Year: 2021

Back to the Study Guide

These have been depressing times, despite good vaccine news, no? Grey winter days do not lift the lockdown spirits. So an unproductive period for me. I don’t think I’m alone in this either.

Regrouping, I realize I’ve been trying to juggle too many balls at the same time recently. So — with apologies to Catarina Dutilh Novaes — I’m going to hang fire on blogging chapter-by-chapter about her interesting The Dialogical Roots of Induction (this is such a wide-ranging book, and it would take me too much time to do the homework to do it detailed justice). I might put together some brisker comments later. I’m also going to back off from the idea of doing some podcasts. I need to focus, and since both are downloaded a lot, I’m going to concentrate over the next few months on completing (i) the new version study guide and (ii) the notes on category theory. Which probably won’t make for many interesting blog posts here!

OK; so I have now uploaded the latest version of the partial Logic: A Study Guide, with a new twelve page chapter on elementary set theory. There is an overview of the topics, and I’ve slightly revised my preference-ordering of recommended texts. It’s been fun (and embarrassingly instructive) to revisit some of those basic set theory book; so I hope that some students will find the results useful!

The Dialogical Roots of Deduction, 2

Moving on to discuss Chapter 2 of Catarina Dutilh Novaes’s The Dialogical Roots of Deduction

CDN has argued in her opening chapter that “deduction remains a puzzling phenomenon. While a number of accounts have been proposed, none of them is entirely satisfactory.” So how to proceed? In §2.1, she proposes to adopt a “more encompassing perspective” than usual, using a wider range of “methodological approaches”. But what, more specifically, does that come to? She writes “the key idea of the project … was to go back to the roots of deduction. It is obviously inspired by Quine’s classic The Roots of Reference (Quine, 1974).”

Now, in principle, that is just fine by me: I’m a fan of Quine’s (underrated?) book, and I’d love to see a broadly comparable project to his, now focused on questions about deduction. But I’m not sure that’s really what we are going to get, as CDN immediately goes on to say that the way she “conceive[s] of the roots in question is broader in scope than Quine”, to the point where we are going to need “sustained engagement with the empirical literature in psychology, cognitive science, and education sciences”, and also “analysis of historical texts … but combined with a broader historical perspective taking into account developments outside philosophy”. I’m open to persuasion that casting the methodological net so very widely will give us in the end a coherently illuminating story: but the project certainly isn’t sounding very like Quine’s sharply focused project.

In fact, §2.2 is titled “The Different Roots of Reference”, and CDN says more about the variety of considerations she wants to bring to bear on her topic. I continue to have a question though — one I had already about Chapter 1 — of what exactly the topic is. She talks of ‘deductive reasoning’. And sometimes this seems to mean elaborated passages of multi-step reasoning (the sort of thing that’s indeed a bit of an acquired taste). For example, she writes “deductive reasoning emerged as a cognitive technology (though arguably, it remains restricted to circles of specialists), in a way similar to literacy.” If we are focusing on that — the sort of thing that, paradigmatically, mathematicians go on for — we’d all agree that there are will be an interesting “historical question: which cultural processes gave rise to the emergence of deductive reasoning [in that sense]” and an “ontogenetic question: how do deductive reasoning skills arise in a given individual?”. And again, when CDN writes “it is not clear that deductive reasoning abilities in fact confer survival advantages on those individuals”, maybe that case is indeed arguable for reasoning-as-extended-passages-of-argument.

But what about, say, the ability to form disjunctive beliefs, and then use such a belief in a step of disjunctive syllogism? What about, say, the ability to form conditional beliefs (in formulating plans), and then use such a belief in a step of modus ponens when we come to believe the antecedent? Are these cultural variables? Some might suppose, I guess, that a culturally contingent practice of extended deductive reasoning has its roots (to borrow a word!) in capactities for simple-minded deductive reasoning which are not so variable. (And the fact that there are some relatively simple deductive tasks we are not very good at — as the Wason test reminds us! — is no reason for thinking that being good at the ones we are good at doesn’t confer evolutionary advantage!) CDN writes that she is going to “survey the main findings on how deductive reasoning skills emerge and develop in individuals, drawing in particular from the literature on the psychology of reasoning and on mathematics education”. That may indeed be an interesting project — but we’ll have to see what it can tell us about deductive inference in general, as opposed more extended passages of reasoning in particular.

