The Many and the One, Ch. 3/i

In Chapter 3, ‘The Refutation of Singularism?’, Florio and Linnebo get down to critical work. As the chapter’s title suggests, the topic is going to be various arguments that have been offered against singularist attempts to render plural discourse in the framework of standard logic. Can we really regiment sentences involving what appear to be plural terms denoting many things at once by using singular terms denoting just one thing — a set, or perhaps a mereological sum? F&L aim to show that “regimentation singularism is a more serious rival to regimentation pluralism than the [recent] literature suggests.”

What is the standard for assessing such formal regimentations? For F&L, as they say in §3.1, the key question is whether or not “singularist regimentations mischaracterize logical relations in the object language or mischaracterize the truth values of some sentences.” But that, presumably, can’t be quite the whole story. If, for example, the purported singularist regimentations turn out to be an unprincipled piecemeal jumble, with apparently logically similar sentences involving plural terms having to be regimented ad hoc, in significantly different ways, in order to preserve the singularist doctrine case by case, that will surely be a serious strike in favour of taking plurals at face value. Or so discutants in this debate have assumed, and F&L don’t give any reason for objecting.

An aside: Not that it matters, but F&L also claim in passing that

Regimentation can also serve the purpose of representing ontological commitments. The ontological commitments of statements of the object language are not always fully transparent. The translation might help clarify them. Following Donald Davidson, one might for instance regard certain kinds of predication as implicitly committed to events. As a result, one might be interested in a regimentation that, by quantifying explicitly over events, brings these commitments to light.

But careful! For Davidson, it is because we (supposedly) need to discern quantificational structure in regimenting action sentences to reflect their inferential properties that we need to recognize an ontology of events for the quantifiers to range over. So while, as F&L say, we want formal regimentation to track already acknowledged informal logical relations, with questions of ontological commitment (at least for a Quinean like Davidson) it goes the other way around — it only makes sense to read off ontological commitments after we have our regimentations (since “to be is to be the value of a variable”).

In §3,2, F&L move on to consider one class of anti-singularist consideration, what they call ‘substitution arguments’. Or rather they briefly consider one such argument, from a 2005 paper by Byeong-Uk Yi. A strange choice, by my lights, since the locus classicus for the presentation of such arguments is of course a 2001 paper by Oliver and Smiley, and then again in their 2013/2016 book Plural Logic. Their Chapter 3, ‘Changing the Subject’, in particular, is a tour-de-force relentlessly deploying such arguments. (F&L wrongly say that “changing the subject” is “[O&S’s] name for singularist attempts to eliminate plurals”. Not so. It is their punning name for one singularist strategy, the one which takes a plural-subject/predicate sentence and tries to regiment it as a singular-subject/predicate sentence. O&S’s following chapter discusses another, different, singularist strategy).

OK. Here’s a very quick reminder of the relevant sections of Plural Logic. In their §3.2, O&S argue for a uniform treatment of plural subjects, whether they are combined with a distributive or collective predicate. Thus, we shouldn’t (as Frege seems committed to do) carve ‘Tim and Alex met in the pub and had a pint’ into two sentences ‘Tim and Alex met in the pub’ [collective predicate, subject referring to some singular thing, the set {Tim, Alex} or mereological whole Tim + Alex] and ‘Tim and Alex had a pint’ [distributive predicate, so this turn is to be carved into the conjunction of ‘Tim had a pint’ and ‘Alex had a pint’]. O&S give two compelling arguments for uniformity. In §3.3, they then argue against a naive version of “changing the subject” where we regiment a plural-subject/predicate sentence by changing to a singular subject (substitute singular for plural) while leaving the predicate unchanged. They give elaborated versions of the familar sort of Boolos objection to doing that: it may be true that the cheerios were tasty, but it seems haywire to say the set of cheeries was tasty, etc., etc.

So in §3.4, O&S discuss the strategy of changing the subject and the predicate in a way that preserves coherence and truth-values. And the first point they press is that initial attempts to do this just move the plural from subject to predicate — for example if we want to regiment the plural subject term in ‘Russell and Whitehead wrote Principia’ by using a singular subject term for a set, we could render that sentence by ‘{Russell, Whitehead} is such-that-its-members-wrote-Principia’. But there are two problems with this sort of regimentation. (1) There are uniformity worries: take the sentence ‘Russell and Whitehead wrote Principia, Wittgenstein didn’t’ (the property denied of Wittgenstein here is surely not the same property of being such that its members etc. etc.). And crucially (2) a singularist will need to get rid of the plural term buried in the complex predicate. And so O&S consider various strategies for various cases. They make some headway in giving more-or-less contorted singular renditions of a number of plural sentences; but they sum up as follows:

The most striking feature of the analyses is their diversity. Although there is a uniform first stage [along the lines of the Russell and Whitehead example] the further analysis required in order to eliminate the residual plurals varies widely from case to case. It appears that we are condemned to a piecemeal and promissory approach, hoping rather than knowing that a suitable analysis can be found for any plural sentence. Such untidiness is unattractive, to say the least.

