# Year: 2014

## Quick book note: Stewart Shapiro’s Varieties of Logic

Stewart Shapiro’s very readable short book Varieties of Logic (OUP, 2014) exhibits the author’s characteristic virtues of great clarity and a lot of learning carried lightly. I found it, though, to be uncharacteristically disappointing.

Perhaps that’s because for me, in some key respects, he was preaching to the converted. For a start, I learnt long ago from Timothy Smiley that the notion of consequence  embraces a cluster of ideas. As Smiley puts it, the notion “comes with a history attached to it, and those who blithely appeal to an ‘intuitive’ or ‘pre-theoretic’ idea of consequence are likely to have got hold of just one strand in a string of diverse theories.” Debates, then, about which is the One True Notion of consequence are likely to be quite misplaced: for different purposes, in different contexts, we’ll want to emphasize and develop different strands, leading to different research programmes. As Shapiro puts it, the notion(s) of consequence can be sharpened in different ways — and taking that point seriously, he suggests, is already potentially enough to deflate some of the grand debates in the literature (e.g. about whether second-order logic is really logic).

And I’m still Quinean enough to find another of Shapiro’s themes congenial. Do we say, for example, that ‘or’ or ‘not’ mean the same for the intuitionist and the classical mathematician? Or is there a meaning-shift between the two? Shapiro argues that for certain purposes, in certain contexts, with certain interests in play, yes, we can say (if we like) that there is meaning shift; given other purposes/contexts/interests we won’t  say that. The notion of meaning is maybe too useful to do without in all kinds of situations; but it is also itself too shifting, too contextually pliable, to ground any grand debate here.

Put it this way, then. I’m pretty sympathetic with Shapiro’s claims that some large-scale grand debates are actually not very interesting because not well-posed. What that means, I take it, is that we’ll in fact find the interesting stuff going on a level or two down, below the topmost heights of cloudy generality, in areas where enough pre-processing has gone on to sharpen up ideas so that questions can be well-posed.

Here’s the sort of thing I mean. Take the very interesting debate between those like Prawitz, Dummett and Tennant who see a certain conception of inference and the logical enterprise as grounding only intuitionistic logic (leaving excluded middle as a non-logical extra, whose application to a domain is to be justified, if at all, on metaphysical grounds), and those like Smiley and Rumfitt who argue that that line of thought depends on failing to treat assertion and rejection on a par as we ought to do. This debate is prosecuted between parties who have agreed (at least for present purposes) on how to sharpen up certain ideas about logic, consequence, the role of connectives, etc.,  but still have an argument about how the research programme should proceed.

Shapiro doesn’t mention that particular debate. Absolutely fair enough (I just plucked out something that happens to interest me!). The complaint, though, is that he doesn’t supply us with much by way of other illustrations of investigations of varieties of logic at a level or two below the most arm-waving grand debates — i.e. at the levels where, by his own account, the real action must be taking place. Hence, I suppose, my general disappointment.

Shapiro does however mention a number of times one interesting example to provide grist to our mills, namely smooth infinitesimal analysis. This, if you don’t know it, is a deviant form of infinitesimal analysis — deviant, at any rate, from the mathematical mainstream. (If you look at Nader Vakil’s recent heavy volume Real Analysis Through Modern Infinitesimals  in the CUP series Encyclopedia of Mathematics and Its Applications, then you’ll find smooth analysis gets the most cursory of mentions in one footnote.) The key idea is that there are nil-potent infinitesimals — at a rough, motivational level, quantities so small their square is indeed zero, even though they are not assumed to be zero. More carefully, we have quantities $$\delta$$ such that $$\delta^2 = 0$$ and $$\neg\neg(\delta = 0)$$, but — because the logic is intuitionistic — we can’t assert $$\delta = 0$$. And then, the key assumption, it is required that for any function $$f$$, and number $$x$$, there is a unique number $$f'(x)$$ such for any nil-potent $$\delta$$, $$f(x + \delta) = f(x) + f'(x)\delta$$. So looked at down at the infinitesimal level, $$f$$ is linear, and $$f'(x)$$ gives its slope at $$x$$ — so is the derivative of $$f$$. Now it turns out that, with enough assumptions in place, this theory allows us to define integration in a correspondingly natural way, and then we can readily prove the usual basic theorems of analysis.

Now that is indeed interesting. But — and here’s the rub — the internal intuitionistic logic is absolutely crucial. The usual complaint by the intuitionist is that adding the law of excluded middle unjustifiably collapses important distinctions (in particular the distinction between $$\neg\neg P$$ and $$P$$).  But in the case of smooth analysis, add the law of excluded middle and the theory doesn’t just collapse (by making all the nil-potent infinitesimals identically zero) but becomes inconsistent. What are we to make of this? In particular, what can the defender of classical logic make of this?

