Year: 2012

Gaps in the truth about truth according to Leon Horsten?

I very much like Leon Horsten’s The Tarskian Turn as an attempt to make more widely available some recent work on formal axiomatic theories of truth,  without scaring readers off by excessive technicality or by unnecessarily spelling out tricky proofs. Students doing an advanced course on the notion of truth really ought to know the headlines about this  formal stuff, for a reason Tim Williamson has stressed:

One clear lesson [of these logical investigations] is that claims about truth need to be formulated with extreme precision, not out of kneejerk pedantry but because in practice correct general claims about truth often turn out to differ so subtly from provably incorrect claims that arguing in impressionistic terms is a hopelessly unreliable method.

So three cheers for Horsten’s efforts at lucid and lively explanation here. But his book is more than a merely expository essay. It is philosophically opinionated, takes sides, and even is prepared to endorse one particular axiomatic theory of truth as philosophically in good shape. This makes The Tarskian Turn engagingly provocative.

As is common in these sorts of investigations, Horsten takes as a test case the enterprise of adding a theory of truth to first-order Peano Arithmetc, PA. And the theory he recommends is PKF (that’s Feferman’s axiomatization of Kripke’s three-valued model, but redone in a Partial, gappy, logic). Roughly the idea is that you augment the language of PA with a truth-predicate T, take a rule version of Kleene’s strong logic, add the non-induction axioms of PA and the rule form of induction. Then we have the T-biconditionals of atomic sentences of PA, plus two-way rules that allow us to commute T with the logical operators: so, in Horsten’s sloppy but readable shorthand, we can infer $latex \neg T(\varphi)$ from $latex T(\neg\varphi)$ and vice versa, and similarly for other operators. Finally we have a two-way rule that allows us to infer $latex T(T(\varphi))$ from $latex T(\varphi)$ and vice versa.

The T-rules here are entirely natural, so PKF has all kinds of nice features. In particular, it is easily seen that everything remains classical for T-free sentences, and so classical PA can proceed undisturbed. So it is only some sentences involving T for which PKF’s non-classicality really matters and where e.g. excluded middle fails (where indeed we might want it to fail). Horsten thinks PKF is a best buy. Is it?

In his much more expansive, much more technically detailed book Axiomatic Theories of Truth Volker Halbach also investigates  PKF but is much less enthusiastic about it. Partly this is because of a general resistance to the idea of departing from classical logic if that can be avoided. And partly this is because of technical observations about the mathematical limitations of PKF (it can’t do much transfinite induction on open wffs involving the T-predicate). However, I don’t see why Horsten need be moved by these considerations. After all, logical regimentation always involves trade-offs of costs and benefits: and Horsten will say that given pure arithmetic remains classical, the cost of having to go non-classical in ones background logic (in ways that only really matter in cases that involve troublesome uses of the T-predicate) is a price worth paying for the benefit of skirting round paradox. And if that leads us to restrict transfinite induction in cases of no ordinary mathematical interest, why care?

But there are other worries about PKF. As Horsten notes, it isn’t that PKF rejects excluded middle in the troublesome cases, but rather it is silent. But is silence really what we want from a theory here? Horsten cheerfully says

The system PKF … is not vulnerable to a strengthened liar attack because it makes no claim concerning the truth value of the liar sentence. PKF simply does not assert the liar sentence, nor its negation, nor that it is true, nor that it is not true.

Indeed. But now the theory is out there, on the table for all to see, can’t we as philosophers stand back and reflect on it, and forcefully raise the question of the truth or otherwise of claims on which PKF fails to give a verdict? And off we go again … (To be sure there are philosophical positions where refusal to affirm or deny a putative proposition P is backed up with therapy that is supposed to massage away our temptation to suppose that P does express a contentful proposition. But Horsten is the business of theory not therapy.)

Horsten himself, then, is remarkably silent on what seems to be an obvious question. Instead he concludes his book by considering whether PKF is at least consonant with a broadly deflationist or minimalist stance. He thinks it is. For the theory treats truth as an insubstantial property without ‘a fixed nature or essence’ in the sense that there is no more to truth than is grasped in grasping some inference rules (though Horsten in fact  doesn’t rule out there being inference rules beyond those codified in PKF). But what about the fact that the compositional theory is non-conservative over arithmetic? Indeed PKF is arithmetically as strong as a transfinitely ramified system of predicative analysis that goes by the label $latex ACA_{\omega^\omega}$, whose first-order arithmetical consequences go far beyond those of PA.

