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.

The Many and the One, Ch. 1

As Louis MacNiece wrote, “World is crazier and more of it than we think, Incorrigbly plural.” Evidently, then, we need a plural logic! Or so say quite a few. And enough has been written on the topic for it to be time to pause to take stock.

I have just now started reading Salvatore Florio and Øystein Linnebo’s The One and The Many: A Philosophical Study of Plural Logic, newly published by OUP with an open access arrangement which means that a PDF is free to download here. The book aims to take stock and explore the broader significance of plural logic for philosophy, logic, and linguistics. What can plural logic do for us? Are the bold claims made on its behalf correct?

I’ll say straight away that Florio and Linnebo write very lucidly in an attractively readable style. Though it is not entirely clear, perhaps, who the intended reader is. The opening pages seem addressed to a pretty naive reader who e.g. may not even have heard of Cantor’s Theorem (p. 3); yet pretty soon the reader is presumed e.g. to understand talk of defining logical notions in terms of isomorphism invariance (p. 22). Again, if the reader really was new to the topic and had never seen before one of the now standard core logical languages for plural logic and its associated core deductive system, the initial brisk outline presentation (pp. 15-20) might perhaps be rather too brisk. But I’m certainly not going to nag about this sort of thing. Whatever F&L’s intentions, I’ll take the likely reader of their book to be someone who has some logical background and in particular has a modicum of prior acquaintance with plural logic and some of the debates about it; and then their brisk early remarks can serve perfectly well as reminders getting us back the swing of thinking about the topic.


So let’s dive in. In the short Chapter 1, ‘Introduction’, F&L highlight three questions which are going to run through their discussion:

  1. Should the plural resources of English and other natural languages be taken at face value or be eliminated in favor of the singular?
  2. What is the relation between the plural and the singular? When do many objects correspond to a single, complex “one” and what light does such a correspondence shed on the complex “ones”?
  3. What are the philosophical and other consequences of taking plurals at face value?

Not, I think, that we are supposed to take these as sharply determinate questions at this stage: take them as pointers to clusters of issues for discussion. F&L also give early spoilers, indicating some lines they are going to take.

In response to (1) they announce they are pluralists, resisting the wholesale elimination of plurals (while, they say, wanting to resist some of the usual arguments against singularism). On (2) they say — surely rightly — that the question is going to entangle us tricky issues in metaphysics, semantics, and the philosophy of mathematics. We can’t, as it were, argue for a particular line on plural logic in isolation; rather we going to have to “chose between various “package deals” that include not only a plural logic but also commitments far beyond”. On (3) F&L trail their view that many of the claims that have been made for plural logic — such as that it “helps us eschew problematic ontological commitments, thus greatly aiding metaphysics and the philosophy of mathematics” — are, in their words, severely exaggerated. Leaving aside the ‘severely’, I’ll probably find myself endorsing a verdict that some of the claims that have been made for plural logic are somewhat overblown. But I’ll be interested to see to see how the detailed arguments pan out.

To be continued.

Postcard from Monmouth

We have been staying in a cottage near Monmouth for a few days. The countryside here is indeed a particularly green and pleasant land; we can sit outside the cottage looking over many rolling miles towards the Black Mountains. The ruins of Tintern Abbey are close by, as is Raglan Castle (both so very well looked after by Cadw for the Welsh Government, and both sites surprising quiet). There is a lot of wonderful walking here, through local woodlands, and in the Usk valley and the Wye valley. The weather has been kind. So a delightful escape from Cambridge.


With brilliant timing, the day before coming away, when I should have been concentrating on things domestic, Logic Matters got hacked. Or rather — since “logicmatters.net” then delivered a Chinese language advert, which hardly looked like a hack aimed at the typical reader here — I suspect some WordPress plugin had been hacked. (I’d been experimenting with different plugins while giving this site a fresh coat of paint). It took a while for me to find the source of the trouble; but then someone kindly recommended the Wordfence security plugin which quickly pinpointed where the evil code been added. Fingers crossed, but I hope the site is now more secure from such exploits. A long afternoon quite, quite wasted though. Not good for the blood pressure.


Calm was restored driving down to Monmouth and stopping at Kiftsgsate Court Gardens and then on to High Glanau, both gardens real works of art. We will visit Coton Manor Garden on the way home, which is even more stunning in its way (though perhaps I prefer the slightly wilder, less perfectly kempt style of Kiftsgate). I have said before, that gardens can be art-works of a kind that the English both do particularly well and particularly love. Is there, I wonder, anything attractively and insightfully written by modern philosophers on the aesthetics of gardens? (A genuine question!)