I guess there is quite a lot to be said here. It is a nice question, for example, how much sense we can make of all this outside the topos-theoretic context where the Kock-Lawvere theory of smooth analysis had its original home. To be sure, as in John Bell’s A Primer of Infinitesimal Analysis, we can write down various axioms and principles and grind through deductions: but how much understanding ‘from the inside’ does that engender? Shapiro says just enough to pique a reader’s interest (for someone who hasn’t already come across smooth analysis), but not enough to leave them feeling they have much grip on what is going on, or to help out those who are already puzzling about the theory. And that’s a real disappointment.

## Philosophers being offensive

If philosophers want to be really offensive, at least do it with style.

I’m reminded of a story about my favourite Cambridge philosopher, C.D. Broad. Not entirely a nice man.

A long time colleague of his at Trinity was the organic chemist Frederick Mann, whom Broad evidently thought an uncultured dullard. One evening at High Table, Broad finds himself unfortunately sitting opposite Mann. Broad beams cherubically. Pauses. And, to no-one in particular, sighs “Ah, where every prospect pleases …”.

## The Very Short Teach Yourself Logic Guide

I seem to have gone full circle! The very first instalment of the TYL Study Guide was a short blogpost here. Then things grew. And grew. Until we get to the current 100 page PDF monster — and that’s only as short(!) as it is because some material has been exported to the Appendix and a supplementary webpage on Category Theory. Of course, the long version is full of good things, explains why the chosen texts are recommended, explains why others aren’t, suggests alternatives and supplementary reading, and more. But still, some might be interested in just getting the headline news.

So now, rather in the spirit of the original post, there’s an encouraging short and snappy page, The Very Short Teach Yourself Logic Guide, which just gives you the winners, the top recommendations for entry-level reading on the various areas of the core math. logic curriculum. (If you want to know why they are the winners, then you will have to look in the corresponding section of the full version of the Guide.)

## Gillian Bennett: on doing the brave, decent and loving thing

If you happened not to see her final testimony and plea for the legalization of assisted suicide, then do read these very moving webpages left by Gillian Bennett (the wife for 57 years of that fine philosopher Jonathan Bennett) who faced with the ravages of dementia, ended her life last month.

## One little “}”

No, you didn’t need new spectacles. One little “}” missing, and the last half of the lovingly crafted TYL version 12.0 was all in the smaller font intended for postscripts and asides. Pah! Sometimes $$\LaTeX$$ is annoying. Just a tinsy bit.

OK, so here’s version 12.0a, which is at least easier on the eyes.

2015 update here

## Teach Yourself Logic, version 12.0

In time for the new semester/new term/new academic year (depending on how things are chunked up in your neck of the woods), there’s a new version of the Teach Yourself Logic Study Guide and a supplementary page on Category Theory, both downloadable from the Guide’s usual page.

The Guide has been restructured into Parts in a different, more logical way, to make navigating though the 95 pages easier. There have been quite a few changes in the recommendations (e.g. on FOL and model theory). The final section on serious set theory has been restored and improved. There’s even now an index of authors’ names. What’s not to like?

The Guide seems to get used quite a lot (one previous version was downloaded almost three thousand times), which is why it seems well worth spending time on it occasionally. But I’m pretty happy with the current structure and content, so I hope that for a while the main Guide will only need minor tinkering to keep it in good shape (though there might be some more supplementary pages still to come).

As usual, please do let me know if you spot typos, or indeed if you have more substantive comments!

## Model theory without tears?

Ah well, you win some and you lose some. I was writing for months about recursive ordinals and proof theory with a view to a short-ish book. And now, quite a way in, I realise that I have to go back to the drawing board, do a lot more thinking and reading if I am to have anything both interesting and true to say, and then (maybe) start over. Well, at least being retired I don’t have ‘research productivity’ (or whatever it is currently called) to worry about. Frustrating, though.

As a distraction, initially just with a view to updating one of the less convincing parts of the Teach Yourself Logic Guide in the next version, I’ve been looking again at some of the available treatments of elementary model theory. One immediate upshot is that there are new pages (briefly) on Jane Bridge’s Beginning Model Theory and (rather more substantially) on María Manzano’s Model Theory linked along with some other recent additions at the Book Notes page.

Now, the books by Bridge and Manzano have their virtues, of course, as do some other accounts at the same kind of level, But still, the more I read, the more tempted I am to put my hand to trying to write my own Beginning Model Theory (or maybe that should be a Model Theory Without Tears to put alongside Gödel Without Tears). The exegetical space between a basic treatment of first-order logic and the rather sophisticated delights of (say) Wilfrid Hodges’s Shorter Model Theory and David Marker’s Model Theory isn’t exactly crowded with good texts, and it would be fun to have a crack at. And unlike the recursive ordinals project, at least I think I understand what needs to be said! Which is a good start …

## Hilbert’s Foundations/Logic Lectures

I’ve just been spending a couple of days looking at the massive volume of David Hilbert’s Lectures on the Foundations of Arithmetic and Logic 1917-1933, edited by William Ewald and Wilfried Sieg (Springer 2013), which has at last arrived in the library here.

