Peter Smith's Blog, page 41

The Many and the One, Ch. 5 & Ch. 6

I confess that I have never been able to work up much enthusiasm for mereology. And Florio and Linnebo’s Chapter 5, in which they compare ‘Plurals and Mereology’, doesn’t come near to persuading me that there is anything of very serious interest here for logicians. I’m therefore quite cheerfully going to allow myself to ignore it here. So let’s move on to Chapter 6, ‘Plurals and Second-Order Logic’. The broad topic  is a familiar one ever since Boolos’s classic papers of — ye gods! — almost forty years ago: though oddly enough F&L do not directly discuss Boolos’s arguments here.

In §6.1, F&L give a sketchy account of first-order logic, and then highlight its monadic fragment. Note, they treat the second-order quantifiers as ranging over Fregean concepts. And they perhaps really should have said more about this — for can the intended reader be relied on to have a secure grasp on Frege’s notion? Indeed, what is a Fregean concept?

The following point seems relevant to F&L’s project. According to Michael Dummett’s classic discussion (in his Frege, Philosophy of Language, Ch. 7), Fregean concepts are extensional items: while (for type reasons) we shouldn’t say that co-extensive concepts are identical, the relation which is analogous to identity is indeed being coextensive. So the concept expressions ‘… is a creature with a heart’ and ‘… is a creature with a kidney’ have the same Fregean concept as Bedeutung. I take it that Dummett’s account is still a standard one (the standard one?). For example, I note that Michael Potter in his very lucid Introduction to the Cambridge Companion to Frege — while noting Frege’s reluctance to talk of identity in this context — writes (without further comment)

Concepts, for Frege, are extensional, so that, for instance, the predicates ‘x is a round square’ and ‘x is a golden mountain’ refer to the same concept (namely the empty one).

But now compare F&L. They write

Two coextensive concepts might be discerned by modal properties. Assume, for example, that being a creature with a heart and being a creature with a kidney are coextensive. Even so, these two [sic] concepts can be discerned by a modal property such as possibly being instantiated by something that lacks a heart.

Which seems to suggest that, contra Dummett and Potter’s Frege, co-extensive predicates can have distinct concepts as Bedeutungen. That’s why I really do want more elaboration from F&L of their story about the Fregean concepts which, according to them, feature in an account of second-order quantification.

§6.2 notes how plural logic and monadic second order logic can be intertranslated (with minor wrinkles). And, analogously to §4.3, a question then arises: can we eliminate pluralities in favour of concepts, or vice versa?

So §6.3 discusses the possibility of using second-order language to eliminate first-order plural terms, as once suggested by Dummett. As F&L note, this suggestion has already come in for a lot of criticism in the literature; but they argue that there is some wriggle room for defenders of (something like) Dummett’s line to avoid the arguments of e.g. Oliver and Smiley and others. I’m not really convinced. For example, F&L suggest that a manoeuvre invoking events proposed by Higginbotham and Schein will help the cause — simply ignoring the extended critique of the manoeuvre already in Oliver and Smiley’s Plural Logic.  In the end, though, F&L think that there is a pretty compelling argument against the elimination of pluralities in favour of concepts on the basis of their respective modal behaviour (but note, F&L are here seemingly relying  on their departure from the standard Dummettian construal of Fregean concepts — or if not, we need to hear more).

§6.4 then looks at the possibility of an elimination going the other way, reducing second-order logic to a logic of plurality. But so far we have only been offered a way of translating monadic second order logic using plurals; the obvious first question is — how can we translate full second-order logic with polyadic predicates, quantifying over polyadic concepts? Perhaps we can do the trick if we help ourselves to a pairing function for the first-order domain (so, for example, dyadic relations get traded in for monadic properties of pairs). F&L raise this familiar idea: but suggest — again very briefly — that there is another modal objection: “while a plurality of ordered pairs can model the extension of a dyadic relation, it cannot in general represent all of its intensional features.” Tell us more! We also get a promissory note forward to discussion of a different objection to eliminating second-order logic.

There’s a short summary §6.5. But, to my mind, this is again a somewhat disappointing chapter. As it happens, my inclinations are with F&L’s conclusion that both plural logic and second order logic can earn their keep (without one being reduced to the other). But I do rather doubt that anyone who already  takes a different line will find themselves compelled to change their minds by the arguments outlined here.

To be continued, but after a break.