Horsten is remarkably unworried. I think he is too swayed by a (quoted) claim of Feferman’s that suggests that systems of predicative analysis only elaborate commitments that are already implicit in accepting PA, a suggestion that runs clean against Isaacson’s well-known thesis (which I’ve defended elsewhere) that PA marks the natural boundary of those truths that can be reached by purely arithmetical reasoning. We can’t examine who is right about that here. But we might complain that neither does Horsten: he fails to acknowledge just how contentious it is to suppose that the progression through systems of predicative analysis stronger than the arithmetically conservative system $latex ACA_0$ can somehow be regarded as insubstantial rather than as involving new infinitary ideas (and so it remains equally contentious to suppose that a theory of truth arithmetically equivalent to a strong system of analysis can still count as deflationary). We might well dissent at the end of the book, then, about Horsten’s philosophical assessment of the merits of PKF.

Still, I think we should still be very grateful for a beautifully structured guided tour, with thought-provoking commentary, making some recent formal work on truth accessible to a wide student audience interested in the truth about truth (and accessible as well as to non-expert colleagues who want to know what the logicians down the corridor have been up to).

The consistency of NF

Randall Holmes has now announced “I believe that I am in possession of a fairly accurate outline of a proof of the consistency of New Foundations.”

He goes on to say “NF has the same consistency strength as TST + Infinity, has the same kinds of extensions as NFU in the same ways, has no interesting consequences for the combinatorics of small sets, etc. No surprises, this is a rather boring outcome in my opinion …” Well, I don’t know about boring! If Randall has indeed cracked this long-standing problem, it’s a major achievement. He is still editing the document, so nothing is released yet. However, Thomas Forster is organizing a conference here in Cambridge in the spring and Randall says he will “certainly be discussing this.”

And Thomas confirms  “I am indeed organising an NF meeting in Cambridge in the spring, current intention is last week of March and first week of April. The idea is that before Randall arrives there will be a warm-up act wherein the background and some preparatory material is set out for people who are not already familiar with it. Thus when Randall arrives we will all be primed and ready to go. … I am not at this stage soliciting other offers of talks, tho’ that may change. If you have something you think I may find irresistible by all means try to twist my arm. And – of course – contact me if you want to come.”

February 2013 concerts: Takacs and Pavel Haas Quartets

Just so you can’t complain if I later post rave reviews of concerts you didn’t know about …  The Takacs Quartet are going to be in Cambridge playing in the Camerata Musica season (the Hagen Quartet and András Schiff are also playing in the season early next year: which would be a quite astonishing line-up were it not for the fact that previous seasons have been equally stellar). And the Pavel Haas Quartet, whom I have enthused about here before, are playing at the Wigmore Hall on February 26 including the Beethoven Op. 130 with the Grosse Fuge. There are tickets left for both concerts as I post (but not for long, I imagine).

Set-theoretic excitements to come …

My friend Thomas Forster has posted an intriguing message to the FOM list, and some readers here might be interested. He writes that he is “in the process of assembling funding for a meeting to discuss and broadcast recent developments in NF.” The evolving plan is to have some meetings next year under the auspices of the  Isaac Newton Institute (who should be able to fund board and lodging for [at least some?] visitors). Thomas continues “At this stage I envisage making a start in late March … featuring introductory material, minicourses, tutorials, continuing into early April with presentations of recent results and discussions of their implications for foundational projects more generally.” My lips are sealed, for the moment, about the set-theoretic excitements which have prompted this occasion. But watch this space. If you wish to be put on the mailing list for this event, please send an email to tf@dpmms.cam.ac.uk.

Teach yourself logic, #4: Beginning set theory (again)

Since my last post, I’ve had second thoughts. Given the number of books on set theory out there which are worth a mention at some point or other even in a very selective reading list, I’ve decided it would plainly be much better to divide the appropriate chunk of the Teach Yourself Logic Guide into two sections, one on the elements of set theory, another carrying the story forward.

What do I mean by ‘elements’ here? The contents of e.g. Enderton’s The Elements of Set Theory. That is to say, the construction of the natural number and real number systems in set theoretic terms, the development of cardinal and ordinal arithmetic, and stuff about the use of axiom(s) of choice and its importance. So understood, the elementary stuff stops before we start discussing the likes of large-cardinal axioms, or get into proving that the Axiom of Choice is consistent with ZF, or the continuum hypothesis is independent of ZFC.

Ok, so with this more restricted brief of giving a reading guide to books covering the elements of set theory at a suitable pace for beginners, what might a sensible list look like? Well, you can see my draft list now as the final section of the updated Guide, which you can download here.

Do have a look: suggestions and comments are extremely welcome!