In §2.3, CDN links the project of looking for “roots” of deduction with the claim that the concept of deduction has significantly changed over time. Which again rather points up the non-Quinean aspects of her notion of “roots”. And this gets me wondering about the variety of philosophical enquiries that might be deemed illuminating enquiries into “roots” in some broad sense (and CDN’s usage is nothing if not broad). So before going on to comment on §2.4, where CDN first sketches her dialogical approach, I’m minded — just to get my own ideas straighter — to pause over this, and say something in my next post about the variety of enquiries into “roots” we find in books I greatly admire by Bennett, Hacking and Craig, as well as by Quine.

To be continued.

The Dialogical Roots of Deduction, 1

There are three recent books on my desk which I’m looking forward to tackling. Two are Joan Weiner’s Taking Frege at his Word (OUP) and Juliette Kennedy’s Gödel, Tarski and the Lure of Natural Language (CUP). But I’m going to start by reading, and blogging about, a third:

Catarina Dutilh Novaes’s The Dialogical Roots of Deduction was published last month by (CUP). Your library could already have e-access via the Cambridge Core system.

CDN aims, says the blurb, ‘to bring together perspectives from philosophy, history, psychology and cognitive science, and mathematical practice … to argue for an overarching conceptualization of deduction as a dialogical practice’. We’ll have to see what this amounts to, and what new light gets thrown on old puzzles about the nature of deductive reason by this approach. So let’s dive in (and it is good to report that the book is engagingly readable).

Chapter 1 is titled ‘The Trouble with Deduction’. There’s a throat-clearing Introduction, and then §1.2 asks ‘What is a Deductive Argument?’. CDN highlights three features: (a) necessary truth-preservation, (b) stepwise structure and perspicuity, and (c) what she calls the bracketing belief requirement.

The first idea is a familiar enough theme: deductive validity is defined as requiring necessary truth-preservation. The question arising is going to be the nature of the necessity here; we’ll return to this.

The second idea is that ‘something else is required of a good deductive argument other than necessary truth-preservation: it must somehow make clear what the connection is between premises and conclusion such that the truth of the premise(s) guarantees the truth of the conclusion(s).’ So, to meet this requirement ‘a deductive argument … will typically contain numerous steps, each of which may be individually simple and thus individually not very informative, but by chaining such steps in a suitable way we may derive non-trivial conclusions from the given premises’.

The third idea is that ‘In its basic form, the game of deduction requires the reasoner to take the premises at face value, no questions asked: the focus is exclusively on the connection between premises and conclusions, not on the nature or plausibility
of the premises or conclusions.’ After remarking on some psychological research, CDN suggests ‘it seems that inferring conclusions from premises while disregarding one’s own doxastic attitudes toward premises and conclusions may require specific training. Yet, it is an integral component of deductive reasoning.’

Now I rather doubt that the third idea of belief bracketing tells us something special about deductive arguments. For isn’t this a feature of lots of (all?) kinds of argumentation? The law student faced with a description of an engagingly complex case, and asked to argue whether e.g. John Doe still has a contract with Jane Roe, entirely disregards whether ‘John Doe’ and ‘Jane Roe’ are pseudonyms for real people, and even disregards the likehood of the two parties getting themselves into the tangle described. The vaccine designer, reasoning about how to modify her vaccine in response to a range of different types of possible future virus mutation; her abductive reasoning brackets at least some questions about the relative plausibility of the scenarios (as she prepares for the worst while hoping for the best).

Perhaps there could be reasoners who can only reason from beliefs. But famously, it grants us an evolutionary advantage to be able to take our reasonings ‘offline’, argue from mere suppositions, and so we are able to send our suppositional hypotheses out to die in our stead. And that’s a general point about reasonings not about deductive reasoning in particular.

What about the second idea, that an epistemically useful deductive argument will typically chain together a number of individually not-very-informative steps? Well, again, isn’t this a point about argumentation more generally? The law student’s expansive legal argument for John Doe’s continuing contractual obligation, the vaccine-designer’s step-by-step argument for tweaking her vaccine design just so, again build up a case (perhaps for a surprising conclusion) by putting together simpler bits of reasoning. What is distinctive about the deductive case is not that there may be numerous steps leading to non-trivial conclusions, but (surely) that each step is necessarily truth-preserving (a property not lost by adding new steps to the argument).