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The Many and the One, Ch. 4

In the next part of their book, ‘Comparisons’, F&L discuss ‘Plurals and Set Theory’ (Chapter 4) and ‘Plurals and Second-order Logic’ (Chapter 6). In between, they also compare ‘Plurals and Mereology’ (Chapter 5). But I confess that I have never been able to work up much enthusiasm for mereology, and F&L’s chapter doesn’t come near to persuading me that there is anything of very serious interest here for logicians; so I’m cheerfully going to allow myself to ignore it.

Here, in bald outline, is what happens in Chapter 4.  §4.1 describes a ‘simple set theory’ framed in a two-sorted first-order language, with small-x quantifiers running over a domain of individuals and big-X quantifiers running over sets of those individuals. The two sorts are linked by an axiom scheme of set comprehension, (S-Comp): ∃X∀x(x ∈ X φ(x)). §4.2 notes that the mutual interpretability of this theory with a simple plural logic. (We can’t just replace big-X set variables by double-x plural variables — we need to work around the usual assumption that there is an empty set in the range of big-X variables but not an empty plurality in the range of double-x plural variables. But that’s minor tinkering.) §4.3 then asks whether this mutual interpretability means we should eliminate plurals in favour of sets or sets in favour of plurals. §4.4 suggests that we need plurals in elucidating the very notion of a set (so don’t eliminate plurals): the root idea is that “For every plurality of objects xx from [a given domain], we postulate their set {xx},” where postulation seems to be tantamount to defining into existence. We are promised more about definitions of this kind in Chapter 12.

§4.5 then notes that mathematical uses of sets crucially involve not just sets of individuals (numbers, perhaps) but sets of sets, sets of sets of sets. etc.; and, for a start, it is very unclear that these can be eliminated in favour of pluralities of pluralities. §4.6 then says more about the iterative conception of set, and §4.7 gives the axioms of ZFC. §4.8 jumps on to wonder whether we can use plurals in explicating the notion of proper classes. The chapter ends with §4.9 which raises a problem:

We have described two very attractive applications of plural logic: as a way of giving an account of sets, and as a way of obtaining proper classes “for free”. Regrettably, it looks like the two applications are incompatible. The first application suggests that any plurality forms a set. Consider any objects xx. Presumably, these are what Gödel calls “well-defined objects”. If so, it is permissible to apply the “set of” operation to xx, which yields the corresponding set {xx}. The second application, however, requires that
there be pluralities corresponding to proper classes, which by definition are collections too big to form sets.

F&L again promise to return to deal with this apparent tension in their Chapter 12.

Does the chapter work? Well, it is pretty difficult to know quite at whom it is aimed. For example, §4.6 very briskly outlines the iterative conception of set, helping itself along the way to the idea that we take unions at levels indexed by limit ordinals (where ordinals are unexplained). But I wonder who is supposed to (a) already be familiar with the notion of a limit ordinal in §4.6, but (b) still need to have the axioms of ZFC given again in §4.7? And won’t the reader who needs §4.7 need more explanation of the role of proper classes in set theory (and the difference between their appearance as virtual classes in e.g. Kunen, versus a more substantive appearance in NBG)?

And to go back to the beginning of the chapter, I would guess that someone with enough logical education to know about limit ordinals would also know enough to want to ask more about the principle S-Comp: does the comprehension principle apply to predicates φ(x) which themselves involve bound set variables? or involve free set variables as parameters? or neither? We are not told, and there is no hint that the issue might matter. And there is no hint at all that the kind of “simple set theory” with two sorts of quantifier might actually be of real interest, e.g. in reverse mathematics when considering subsystems of second-order arithmetic. This lack of development is typical.

As it happens, I am in sympathy with F&L’s overall line that (i) plural logic is repectable and can earn its keep in certain important contexts, and (ii) set theory is just fine in its place too! But I can’t see that this arm-waving chapter really advances the case for either limb (and I could nag away more at some of the details). In so far as there are hints of novel argumentative moves, the work of elaborating them is left for much later. So I did find this chapter frustratingly rather superficial.

To be continued.

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“The truest thing that I know in this world”

Portrait by Felix Broede

“Mozart’s music is, for me, the truest thing that I know in this world. And playing his music has truly changed me, and I believe it can change anyone who is open to miracles.” Moving words from the young German pianist Elisabeth Brauss, and very moving playing of Mozart’s No. 23 in A major, K488 at her Proms debut. You can listen here (from 37 minutes in).