Teach yourself logic, #4: Beginning set theory

Here, after rather a long gap, is another instalment in the “Teach Yourself Logic” series. For new readers: given the dire state of logic teaching in some grad schools in philosophy (especially in the UK), I’m trying to put together a helpfully annotated reading list. The aim is to give students needing to teach themselves some logic a Guide through the daunting yards of books that are (or ought to be) in their university library. The reading list might be helpful to some mathematicians too. What we’ve covered up to now falls into four sections:

1. Back to the beginning
2. Getting to grips with first-order logic
3. Modal logic
4. From first-order logic to model theory

And here’s an edited version of the list so far. But I’m next going to jump out of sequence to what is planned to be §9 of the Guide, covering set-theory, since that’s what I happen to be thinking about right now. So now read on …

Where to start?

• Derek Goldrei, Classic Set Theory (Chapman & Hall/CRC 1996) is written by a lecturer at the Open University in the UK and has the subtitle ‘For guided independent study’. It is as you might expect extremely clear, and it is quite attractively written (as set theory books go!).
• Winfried Just and Martin Weese, Discovering Modern Set Theory I: The Basics (American Mathematical Society, 1996). This covers overlapping ground, but perhaps more zestfully and with a little more discussion of conceptually interesting issues, though also it is at some places more challenging (the pace can be uneven). But this is evidently written by enthusiastic teachers, and the book is engaging.

My next suggestion some might find a bit surprising, as it is a blast from the past. However, both philosophers and mathematicians ought to appreciate the way it puts the development of our canonical ZFC set theory into some context, and also discusses alternative approaches:

• Abraham Fraenkel, Yehoshua Bar-Hillel and Azriel Levy, Foundations of Set-Theory (North-Holland, 2nd edition 1973). This really is attractively readable, and should be very largely accessible at this early stage. I’m not an enthusiast for history for history’s sake: but it really is worth knowing the stories that unfold here.

One intriguing feature of that last book is that it nowhere mentions the idea of the ‘cumulative hierarchy’ — the picture of the universe of sets as built up in a hierarchy of  levels, each level containing all the sets at previous levels plus new ones (so the levels are cumulative). This picture — nowadays familiar to every beginner — comes to the foreground in

• Michael Potter, Set Theory and Its Philosophy (OUP, 2004). For philosophers (and for mathematicians concerned with foundational issues) this is — at some stage —  a ‘must read’, a unique blend of mathematical exposition and conceptual commentary.  Potter is presenting not straight ZFC but a very attractive variant due to Dana Scott whose axioms more directly encapsulate the idea of the cumulative hierarchy of sets. However, it has to be said that there are passages which are pretty hard going — sometimes because of the philosophical ideas involved, but sometimes because of unnecessary expositional compression. In particular, at the key point at p.  41 where a trick is used to avoid treating the notion of a level (i.e. a level in the hierarchy) as a primitive, the definitions are presented too quickly, and I know that real beginners can get lost. However, if  you have read Just and Weese in particular, you should be able to work out what is going on and read on past this stumbling block.

The books mentioned so far don’t mention but don’t treat (1) independence and consistency results, and though Potter mentions (2) large cardinals, again this is without any development. For a first look at (1), a possibility is

• Keith Devlin The Joy of Sets (Springer, 2nd end. 1993). This is again well written, but goes significantly faster than Goldrei or Just/Weese,  Chs 1–3 giving a fast track coverage of some of the material in those books. Later chapters in this compact book introduce more advanced material. In particular, Ch. 5 discusses Gödel’s notion of constructible sets, Ch. 6 uses “Boolean valued” sets to prove the independence of the Continuum Hypothesis, and Ch. 7 considers what happens if you allow non-well-founded sets (infinite downward membership chains). But all this is done pretty speedily, which may or may not appeal.

But you could well jump over Devlin and go straight to a more comprehensive treatment of independence proofs, the self-selecting

• Kenneth Kunen Set Theory: An Introduction to Independence Proofs (North Holland, 1980), rewritten as his Set Theory* (College Publications, 2011). The first version is a modern classic, used in many university courses: the new version is a timely update and — on a fairly brief inspection — looks to have all the virtues of the earlier version, but acknowledging thirty years of progress.

And then of course — if you’ve got the set-theoretic bug — the modern bible awaits you in the form of the rather monumental

• Thomas Jech, Set Theory: The Third Millenium Edition (Springer, 2003).

Which is more than enough to be getting on with!