I think we are supposed to read ‘unattractive’ as indeed a radical understatement!

Now back to Florio and Linnebo. As I said, they consider just one observation by Yi, namely that there are contexts where we can’t intersubstitute ‘Russell and Whitehead’ and ‘{Russell, Whitehead}’ salva veritate (without changing the predicate). And F&L in effect note that changing the predicate in an appropriate way will rescue the day for Yi’s particular examples — though they cheerfully allow different changes in a couple of different contexts. But how piecemeal do they want to be? What about Oliver and Smiley’s further examples? F&L just don’t say.

Snap verdict: F&L’s two page jab gives no good reason to dissent from O&S’s extended trenchant arguments against singularism based on substitution considerations, broadly understood.

To be continued.

6 thoughts on “The Many and the One, Ch. 3/i”

  1. Which cheerios is ‘the cheerios were tasty’ about? It seems to me it will be some particular cheerios, such the ones someone put in a bowl for their breakfast. It doesn’t seem ‘haywire’ to me to say it was that bowlful of cheerios, or that bunch of cheerios, that was tasty. (Indeed, the rest of the box might not be.) And if ‘bowlful’ or ‘bunch’ is ok, why not ‘set’?

    There seems to be a tendency among people antipathetic to set theory, to read ‘that set of cheerios was tasty’ as if it’s about the set, not its contents (the cheerios), being tasty — as if ‘that box of cheerios was tasty’ would be about about the box itself (the cardboard) being tasty, rather than its contents (the cheerios). Yet they *wouldn’t* read ‘that box of cheerios was tasty’ in that way. They’re perfectly willing and able to understand that ‘that box of cheerios was tasty’ isn’t about the cardboard, and they wouldn’t think saying it was ‘haywire’.

  2. You are right, of course. Lots of set talk (superficially singular) is a just way of talking about the set’s members (plural) and, when the wraps are off, is not committing us to the existence of something else in addition to the members. But then set talk, so construed, is not apt for use in a singularist strategy which would have us replace a plural term referring to the members (plural) by a singular term reference to some singular thing. (And the Boolos jab about the cheerios is aimed, I take it, at those who would replace reference to the cheerios, plural, by reference to some singular other thing, such as a set in the official Cantorian sense.)

    1. I don’t think such talk is only superficially singular (not really singular); I think it is singular and does let us replace a plural term by a singular term.

      However, that doesn’t entail that ‘the set of cheerios was tasty’ means the set, as opposed to its members, was tasty, just as ‘the box of cheerios was tasty’ doesn’t mean the box itself (the cardboard) was tasty. People are perfectly able to understand that in the ‘box’ case without saying it’s only superficially singular or that it doesn’t involve the existence of something else (the box) in addition to the cheerios it contains.

      Of course, there are differences between sets and boxes. A box is another physical object, for instance; a set isn’t. If anything, though, that ought to make it easier, rather than harder, to understand ‘the set of cheerios was tasty’ as saying the cheerios were tasty.

      And of course, ‘set’ isn’t a word we normally use with cheerios. That does make ‘the set of cheerios was tasty’ seem a bit strange, in English. But that doesn’t seem to be what’s supposedly ‘haywire’.

  3. BTW, I find it quite difficult to make sense of the ‘Genie’ argument in §3.2, and I think it illustrates the weirdness of the way advocates of plural logic sometimes use English.

    We’re told ‘Genie’ abbreviates the set term ‘{Russell, Whitehead}’, and then that

    (3.1) Genie is one of Genie

    is ‘arguably true (and logically so)’. True? Seriously? It’s not true in ordinary English. Nor does unabbreviating to get

    (3.1′) {Russell, Whitehead} is one of {Russell, Whitehead}

    make it look any more true. F&L later say ‘One might even complain that (3.1) is ungrammatical’. Indeed.

      1. Is the translation of (3.1), ‘Genie is one of Genie’, as

        (3.3) Genie = Genie.

        there in Yi’s presentation? It occurs to me that (3.1) could instead be translated as

        (3.3′) Genie ∈ {Genie}

        which avoids the problem of translating ‘is one of’ in two different ways.

        And for ‘Genie is one of Genie’ to be true, it seems the second ‘Genie’ must be being understood as a plural (plurality?) that contains only one element. I haven’t been able to think of any other way to make sense of (3.1) as true.

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