LaTeX for Logicians — a new look (and time for new content?)

Here are the new-look pages for LaTeX for Logicians.

The LaTeX for Logicians front page got over 35K visits last year, with some of the other individual pages getting 15K visits. So these pages are evidently still being found useful. I haven’t updated some of them  for well over two years, and I am certain to have missed some more recently added LaTeX resources that will be of interest to users of these pages.

So this is your moment: as I update these pages, do please let me know what’s missing!

… and back again

Update on the Logic Matters website. You can get lost down the infinite rabbit hole of WordPress customizations. But I’ve managed to escape, fixed on a theme, suppressed most of its fancy options, aiming for simplicity verging on starkness, and have got to work … Lots still to be done (for a start, in making more tablet and phone friendly), but you’ll get the basic idea. Any helpful comments/suggestions will of course be welcome.

Update on the hardback of Gödel Without (Too Many) Tears. Hooray! — a copy (from UK Amazon) has arrived at last, and another copy (from Blackwell’s, one to send on to the British Library) arrives tomorrow. UK and US Amazon are both now promising very speedy delivery.

I must say that I am very pleased with the result, it is really decently produced. So that, together with the sales figures, encourages me to organize hardback library copies of IFL and IGT. More about that anon. But for the moment, do please remember to get your local friendly librarian to order the hardback GWT for the library! — details here.

Down the rabbit hole …

This Logic Matters site currently lives on Bluehost. But for various reasons, I’m in the middle of moving to a different hosting provider, Siteground: significantly more expensive (after the initial year’s discount) but by very many accounts also significantly better. Certainly, an experimental test version of the site runs there a lot faster, both on my iMac and even more so on an iPhone. As I’ve said before, the whole site could do with a good deal of tidying under the bonnet. So the needed update will keep me from fretting about the state of the world for a week or two.

I’ll need then to chose a modern WordPress theme that maintains the uncluttered look I like. I’ll ignore the pricey paid options. That only leaves about eight thousand free themes to choose from. So this is going to be dead easy. Down the rabbit hole I go …

Beginning Mathematical Logic again

I have uploaded a slightly revised version of Part I of the Study Guide, with just a few changes to the arm-waving chat and a couple of additions to the recommendations in the Computation/Arithmetic/Gödel’s Theorem chapter. You can download it here.

I’m working away at Part II, mostly enjoying the (re)reading around. An earlier time-slice of myself might have persisted in reading the less fun books out of a misplaced sense of duty. Now I tend to think that if someone really can’t be bothered to write with transparent clarity and make some honest attempt to take their reader along with them by e.g. providing enough signposts along the way, then maybe I can’t be too bothered about struggling with their ill-written texts. So I move on much more quickly to find something more logically entertaining.

Big Red Logic Books: now available in Australia!

Short version: paperbacks of An Introduction to Gödel’s Theorems,  An Introduction to Formal Logic, and Gödel Without (Too Many) Tears are now available from Amazon in Australia.

Slightly longer version: An Australian version of Amazon’s KDP print-on-demand service has been up and running since the beginning of the year. Initially, however, it couldn’t handle books in the format of the Big Red Logic Books. But (though they haven’t told authors!) I have just discovered in the last hour that the books are now available locally. The prices are set to the minimum possible (the fixed printing and distribution charges are higher in Oz, but I’ve set the royalties to zero to compensate).

So please spread the word Down Under. The books have been available as PDF downloads for a year, but there are quite a few who much prefer to work from printed books. And do tell local librarians (you might need to do a bit of explaining/cajoling too, as librarians tend to hold their professional noses over self-published books, and don’t approve of Amazon either! — but other publication routes would have been much more expensive).

I’d be interested to hear how the physical copies turn out  (the UK printed ones are really surprisingly good, apart from slightly flimsy covers, given the price point).

Big Red Logic Books: half year report

Self-publishing seemed exactly appropriate for the Big Red Logic Books. They are aimed very particularly at students, so why not make them available as widely as can be? — free to download as PDFs, for those happy to work from their screens, and at the smallest possible cost for the significant number who prefer to work from a physical copy.

I’d certainly warmly encourage others to self-publish, if it is appropriate. However, as Bishop Butler didn’t quite say, every book is what it is and not another book. And every author’s situation is what it is. But if you want info about the whys and hows and wherefores of self-publishing do get in touch.