So, until I hear more, I’m inclined to think that what is going to really matter for an account of distinctively deductive argumentation is going to be (as usually supposed, perhaps) the story about necessary truth-preservation (or warrant-preservation if you are constructively minded). Though this is consistent, of course, with that story being best told within a wider account of dialogical procedures of joint reasoning: we’ll see.

In §1.3, ‘The Issues’, CDN presents three philosophical questions about deductive
reasoning. One of them we have already noted, the nature of the necessity supposedly involved in deduction. Can we cash out the notion of necessity here in terms e.g. of a quantification over models? Is a proof-theoretic approach viable? There is a familiar bunch of questions here, and CDN notes some of the inconclusive recent debates in the literature.

But a prior issue is ‘Where Is Deduction to Be Found?’ — just what role does deductive reasoning play in our conceptual economy? CDN leans to the view that ‘deductive
reasoning is predominantly instantiated in mathematics and in some other regimented contexts of argumentation, such as philosophy’. Really? Maybe extended stretches of deductive reasoning are principally to be found there. But what about one-step syllogisms in Barbara? What about the instant one-step inference from a background general belief that No As are Bs and the new discovery that Jo is A to the conclusion Jo is not B? What about the jump from the background beliefs that ‘John is taller than Jo’ and ‘Jo is taller than Jane’ to the conclusion that ‘John is taller than Jane’? I’d have thought that such bits of local mini-inference were pretty common outside our mathematical activities! CDN’s view seems, then, to apply not to deductive reasoning in general but rather to extended passages of deductive argumentation in particular.

The third issue CDN raises is ‘What Is the Point of Deduction?’. The worry here is the old one — how to resolve the supposed inherent tension between the justification of deduction (the conclusion is already somehow there in the premisses) and the utility of deduction (we can get new knowledge by deductive reasoning). I confess I’ve always found this difficult to get excited about once we’ve noted that, while ‘entails’ may be transitive, ‘obviously entails’ certainly isn’t. Still, CDN remarks that ‘Deduction does not seem to be a particularly suitable way to produce new information … and it does not seem to be a reasonable guide for managing our beliefs and thoughts either’ (after all, it can’t be a sensible instruction to adopt every deductive consequence of our beliefs). So, she asks, ‘What, then, if anything, is the ‘point’ of deduction?’ And the promissory note is that her dialogic approach will give us a grip on this.

To be continued.

Luca Incurvati’s Conceptions of Set

Almost a year ago, I started a sequence of what turned out to be fifteen blog posts on Luca Incurvati’s book Conceptions of Set. I never got round to returning to put everything together (after some comments from Luca himself) into a more careful and re-thought review. One day …! But meanwhile, if you were interested in those posts, you might enjoy Øystein Linnebo’s new review of the book.

World Logic Day — a small contribution!

A small contribution for World Logic Day 2021 today: Gödel Without (Too Many) Tears is now available as a free PDF download — linked here.

Later in the year, I plan to put up a series of short podcasts, where I give introductory chapter-by-chapter chats about book. Many students are stuck in front of video lectures for far too long at the moment anyway, so I’m very reluctant to adding to the catalogue of full-scale lectures, quite apart from the time it would take me to record decent ones. So brisk arm-waving talks sketchings some Big Ideas which you can listen to while walking around or staring out of the window, followed by readings at your own pace of comparatively content-rich chapters, seems a format for teaching/learning worth trying out.

Lea Desandre and Ensemble Jupiter, Lettres Amoureuses

Music for lockdown. And “without music, life would be a mistake”, as Nietzsche said.