However, as an afterthought in small print for the list, we might usefully add a handful of other books that seem to me of particular interest for one reason or another. These could all be read after Goldrei or Just/Weese before or alongside Kunen. In no special order

• Thomas Forster, Set Theory with a Universal Set: Exploring an Untyped Universe (OUP, 2nd end. 1995). Focuses on Quine’s NF and related systems. It is worth knowing something about alternatives to ZFC.
• Thomas Jech, The Axiom of Choice* (North-Holland, 1973: reprinted by Dover Books 2008). Readable, attractively short, and will tell you about a variety of constructions (including Fraenkel-Mostowski models and Cohen forcing).
• Raymond Smullyan and Melvin Fitting, Set Theory and the Continuum Problem*  (OUP 1996, revised edition by Dover Books 2010). Famously lucid authors trying to make hard proofs accessible and doing a good job.
• If you were intrigued by some of the historical material in Fraenkel/Bar-Hillel/Levy then you should enjoy José Ferreirós, Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics (Birkäuser, 2007). This is slightly mistitled — it is the history of early set theory, stopping around Gödel’s relative consistency results. But a very interesting read.

I could go on! Should I have mentioned Moschovakis’s nice Notes on Set Theory earlier on in the main list? Added Bell on Boolean-Valued Models to the small print?? Even added Halbeisen’s interesting new book on Combinatorial Set Theory??? But let me pause here.

Added later I surely should have mentioned at the outset Enderton’s book Elements of Set Theory (what a pity it isn’t available as a cheap Dover reprint!). And I am beginning to wonder about splitting this burgeoning section of the Guide into two sections — ‘Beginning Set Theory’ and ‘Continuing Set Theory”.

Postcard from Cambridge

Sometimes, at the beginning of the academic year, Cambridge goes into a grey sulk. But this year the weather has been wonderful, and walking or cycling across the common into town has been a delight. (Note the Red Poll cattle, back grazing a few hundred yards from the centre of town, shortly no doubt to appear at the Sunday market …)

OK: back to business! Having been distracted from other projects by getting Gödel Mark II off to press, I’m slowly getting into thinking about ordinals again now I’m back from Berlin. So (although the intended book is provisionally titled Ordinals Without Sets) I’m doing an amount of set-theoretic homework. No doubt I am doing far more than I really need, somewhat painfully scraping off years of rust. But I’m sure this will be Very Good For Me (gym for the mind).

I’ve also just started to get back to that stuttering project of putting together a Teach Yourself Logic reading list, and will post the next instalment of that very shortly. Here in the meantime is an updated version of the list so far.

Postcard from Berlin

We have been staying in Berlin with The Daughter (my first visit there since I crossed Europe in an A30 van with two friends as a student, driving through Checkpoint Charlie en route to Warsaw … but that was in another life). The Daughter’s temporary apartment is in old East Berlin, fairly central, near the Senefelderplatz U-bahn station. Rents are still low in Berlin, so little shops and small local cafés with just a handful of tables abound. And the cafés, at least, seem to be thriving: all different, offering different little menus, run by very welcoming young people. We’ve had some great coffee, nice meals, and love the atmosphere. And we much regret the lack of anything comparable here in Cambridge. It has been warm enough to sit at outside tables often enough, usually accompanied by crowds of chattering sparrows pecking at dropped crumbs (a much rarer sight in England than they used to be).

Getting around Berlin on trams and the U-bahn is fun (very cheap, very efficient — city public transport as it should be). And we’ve been bowled over by the museums and galleries: in six days we’ve perhaps had our fill for this trip, but we’ll have to be back soon. The Pergamon Museum is, as everyone says, amazing. But we’ve also especially enjoyed a couple of visits to the stunning collection of paintings at the Gemäldegalarie (and on both occasions the gallery seemed almost empty, at least by the oppressive standards of the National Gallery in London). You can see something of those paintings in the Google Art Project pages here. Simply wonderful.

Whoosh …!!!

No, no, that isn’t the sound of another deadline flying by. Can’t you hear the difference? It’s the cheerier noise made as a weighty email takes off and departs to the publisher, freighted with a PDF of Gödel Mark II.

Of course, that’s not the end of the matter. The wheels at CUP will start turning, and eventually I’ll get a report from their proof-reader, and have to get back to tinkering again. But for a number of weeks I can put that book aside. I feel oddly bereft. I’ll have to get working on another …

With a little help from my friends

All editing and no play makes Jack a very dull blogger. Sorry! Must do better. But I’ve promised to get the Gödel book off to CUP in ten days, so I’m doing a last read-through, mostly obsessing about minutiae, though I also have to sort a few minor bugs in the last chapters caught by sharp readers. And that is rather all-consuming.