With minor hiccups, my experience with the Amazon KDP system has been very easy and straightforward. I started writing up some detailed techie notes to post here, but quickly realized they were likely to be of very little general interest! But some might be interested, and a few encouraged to follow my example, by the raw figures for the first half of 2021 (so this is the period after the initial flurries of downloads and sales last year, and without any particular advertising efforts).

PDF downloads Paperback sales
Intro Formal Logic 5247 453
Intro Gödel’s Theorems 3730 352
Gödel Without Tears 995 432

I’m rather surprised by the very different ratio of downloads to sales for GWT. Overall,  I am more than happy with these figures. And no, I’m not making a fortune! — the royalties are set at pennies per book. The hope was just to cover set-up costs, and to defray some of the hosting costs for Logic Matters.  I’ll re-do the sums from time to time and lower the prices further if I can.

As I’ve noted before, there is now a hardback for libraries of GWT which has, to my surprise, sold 62 copies in the first two weeks. Or so say Ingramspark who provide the hardbacks. Though whether anyone has yet seen a copy out there in the wild, I have yet to hear! I’m still myself waiting for a copy from Amazon and a copy from Blackwell’s (two copies, so I can send one off to the British Library for posterity …).

The Pavel Haas Quartet again: two online concerts

In these days of Covid, our chances of seeing live concerts from our favourite musicians are much reduced. In particular,  since we don’t live in the Czech Republic or nearby, I’m not going to get to see the Pavel Haas Quartet live again for a good while yet. But I will be catching a couple more online concerts, this time recorded for the West Cork Chamber Music Festival. The first is tonight (29th June), and then available on demand for 48 hours. The second is on Saturday (3rd July) and then again available on demand for 48 hours.

I don’t know quite how many readers here ever follow up the musical posts. However,  I do occasionally get ‘thank you’ emails! So let me say more about these concerts. Who knows, it might tempt a few readers to catch one or other of them. I hope so!

The first features Martinů, Quartet No.2; Schulhoff, Quartet No.1; Janáček, Quartet No.2 ‘Intimate Letters’. From the West Cork website:

Pavel Haas was a Jewish Czech composer like Schulhoff. They both perished in the Holocaust and the Nazis set out systematically to suppress their music by taking over music publishers and banning all performances of their music. This was lethally effective and it took decades to rediscover their music and to return it to its rightful place in the repertoire. This concert features a Prague-based Czech Quartet playing the music of three well-known Czech composers. Martinů was a wonderful composer of chamber music. He wrote: ‘It is hard for me to express the happiness I feel when I start composing chamber music – the delight of leading the four voices, in a quartet one feels at home, intimate, happy.’ Martinů wrote seven quartets that are seldom heard, hopefully a future Festival will include the full cycle. Schulhoff fought in the First World War and post-War turned away from traditional musical forms, associating them with the decadence of the old order that had led to the catastrophe of world war. He spent the Twenties experimenting with different forms both musically and politically. His First Quartet, dating from 1924, is an explosion of energy but otherwise follows a surprisingly conventional path. Janáček’s two quartets are well-known as they trace in music his obsessional love affairs. Milan Kundera wrote: ‘His music is a breathtakingly close confrontation between tenderness and brutality, madness and peacefulness; it condenses the whole of life, with its hell and its paradise.’

The second concerts features Dvořák, Piano Trio No.3 in F minor Op.65 and Piano Quintet No.2 in A major Op.81.  From the concert website again:

For this special concert, Pavel Haas Quartet is joined by Boris Giltburg for two major works by Dvořák, his tempestuous F minor Piano Trio and the infectious delight of his second A major piano quintet. For the Trio the Quartet’s leader, Veronika Jarůšková, and cellist, Peter Jarůšek, join Boris Giltburg. This concert was recorded at the Martinů Hall in Prague. Dvořák’s Third Piano Trio was composed shortly after his mother’s death, it opens in passionate agitation and ends in an emotional tempest. In between comes a tuneful and light-hearted Allegretto leading to a calm and meditative Adagio. Despite his personal loss, Dvořák is able to swathe his distress in a succession of the glorious melodies for which he was so renowned. The opening of the Piano Quintet never loses its magic however often we hear it, while the Andante gives the languid voice of his own instrument, the viola, a leading role. There are no shadows in this unblemished music.

Need I say more? These concerts should indeed be wonderful. Very inexpensive online tickets for the Tuesday concert available here and for the Saturday concert available here.

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