Peter Smith's Blog, page 66

IFL2: Natural deduction for quantifier logic

Here is a tranche of draft chapters for IFL2. Chapters 25–30 are revised chapters on QL languages (informal syntax and semantics, how to translate in and out). Chapters 31–34 are newly drafted chapters on natural deduction for QL proofs (the system is standard, and I hope contains no surprises).

The following chapters will give a more formal semantic story, define q-validity etc.; but I think it is good to get natural deduction proofs up and running in an intuitive way first (after all, they are supposed to be fairly natural!).

All comments, especially on the new Chapters 31–34. In particular, chapter 34 is a stand-alone short chapter on empy domains, which significantly revises an earlier draft section in response to some very just comments.

The post IFL2: Natural deduction for quantifier logic appeared first on Logic Matters.

IFL2: Chapters on propositional natural deduction (revisited)

Here is an improved version of the five chapters on propositional natural deduction for IFL2. (Exercises to be added, which will fill some gaps, like noting the equivalence of DN and Classical Reductio, given the other rules, or dealing with biconditionals.)

As I have said before, one advantage of basing an intro logic book on trees (as in IFL1) is that people don’t get very exercised about how a tree system should be developed. By contrast, people get decidedly heated about the best form of natural deduction to adopt. And I have changed my mind more than once about how to do things (having explained things rather differently in classes over the years). I have ended up with a more conservative Fitch-style system than in previous drafts. But that is still probably not going to satisfy half of those who urged me to go for natural deduction in the second edition: it doesn’t even fully satisfy  me. But just is there’s no meeting all the desiderata!

As always, all comments (other than variants on “you have written the wrong book!”) and all corrections will of course be very gratefully received.

The post IFL2: Chapters on propositional natural deduction (revisited) appeared first on Logic Matters.

Postcard from Cornwall

Evening sun at St Mawes in Cornwall, where we have been again for a while. So no logic matters for the last fortnight (the book has not been nagging away insistently enough to the break the spell). And we’ve been trying to ignore Brexit too (which all looks even madder than ever). So a lot of walking along coastal paths, visits to some great gardens, and more walking. Or just sitting in the apartment watching the quiet comings and goings in the harbour below. All balm to the soul.

Back to real life and builders, not to mention IFL2, in a couple of days …

The post Postcard from Cornwall appeared first on Logic Matters.

Schiff’s Schubert

I can’t let one standout recent double CD release pass without the warmest recommendation. András Schiff has recorded more late Schubert — this time the Four Impromptus D899, the Drei Klavierstücke D946, and the Piano Sonatas D958 and D959. Schiff of course recorded all these about thirty years ago in much admired performances. But this time, as with another double CD a couple of years ago, he is playing on a Brodmann fortepiano of about 1820. And again the effect is quite magical.

Here is Andrew Clements writing in the Guardian about a live performance at Wigmore Hall on this instrument:

Such a piano may lack the tonal power of its modern counterpart, but as Schiff’s Schubert performances demonstrated so eloquently, the illumination it brings to works to which it’s exactly matched chronologically and geographically is extraordinary. The distinct characters of the top, middle and bass registers, together with the effects produced using the pedals (four on Schiff’s instrument) added extra layers of articulation and transparency to the music, while the intimate, contained soundworld complemented Schiff’s introspective view of these works perfectly.

And here is the end of Katherine Cooper’s fine review of the new recordings

In [an interview], Schiff defines Schubert by ‘his modesty, his humility, his lack of ego’; the same qualities are evident in every bar of these performances, which quietly command absolute attention rather than clamouring for it. These are profoundly affecting interpretations born out of a long-term loving relationship with both music and instrument, and ones to which I can see myself returning for years to come.

Me too! If you are sceptical — as indeed I confess I was before hearing Schiff’s previous fortepiano recording — then do try [from your favourite streaming service!] one of my favourite of Schubert’s shorter pieces, the second of the Drei Klavierstücke. I find Schiff’s touching performance of a piece I thought I knew so well to be wonderful beyond words, or at least, beyond any words I can muster.

The post Schiff’s Schubert appeared first on Logic Matters.

