Logic

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.

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).

Beginning Mathematical Logic: A Study Guide

I’ve renamed the old Teach Yourself Logic study guide; it is now more aptly called Beginning Mathematical Logic: A Study Guide. And there is now a new version of Part I of the Guide (all 95 pages of it) which you can download from here. It’s taken some time to settle on a style for the expanded Guide (though in the end I have not worried too much about keeping the level of the “overviews” of various topics consistent in level), and also it’s a judgement call where to place e.g. a quick introduction to second-order logic.

If you read the PDF from within a browser (as opposed to downloading it and using a PDF reader) it seems best to use Firefox on a Mac. Because then, if you go back after clicking a link, you are returned to your place in the Guide: Safari returns you unhelpfully to the beginning of the Guide.

All comments/corrections gratefully received as always — but perhaps better to use email until I can sort out the comment handling on the blog. The comments will arrive in my admin dashboard but won’t be visible.

New look, new hardback

The three Big Red Logic Books have a new look. I’m staying with the red theme, but no longer using the free Amazon KDP online cover-builder which produced their rather muddy colour and muddy swirls. The outsides, then, are a bit less dull. I’m afraid the insides of the paperbacks stay just the same!

More importantly, perhaps, a hardback of Gödel Without (Too Many) Tears is published today. It is currently available from e.g. Barnes and Noble (in the US), Gardners (the UK library book suppliers), Booktopia (in Oz), as well as the local Amazons. I hope it will soon propagate to other sellers like Blackwells. It won’t just appear on bookshop shelves, however: you’ll have to order it.

I’m not really expecting anyone reading this blog to buy it for themselves! However, you might like to recommend the hardback to your local friendly university or college librarian. Some librarians are pretty resistant to buying from Amazon (especially self-published paperbacks). That’s why I’m experimenting with hardback publication. We are going a step up here, using the same print-on-demand providers now used e.g. by CUP for some of their books, with a “proper” ISBN officially assigned to the Logic Matters imprint. It is still pretty cheap as academic hardbacks go — £14 in the UK and comparable prices elsewhere (so this isn’t going to make my fortune: to be honest, I’ll be pretty surprised if I even recoup the set-up costs.)

Of course the PDF version is still freely available, and I’ve kept the paperback version as Amazon-only as that absolutely minimizes the price to students. But I like to think that the book should be available on the shelves in university libraries, so please take a moment to recommend the hardback! Its ISBN is 978-1916906303. 

(Apologies by the way to readers down under that the paperback is still not locally printed and hence not cheaply available to you: Amazon say they are working on being able to produce paperbacks in the relevant format “soon” …)

Update: At the moment Amazon UK are giving a very long delivery date for the hardback, but I hope that’s temporary. Amazon US by contrast are giving a relatively short delivery date.

Roman Kossak’s Model Theory for Beginners

Following on from his Mathematical Logic (2018), Roman Kossak has now published Model Theory for Beginners: 15 Lectures (College Publications, 2021). As the title indicates, the fifteen chapters of this short book — just 138 pages — have their origin in introductory lectures given to graduate students in CUNY. Roughly speaking, the topics of the first half of this new book overlap quite closely with the second half of his previous book. And after grumbling a bit about Part I of that earlier book, I did warm considerably to the model-theoretic Part II, which I think makes for a very approachable elementary introduction to a cluster of issues about definability.

The new treatment is aimed at a rather more sophisticated reader, the writing is a bit less relaxed, and indeed becomes increasingly terse as the book progresses (in later chapters, I could often have done with a sentence or two more motivational chat). But overall, this strikes me as a welcome book. Though I’m at all not sure it is all suitable for beginners.

In a little more detail, after initial chapters on structures and (first-order) languages, Chapters 3 and 4 are on definability and on simple results such as that ordering is not definable in (Z, +). Chapter 5 introduces the notion of types, and e.g. gives Cantor’s back-and-forth proof that countable dense linearly ordered sets without endpoints are isomorphic to (Q, <). Chapter 6 defines elementary equivalence and elementary extension, and establishes the Tarski-Vaught test. Then Chapter 7 proves the compactness theorem, Henkin-style, with Chapter 8 using compactness to establish some results about non-standard models of arithmetic and set theory.