The original material is all in German (sadly but understandably untranslated), and since my grasp of the language is non-existent, what I have been reading is in fact the general editorial introduction, and the introductions to the various chapters of book covering different lecture courses and supplementary material (like Bernays’s Habilitationsschrift published here for the first time). There’s a lot of other editorial apparatus (the whole project seems to have been done to an extremely high standard), but these discursive introductions themselves amount to upwards of 130 pages. They are extraordinarily interesting and illuminating even if you can’t (or simply don’t) read the texts they are introducing. True, some of the material in these introductions overlaps heavily with Sieg’s essays already published in his Hilbert’s Programs and Beyond, but it is all still well worth reading (again). So this is a volume your university library should certainly get: and not just for you to leave it on the shelf and admire from afar!

I take it that few people by now cleave to the old myth — propagated e.g. by Ramsey —  that Hilbert was a gung-ho naive (or even not-so naive) formalist. But that myth (already surely fatally damaged by Sieg and others) should be well and truly buried by the publication of these various lecture notes which witness Hilbert’s developing positions over the 1920s as he explores the foundations of arithmetic.

Let me just remark on two things that struck me — not about the development of Hilbert’s program(s) and the search for consistency proofs, however, but about his contribution to the modern logic. First, it is now clear that the wonderful book by Hilbert and Ackermann published in 1928 isn’t the fruit of a decade’s intensive work after Hilbert’s 1917 return to thinking hard about foundational matters. Rather, the early sections of that book are based very closely on notes for a 1920 lecture course ‘Logik-Kalkül’ prepared by Bernays, and then the core of the book is equally closely based on Bernays’s notes for a 1917/1918 course ‘Prinzipien der Mathematik’ (Ackermann’s contribution to the substantive content of the book indeed seems to have been markedly less than that of Bernays). So suddenly, within seven years of the first volume of Principia (which seems now to belong to a remote era), Hilbert has the makings of a logic text of a recognisable form can still be read with profit. That really is an astonishing achievement.

But second, there are more key ideas about logic in those early lectures which are left out of the book. Thus, in the 1920 lectures, the editors tell us

The logical calculus seems to have been designed to present propositional and first-order logic in a purely rule-based form which allows logical calculations to be presented as they naturally arise within a mathematical proof, and thus to furnish an analysis of logical inference and of the activity of mathematical reasoning.

This aspect of Hilbert’s logical investigations is lost from view in the later book Hilbert and Ackermann 1928 , where the rule-based version of quantificational logic is omitted altogether, and where the canonical versions of both sentential and first-order logic are presented axiomatically. But in these lectures the goal is to obtain a more direct representation of mathematical thought. In the 1922/23 lectures, Hilbert would formulate a calculus that presents, axiomatically, the elimination and introduction rules for the propositional connectives. Here, in early 1920, … Hilbert describes his rules for quantificational logic as ‘defining’ or ‘giving the meaning’ of the quantifiers. (pp. 279-80)

So here then already are intimations of ideas that Bernays’s student Gentzen would bring to maturity a dozen years later. Remarkable indeed.

There’s a lot more of equal interest in the editors’ wide-ranging introductory essays. Warmly recommended.

## Cutting the TYL Guide down to size

The Teach Yourself Logic Guide was getting rather ridiculously bloated — 138 pages in the previous version. Oops. That was getting distinctly out of hand. I was losing sight of the originally intended purpose of the Guide.

Time to re-boot the project!

So there’s now a new version of the Guide available that weighs in at a much trimmer 78 pages (OK, that probably still sounds a lot, but the layout involves small pages and largish print for on-screen reading, and the first quarter is very relaxed pre-amble). Some of the now deleted material has been re-packaged as supplementary webpages. So I hope that the resulting Guide looks a lot less daunting both in size and coverage. It should certainly be easier to maintain, having divided the core Guide from the supplements which can be updated separately.

The Guide and the add-ons can be accessed here. Spread the word to your students (or if you are a student yourself, I do hope you find something useful here).

## Category theory in two sentences

Tom Leinster’s book Basic Category Theory  arrived today on the new book shelves at the CUP bookshop.  I just love the opening two sentences, which seem about as good a minimal sketch of what category theory is up to as you could hope for:

Category theory takes a bird’s eye view of mathematics. From high in the sky, details become invisible, but we can spot patterns that were impossible to detect from ground level.

That’s a brilliantly promising start: and thirty pages in, the book is still proving a really good, if moderately taxing, read.

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