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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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New book: Wittgenstein on mathematics.

There’s a new book in the Cambridge Elements series on Wittgenstein’s philosophy of mathematics by Juliet Floyd. And for a few days it is freely available to read (and indeed download) here. I really rather doubt that it will appeal, though. Unless you like this sort of writing (the sixth paragraph, not at all untypical):

Aspects are modal, attaching to possibilities and necessities: fields of significance, opportunities for projecting and instantiating our concepts. We see through the picture to our own seeing of it as realizing one way among others. What we see is seen, but also we see. We rearticulate what we see, sometimes seeing it thereby anew. There is an active and a passive aspect to this. Aspects show themselves (the middle voice). What we are seeing is not simply an actual drawing on a page. We can also “see” in these drawings possibilities of projecting our concepts. Here we take modality as primitive, though up for investigation.

This ex-editor of Analysis most certainly wouldn’t have let that pass as acceptable.

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Vaguely distracted

I’ve been distracted from plurals — just for a day, I tell myself — by the arrival of Crispin Wright’s collected papers on vagueness (a 450 page book at a very decent price, by the way, and a must-read for anyone half-interested in the topic). Richard Heck contributes a forty page introduction, picking out some main themes: and on a quick first read this seems pretty insightful and really very helpful for (re)orientation. And then jumping to the end, I’ve been reading Wright’s most recent piece on “Intuitionism and the Sorites Paradox” (which was in the Oms/Zardini edited volume of essays on the Sorites). This is, needless to say, an impressive, challenging, imaginative essay. But I’m going to have to just mull over the ideas for now, and return to this, and to many of the other essays in the book, more seriously later. I’m sure, though, that my quarter-baked thoughts on vagueness won’t be fit for public consumption here! And for now it is back to plural logic (because getting a bit clearer about that is more directly relevant to other projects …).

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Settling in …

Logic Matters seems to be settling comfortably into its new home with a different hosting company, though a fair bit remains to be done.

For a start, pages now load much faster (mostly, within a second or two). The laggardly behaviour on Bluehost was an annoyance, a major reason for moving in the first place, so it is good that Siteground is indeed considerably snappier.The new Astra theme is, as they say, “responsive” (meaning that it knows whether you are on a computer, tablet or mobile, and makes visual adjustments accordingly).To help keep pages loading fast, I have kept things simple (with e.g. a minimal number of plugins.).  And I’m really pretty happy with the understated look’n’feel of Logic Matters on a tablet or mobile. After all, this is basically a wordy text-based site, so visual drama seems inappropriate. On a desktop, if you open a big window then you do get a really boring expanse of grey around the content areas. But replacing that with even a very low-key abstract background gives a surprisingly distracting effect; so I’m inclined to keep things as plain as they are. So I hope people like the simple layout (though I’m certainly open to suggestions for fine-tuning. the design)The content of the footer area on tablet/computer pages is a bit unimaginative: but I’m not sure what would be more interesting/useful (again I’m open to suggestions).The blog is now 15 years old (I missed its birthday), but I’m for the moment keeping all the legacy posts online, if only because they are the nearest I have to a diary of recent years. A link-checker tells me that a lot of old links within the blog no longer work but I’m not going to worry about that. I might set up a page, however, linking to past blog highlights (i.e. linking to posts which are mini-essays of one kind or another that might have more than ephemeral interest).One bit of functionality I still need to restore is being able to sign up to notifications of posts. It’s on my to-do list.Meanwhile, I’m rather slowly and sporadically checking/reworking the other, static, pages. In particular, the much-visited LaTeX pages are more than overdue for a bit of attention: but don’t hold your breath — updating these and some other pages always seems to take an inordinate amount of time (if only because web-searches for relevant materials take me off down distracting byways).

I’m glad I made the hosting change and updated to a modern WordPress theme. It’s been a bit of a palaver, because of course you always underestimate the amount of fiddly work involved in this sort of thing. But it’s been worth it, and we’re getting there …

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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.

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The Many and the One, Chapter 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.

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The Many and the One, Ch. 2

Chapter 2, ‘Taking Plurals at Face Value’, continues at an introductory level.

Oddly, Florio and Linnebo give almost no examples of the full range of plural expressions which they think a formal logic of plurals might aim to regiment (compare, for example, the rich diet of examples given by Oliver and Smiley in §1.2 of their Plural Logic, ‘Plurals in Mathematics and Logic’). Rather F&L start by immediately sketching three singularist strategies for eliminating plurals, starting the with familiar option of trading in a plural term denoting many things for a singular term denoting the set of those things.