So there is a somewhat different arrangement of initial topics here, compared with books whose first steps in model theory are applications of  compactness. But the early chapters are indeed nicely done. However, I don’t think that Kossak’s Chapter 8 will be found an outstandingly clear and helpful first introduction to applications of compactness — it will probably be best read after e.g. Goldrei’s nice final chapter in his logic text.

Chapter 9 is on categoricity — in particular,  countable categoricity. (Very sensibly, Kossak wants to keep his use of set theory in this book to a minimum; but he does have a section here looking at κ-categoricity for larger cardinals κ.) And now the book starts requiring rather more of its reader. Chapter 10 is on indiscernibility and the Ehrenfeucht-Mostowski Theorem: but it is difficult to get a sense from this chapter of quite why this matters.

Up to this point, the structures we’ve been looking at are all officially relational. Chapter 11 adds functions, and discusses Skolem functions and Skolemization (this could have been more relaxed and helpful). We return to arithmetic in Chapter 12; there’s a compressed  discussion leading up to a version of Robinson’s model-theoretic proof of Tarski’s theorem of the arithmetic undefinability of arithmetic truth. But I rather doubt that this will be readily accessible to someone who hasn’t already read e.g. some of Kaye’s book on non-standard models of PA and met ideas like overspill.

The last three chapters are more advanced still, on saturation, automorphisms of recursively saturated structures, and (very briefly) stability. Are these topics for those just starting out on model theory? That’s a judgement call. But I suspect that the mode of presentation could be found quite challenging by many beginners — for me, more classroom asides in later chapters would have been welcome.

So as with Kossak’s earlier Mathematical Logic, then, I have rather different reactions to the two halves of Beginning Model Theory. But I’d say that the first eight or nine chapters do work very well under the advertised title (and I’ll be recommending them in the Study Guide). Later chapters are probably to be read in parallel with familiar moderately advanced texts like Marker’s classic.

Finally, bonus points for publishing very inexpensively with College Publications, and with tidy LaTeX layout too (however, they still can’t design a nice title page and verso!). But dock half a point for the number of minor typos …

Mathematical logic from a compsci angle?

In the Study Guide entry on First-Order Logic, after the list of main reading suggestions, there is a further list of suggested parallel/additional reading which ends by warmly recommending Melvin Fitting’s First-Order Logic and Automated Theorem Proving (Springer, 2nd edn 2012). Although published in a series aimed at compsci students, this should certainly appeal to logicians who are primarily philosophers or mathematicians but who want to know a little more about some themes (e.g. resolution proof systems) of special interest to computer science, and who want to re-encounter some familiar ideas approached from a slightly different angle. You can, if you want, skip some of the bits on Prolog and still get a particularly elegantly and clearly written account.

I’ve been wondering whether there are other books also written for computer scientists which could similarly appeal to the Guide’s intended readers. I know, of course, Michael Huth and Mark Ryan’s Logic in Computer Science (CUP, 2nd edn 2004). I haven’t reread this recently, but I recall it as being attractively and clearly written — the long first two chapters on propositional and predicate logic are well done, with a few interesting extras for the philosophical or mathematical reader (e.g. on SAT solvers). But then the later chapters go off in directions no doubt of key concern to computer scientist, but less interesting for the rest of us (for me, anyway!).

I’ve had Mordechai Ben-Ari’s Mathematical Logic for Computer Science (Springer 3rd edn 2012) recommended to me. But I thought this pretty second-rate.   The level of exposition is poor, and indeed at points seemingly outright confused (e.g. about the status of the Deduction Theorem for a Hilbert system). Someone who already has a grip on the standard math logic approaches could, I guess, get something out of the book by diving straight into the chapters on propositional resolution, SAT solvers, and first-order resolution, for example. But I didn’t find this material well explained: it is surely treated more pleasingly elsewhere.

To repeat, then, I’m interested in locating logic books coming from a compsci angle which will however also appeal to someone whose main interest remains in philosophy or mathematics. Luis Augusto’s book is advertised as aimed at such readers, but you know what I think of that. So are there other options?  I’d be very interested to hear!

Luis Augusto’s Formal Logic

“Of making many logic books there is no end; and much study is a weariness of the flesh.” The author of logical Ecclesiastes had probably just been reading the likes of Luis M. Augusto’s unnecessary Formal Logic: Classical Problems and Proofs (College Publications, 2019). I’ve noted before that this publisher’s quality control is lousy. Fortunately, because its books are relatively inexpensive, you can take the chance and order one which has a tempting-seeming  blurb, without cursing too much if the punt doesn’t come off.

