The Many and the One, Ch. 3/ii
In Chapter 3, recall, Florio and Linnebo are discussing various familiar arguments against singularism, aiming to show that “the prospects for regimentation singularism are not nearly as bleak as many philosophers make them out to be”.
Now, it has always struck me that the most pressing challenge to singularism is actually that the story seems to fall apart when it moves from programmatic generalities and gets down to particulars. If the plan is, for example, to substitute a plural term referring to some Xs by a singular term referring to the set of those Xs, then how does work out in practice? How do we substitute for the associated predicate to preserve truth-values (without burying a plural in the new predicate)? Is the same treatment to apply to a plural term when it takes a distributive and collective predicate? The anti-singularist’s contention is that trying to substitute for plural terms ends up with (at best) ad hoc, piecemeal, treatments, and the resulting mess smacks of a degenerating programme (as Oliver and Smiley remark, having noted that e.g. Gerald Massey ends up giving four different treatments for four kinds of collective predicate, “where will it end?”). Now, this line of anti-singularist criticism might be more or less compelling: but in the nature of the case, that can’t be settled by a single counter-jab at one example. The devil will be in all the details — which is why I found F&L’s very brief treatment of what they call substitution arguments quite unsatisfactory.
But now let’s move on to consider another familiar anti-singularist line of argument that goes back to Boolos in his justly famous paper ‘To Be is to Be a Value of a Variable’. Here’s an edited version:
There are certain sentences that cannot be analyzed as expressing statements about sets in the manner suggested [i.e. replacing plural forms by talk about sets], e.g., “There are some sets that are self-identical, and every set that is not a member of itself is one of them.” That sentence says something trivially true; but the sentence “There is a set of sets that are self-identical, and every set that is not a member of itself is a member of this set,” which is supposed to make its meaning explicit, says something false.
F&L consider this sort of challenge to singularism in their §3.4.
One point to make (as F&L note) is that the argument here generalizes. Suppose we replace plural talk about some Xs with singular talk (not about the set of those objects) but by singular reference to some other kind of proxy object; and we correspondingly replace talk about some object o being one of the Xs by talk of o standing in the relation R to that proxy. Then it is easy to see that R can’t be universally reflexive if it is to do the intended work. So there will be some proxy objects such that any of the proxies which are not R to themselves is one of them. But this truth supposedly goes over to the claim that there is a proxy which is R to just those proxies which are not R to themselves. And it is a simple logical theorem that there can be no such thing.
But a second point worth making (which F&L don’t note) is that the quantificational structure of the Boolos sentence isn’t essential to the argument. Revert for ease of exposition to taking a singular term which refers to a set as the preferred substitution for a plural term, with membership as the R relation. Then consider the simple truth ‘{Jack, Jill} is one of the sets which are not members of themselves’. Supposedly, this is to be singularized as ‘{Jack, Jill} is a member of the set of sets which are not members of themselves’. Trouble!
OK. So how do F&L propose to blunt the force of this line of argument? They have two shots. First,
The paradox of plurality relies on the assumption that talk of proxies is available in [the language we are trying to regiment]. The lesson is that, if [the language to be regimented] can talk not only about pluralities but also about their proxies, then the regimentation validates unintended interactions of the sort just seen. To block the paradox, we would therefore have to prevent such problematic interactions. One possibility … is to refrain from making a fixed choice of proxies to be used in the analysis of all object languages. Instead, the singularist can let her choice of proxies depend on the particular object language she is asked to regiment. All she needs to do is to choose new proxies, not talked about by the given object language. In this way, the problematic interactions are avoided.
But hold on. I thought the the singularist was trying to give a regimented story about our language, using some suitably disciplined fragment of our language with enough singular terms but without the contended plurals? The proposal now seems to be that we escape paradox by introducing proxy terms new to our language, which we don’t already understand. Really? Usually singularists talk of sets, or mereological wholes, or aggregates, or whatever — but now, to avoid paradox, the idea is that we mustn’t talk of them but some new proxies, as yet undreamt of. It is difficult to see this as rescuing singularism as opposed to mystifying it.
F&L’s second shot is more interesting, and suggests instead that we discern “a variation in the range of the quantifiers involved in the paradoxical reasoning.” Thus, in the Boolos sentence “There is a set of sets that are self-identical, and every set that is not a member of itself is a member of this set” the proposal is that we take the ‘there is’ quantifier to range wider than the embedded ‘every set’ quantifier, and this will get us off the hook. On the face of it, however, this seems entirely ad hoc. Still, this sort of domain expansion is often put on the table when considering puzzles about absolute generality, and F&L announce they are going to return to discuss such issues in their Chapter 11. Fine. But so far, we have no hint about how the story is going to go.
And, more immediately, how do considerations about domain expansion engage with the not-overtly-quantified version of the Boolosian challenge that involves only a plural definite description. F&L just don’t say. They are, indeed, so far remarkably silent about plural terms and plural reference which, you might have supposed, would need to be a central topic in any discussion of plural logic.
We’ll have to wait to see what, if anything, F&L have to say later about e.g. plural descriptions. But for the moment, I think most readers will judge that the singularist’s prospects of escaping Boolos’s type of Russell-style paradox still look pretty bleak!
To be continued.
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