00:00 – 01. Monteverdi – Si dolce è’l tormento
03:56 – 02. Frescobaldi – Se l’aura spira
05:48 – 03. Kapsberger – Toccata VI
09:30 – 04. Dalza – Calata ala Spagnola
11:20 – 05. Handel – Lascia la spina
16:05 – 06. Falconieri – Folias
19:35 – 07. Monteverdi – Lettera amorosa
26:37 – 08. Frescobaldi – Cosi mi disprezzate, Toccata prima
29:23 – 09. Handel – Ombra mai fu
32:46 – 10. Kapsberger – Toccata I
35:12 – 11. Merula – Intonazione cromatica del terzo tono
37:43 – 12. Merula – Canzonetta spirituale sopra alla nanna
45:15 – 13. Vitali – Toccata et Bergamasca
48:15 – 14. Merula – Folle è ben chi si crede
53:07 – 15. Merula – Ciaccona
55:50 – 16. Merula – Quel sguardo sdegnosetto

Monteverdi’s Lettera amorosa is a particular delight.

What’s the use of video lectures?

I wrote this back in January 2021: but left it unpublished — not because I changed my mind after writing it, but because I thought it was rather too tactless and unhelpful at a time when so many were being forced to produce such lectures. But many months have past, so let’s ask again: what’s the use of video lectures?

Probably not a lot.

Or at least, not nearly as much as university admin (or some lecturers) like to think.

There’s a rightly famous book by Donald Bligh, first published fifty(!) years ago, What’s the Use of Lectures?  Bligh took a serious look at the available research and concluded that traditional lectures are just not very effective at promoting critical thought, inspiring students, or changing attitudes. What they are potentially good for is conveying  information. But only if very carefully organized, taking account of people’s capacity for concentrating over a period of time while absorbing new info. And even then, lectures are no more effective than other ways of teaching information — giving careful guidance pointing the student to appropriate chunks of decent textbooks, say.

Why suppose recorded video lectures — which so many university students under lockdown are being subjected to  — are any better than traditional lectures? After all, there’s not even the small sense of occasion, or the pleasure of being in the classroom with your friends with the chance to chat after. And more seriously, there’s not the useful discipline of being forced to pay close attention and organize your note writing in real time, because you can always rewind the video. So why suppose — and I guess it is university managers who are driving this — that very hard-pressed university teachers’ time is being best used in recording these lectures? It is very difficult to believe it is. (I’d probably say the same for ‘live’ large-class Zoom lectures too.)

To be sure, if you find some well-done online videos out there which you’d like to recommend to students as a possible resource, fine. But is it worth producing your own? The answer isn’t obviously ‘yes’.

It may vary from subject to subject. But when I’ve been trying to learn a new area of mathematics in recent years, I’ve found (of course) good textbooks invaluable. But I’ve also learnt a great deal from less formal lecture notes or handouts or relaxed expository papers which people have posted online,  from blog posts, and not least from the question-and-answer sites math.stackexchange and mathoverflow. I’m immensely grateful to all those who have put effort into these sorts of resources (and I’ve tried to do my bit in turn). But video lectures come way down the list of what I’ve found useful. For a start, they are usually a tediously slow way of conveying information; and to be honest, most people aren’t terrifically good at giving them (a student view too, in my admittedly very limited experience).

I’ve sometimes been asked whether I’d do some videos myself. But if I have any pedagogic skills, I’m quite sure they are better deployed elsewhere than adding to the pool of not-very-good straight lectures. The one video enterprise that I might be tempted to take on is examples classes. Yes, talking through examples — explaining why so-and-so might be a good proof strategy, backtracking from dead ends, with the proof developing onscreen in real time  — could indeed be useful and instructive for students. Video lectures, not so much.

Logic: A Study Guide now “launched”

I’ve now retired the mid 2020 version of the TYL guide, and marked the new year by officially launching a replacement, retitled as Logic: A Study Guide. (Although the Guide’s title has changed, its webpage address stays the same, so as not to break external links.)

The Guide now takes the form of two PDFs. The first contains the rewritten Chapters 1 to 7. That’s three preliminary chapters about the aims and structure of the Guide, and then four chapters on very-elementary (“naive”) set theory, FOL, elementary model theory, and arithmetic/Gödel’s theorems. These have already been posted here, and many thanks for the most useful comments so far.

The second PDF contains Chapters 8 onwards, renumbered but otherwise unrevised from the last version of TYL. As the weeks and months go by, the first PDF should grow as more newly revised chapters are added, and the second PDF will correspondingly shrink. Or at least, that’s the plan!

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