This particular book aims to highlight problems which, though they “feature in introductory logic textbooks aimed at computer science students, … are largely or wholly absent from textbooks targeting a mathematical or philosophical studentship.” Looking for books with a compsci angle for the Logic Guide, I was intrigued.  But, apart from being written in poor almost-English, the technical exposition here is unappealingly hard going, and the level of motivational explanation third-rate. It could be so much better. So this is just a warning note: if you are similarly tempted by the blurb for this book, simply resist. And if that sounds a bit tetchy, it could be because my flesh is more than a bit weary after trying to study it for a day.

Roman Kossak’s Mathematical Logic

What do you make of this?

Think of a number, say 123. What is 123? It is a sequence of digits. To know what this sequence represents, we need to understand the decimal system. The symbols 1, 2, and 3 are digits. Digits represent the first ten counting numbers (starting with zero). The number corresponding to 123 is 1 . 100 + 2 . 10 + 3 . 1. In this representation, the number has been split into groups: three ones (units), two tens, and one hundred.

I’d worry that someone who wrote that is hopelessly confused between numerals (expressions which reprsesent) and numbers (what the numerals represent). And just how can a number (that very thing which is represented) be “split into groups”?

Or what about this, following some examples of equinumerous collections?

All those equinumerous collections have their individual features, but there is one thing that they all have in common. That is this one thing that we call the size. This common feature is the size of the collection and of all other collections that are equinumerous with it. Now we can introduce the following, more formal definition: a counting number is the size of a finite collection.

So numbers are “features”, i.e. properties? A moment ago they were things that can be split into groups! (Great-uncle Frege is not resting quietly …)

Let’s put this sort of thing down to a certain arm-waving carelessness rather than confusion: still, it doesn’t exactly inspire confidence in the more conceptual/philosophical remarks in Roman Kossak’s Mathematical Logic (Springer 2018).

Actually, this book is mis-titled. There is little core logic here. The early chapter entitled ‘First-Order Logic’ is a fleeting introduction to first-order languages, too fast for real newbies, and the idea of a formal deductive system is only mentioned, and then without elaboration, at p.132 (and the book has just 155 pages before the final summary chapter and the appendices start). What the book is in fact centrally about is signalled by its subtitle: “On Numbers, Sets, Structures, and Symmetry”.

As Kossak says in his final summary, his “aim in this book was to explain the concept of mathematical structure, and to show examples of techniques that are used to study them. It would be hard to do it honestly without introducing some elements of logic and set theory.” The examples of structures are near all numerical ones. So Part I is an introduction to the construction of the integers, rationals and reals from the naturals, and a lightning tour of some of the presupposed set theory. This is done in with a fair amount of motivational chat, so some credit for that. But I still think the beginner would be notably better off reading one of the usual introductions to these ideas in elementary set theory books like Enderton’s or Goldrei’s. Charitably, the author is rushing on to get to what really does interest him, the book’s distinctive content in the seventy-odd pages of Part II.

And this Part, to get much more positive, is a very approachable introduction to some simple model theoretic ideas, but taking a rather different route to that of some familiar texts.  So Kossak explores the first-order definable features of various structures defined over the natural numbers, the integers, the rationals, the reals, and the complex numbers, and he nicely brings out some of the perhaps unexpected complications. He helps himself to the compactness theorem and e.g. the Tarski–Seidenberg theorem (which are not proved) to give partial demonstrations of various results. And along the way, the reader is introduced not only to basic notions like that of an elementary extension but also somewhat more sophisticated ideas like being a minimal structure. This is done with a light touch, helpful examples, and again a good amount of motivational chat. I’m not sure that some of the sketched proofs are quite as clear as they could be, and there can be philosophical wobbles in the commentaries. But this Part of Kossak’s book does, I think, get across some basic model theoretic ideas without too many tears, in an unusually accessible way, making connections that aren’t often brought out; and (those wobbles apart) I enjoyed it and learnt from it.

Added: However, since writing this, Kossak’s 2021 short book Model Theory for Beginners has landed on my desk, which covers the same ground as Part II and significantly more but in the same accessible style. At first glance, this looks very likely to be the book to go for, and I’ll report back in a few days.

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