They will be returning to discuss these singularist strategies in detail later. But for now, in their §2.2, F&L introduce the rival idea that “plurals deserve to be understood in their own terms by allowing the use of plural expressions in our regimenting language”. §2.3 then announces “the” language of plural logic. But that’s evidently something of a misnomer. It is a plural formal language, but — for a start — it lacks any function expressions (and recall how central it is O&S’s project to have a workable theory account of function expressions which take plural arguments).

F&L leave it open whether one should “require a rigid distinction between the types of argument place of predicates. An argument place that is open to a singular argument could be reserved exclusively for such arguments. A similar restriction could be imposed on argument places open to plural arguments.” But why should we want such selection restrictions? O&S remark very early on (their p. 2) that — bastard cases aside — “every simple English predicate that can take singular terms as arguments can take plural ones as well.” Are they wrong? And if not, why should we want a formal language to behave differently?

F&L seem think that not having selection restrictions would depart from normal logical practice. They write

In the philosophical and logical tradition, it is widely assumed that if an expression can be replaced by another expression salva congruitate in one context, then it can be so replaced in all contexts. This assumption of “strict typing” is true of the language of first-order logic, as well as of standard presentations of second-order logic.

But that’s not accurate. For example, in a standard syntax of the kind F&L seem to assume for singular first-order logic, a name can be substituted salva congruitate for a variable when that variable is free, but not when it is quantified. (As it happens, I think this is a strike against allowing free variables! — but F&L aren’t in a position to say that.) Any anyway, there is a problem about such selection restrictions once we add descriptions and functional terms, as Oliver and Smiley point out (Plural Logic, p. 218). If we allow possibly plural descriptions and possibly multi-valued functions (and it would be odd if a plural logic didn’t) it won’t in general be decidable which resulting terms are singular arguments and which are plural; so having singular/plural selection restrictions on argument places will make well-formedness undecidable. (If F&L don’t like that argument and/or have a special account of ‘singular’  vs ‘plural argument’, which they haven’ previously defined, then they need to tell us.)

Moving on, §2.4 presents what F&L call “The traditional theory of plural logic”. I’m not sure O&S, for example, would be too happy about that label for a rather diminished theory (still lacking function terms, for a start), but let that pass. This “traditional” theory is what you get by adding rules for the plural quantifiers which parallel the rules for the singular quantifiers, plus three other principles of which the important one for now is the unrestricted Comprehension principle: ∃xφ(x) → ∃xx∀x(x ≺ xx φ(x)) (if there are some φs, then there are some things such that an object is one of them iff it is φ).

Evidently unrestricted Comprehension gives us some big pluralities! Take φ(x) to be the predicate x = x, and we get that there are some things (i.e. all objects whatsover) such that any object at all is one of them. F&L flag up that there may be trouble waiting here, “because there is no properly circumscribed lot of ‘all objects whatsoever’.” Indeed! This is going to be a theme they return to.

§2.5 and §2.6 note that plural logic has been supposed to have considerable philosophical significance. On the one hand, it arguably is still pure logic and ontologically innocent: “plural variables do not range over a special domain but range in a special, plural way over the usual, first-order domain.”
And pressing this idea, perhaps (for example) we can sidestep some familiar issues if “quantification over proper classes might be eliminated in favor of plural quantification over sets”. On the other hand, a plural logic is expressively richer than standard first-order logic which only has singular quantification — it enables us, for example, to formulate categorical theories without non-standard interpretations. F&L signal scepticism, however, about these sorts of claims; again, we’ll hear more.

The chapter finishes with §2.7, promisingly titled ‘Our methodology’. One of the complaints (fairly or unfairly) about O&S’s book has been the lack of a clear and explicit methodology: what exactly are the rules of their regimentation game, which pushes them towards a rather baroque story?  Why insist (as they do) that our regimented language tracks ordinary language in allowing empty names while e.g. cheerfully going along with the material conditional with all its known shortcomings? (If conventionally tidying the conditional is allowed, why not tidying away the empty names?) Disappointingly, despite its title, F&L’s very short section doesn’t do much better than O&S. “We aim to provide a representation of plural discourse that captures the logical features that are important in the given context of investigation.” Well, yes. But really, that settles nothing until the “context of investigation” is articulated.

To be continued.

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