no super- plurally referring subject terms in English (his paper predates Nicholas and Linnebo’s), but if we grant he’s wrong about this, we can give the reference condition for our ’Those people, the Smiths, and we’ similar to the one we gave for ’are scattered’. For the sake of explicitness:

• ∃xxx.(∀yy(yy « xxx iff Those People(yy) or The Smiths(yy) or We(yy)) & Refers(’Those people, the Smiths, and we’, xxx))

We can then suggest truth conditions for simple sentences that look as so:

• The marbles are scattered iffJthe marblesK«Jare scatteredK

One final detail will be useful, as I think it helps bring out what exactly is going on when we appeal to these superlative plurals, and that’s the account of intransitive verbs, like ’loves’. Although Rayo doesn’t describe it in much detail, he notes two

ways to capture such things. Either one can say that it refers to a plurality of ordered pairs, or one can say that it superduperplurally refers. This is another sui generis type of reference. Unfortunately it’s more or less impossible to express what it involves without nominalising devices like those Rayo mentions. But if we heed his warning not to take them ontologically seriously, we can say that ’loves’ refers to a superduper- plurality, consisting of all those superpluralities which consist of pluralities consisting of lover and lovee, and (assuming singles are limiting case of plurals) lover.

We’re here making use of a technical trick of encoding the subject of the verb by having the subject occur twice: once in a plurality consisting of lover and lovee, and again in a limiting plurality consisting solely of the lover. Assuming Romeo loves Juliet, but not vice versa, and Charles loves Emma, but not vice versa, we could represent the semantic value as so:

| | |Romeo, Juliet|, |Romeo| | | |Charles, Emma|, |Charles| | |

Now the reader might start getting impatient at this stage. Even if you’re happy with plurals, and maybe with superplurals, you might think that supersuperplurals are be- yond the pale: they, one might think, have no natural analogue, and would require a complexity of attention that most humans probably aren’t capable of. However, I think this worry is misplaced, as I think the fact that to account for intransitives may require going up the superlative hierarchy actually tells us a bit about how we should think of that hierarchy.

In particular, and returning to the point I made above, I think we should think of these different types of reference as just alternative ways of packaging the complexity any compositional semantics will need. After all, recall the standard treatment of in- transitive verbs, according to which they are <e,<e,t». While perhaps not as mentally taxing as superduperplural reference, nevertheless in saying that these are the denota- tions of intransitives we have gone a long way from intuition. Not only that, but some views don’t stop there. Consider in situ treatments of quantifiers in object position, which posit a type raising operation that takes the intransitive to the massively com- plicated <e,<e,t»,<e,t>. The only way standard theories manage to avoid such positing is by committing to not uncontroversial positions in syntax concerning movement and covert elements. The point is that semantics for natural language requires a certain degree of complexity. One can put it into the semantic types, or one can put it into the syntax, or one can, with superlative reference, put it into the relation of referring. But one must put it somewhere, and so complexity alone is not sufficient reason to reject approaches like the one we are considering.

Let me say one more thing about this. I think it’s instructive to compare what we’re doing when we assume these different types of reference, on the one hand, and when we posit higher order semantic types, on the other. Indeed, I think we should view the two activities as similar: we should think of these types of reference just as we think of the types of denotation. Some of them, like singular and plural and–in my view–superplural reference reflect an important fact about speaker’s and thinker’s un- derstanding of language. In that way, they are like the basic e type expressions assigned as the denotation of names, or perhaps the idea of intensions as functions from worlds to truth values (if one thinks that the core notion of content is that which distinguishes between possibities). On the other hand, there are higher order functions and higher level reference relations, which don’t reflect an aspect of speaker’s understanding, but are posited because they get the job done.

With that apologia for superplural reference out of the way, let’s see how to make use of it in the semantics I am developing.