Nietzsche vs the philosophical philistine

Tom Stern, who was a graduate student here in Cambridge while I was still in the Faculty, has edited The New Cambridge Companion to Nietzsche, newly published by CUP. I’ve dipped into it. The editor’s “Introduction: Nietzsche’s Life and Works” is characteristically lucid. But I was very struck by his peroration:

Looking at the different ‘Nietzsches’ described in this introduc- tion – the phases of his writing, the varying interpretations, textual complexities, stylistic challenges and the likely unfamiliarity of his historical context – the non-specialist reader may be tempted to despair of ever finding a stable, satisfactory view of his ideas. One could offer many responses to such perfectly understandable despair: that Nietzsche may have cultivated it, and certainly to some degree deserves it; that some ideas nonetheless appear often enough, and with sufficient force, to be ascribed to him; that often there is, if not critical consensus, at least a shared sense of the available options, with their strengths and weaknesses. But perhaps the best reply would be that, whatever Nietzsche thought, the confrontation with his texts and his interpreters has repeatedly proven itself to be enormously fruitful. When reading his works, or a Companion such as this, you will probably meet some thought which lights you up. And it might even be one of Nietzsche’s.

And there you have why I can’t find much enthusiasm for writers like Nietzsche. I do read a fair amount of literary fiction and poetry. But when it comes to philosophy, I really don’t want to be “confronting” texts, I don’t want “stylistic challenges”, I don’t want to tangle with “varying interpretations”. I just want the directest plain talk, as explicit as possible, with reasoning  laid out on the table and the steps signalled as clearly and frankly as may be. I’m not sure I particularly want to be “lit up” — I much rather want to see honest toil in working through the details of hard arguments. So in philosophy, I guess I’m just a bone-headed philistine. (Or still a mathmo at heart, and not a real philosopher at all.)

The post Nietzsche vs the philosophical philistine appeared first on Logic Matters.

Empty domains: Must do better

In §11.1 of their Plural Logic, Alex Oliver and Timothy Smiley argue — with characteristic vigour — that an acceptable singular logic (as well a plural logic) should allow empty domains. Their two related headline thoughts:

Logic should be topic-neutral, and one thing it should be neutral about is whether the domain is empty or not (after all, we can want to make deductions about e.g. sets or superstrings even if we doubt that there are such things).
Standard logic makes the likes of a logical theorem:  but a wff of that kind is not in general a logical truth.

Now, you don’t have to buy their defence of free logic to acknowledge that Oliver and Smiley are right about one thing — namely that many of the textbooks are pretty feeble in what they say by way of explanation/defence of the standard quantifier rules and their presumption that domains are populated. They give half a dozen examples. Let’s also take  two further widely adopted textbooks which they don’t mention.

Bergmann, Moor and Nelson in The Logic Book just tell us that domains are non-empty sets, without comment (unless my eye has skipped over the relevant discussion). They note that the likes of are — in their phrase — quantificationally true. But they don’t discuss (again, unless my eye has skipped) the relation here between being quantificationally true and being logically necessary.
While Barker-Plummer, Barwise and Etchemendy in Language, Proof and Logic offer just this:

In FOL we always assume that the domain of discourse contains at least one object and that every individual constant in the language stands for an object in that domain. (We could give up these idealizations, but it would complicate things considerably without much gain in realism.)

Oliver and Smiley might reasonably protest: Why isn’t being able to argue about some domain while being neutral about whether it is empty not a major gain in realism? And it is just a fib that allowing empty domains, at least, complicates things “considerably”.

Now, I think there should be a strong bias towards sticking to standard rules in an introductory text.  But I do think a text Must Do Better that is often that case, in at least acknowledging that there are issues here, and it should be frank that debatable choices are being made. So here is my effort at writing something on empty domains for the second edition of IFL. Comments? Thoughts? Suggestions for improvements?

The post Empty domains: Must do better appeared first on Logic Matters.

Category theory page updated

For those interested in category theory, this is just to say that the page here of links to online resources (at intro and mid-level) has been updated, broken links repaired, a couple of links added. Please spread the word, and also please let me know about recent additions to the available materials. Since right now I’m not working in this area, I could very well have missed things.

For those not interested in category theory, here’s the view from Garret Hostel Bridge walking through town on Easter Sunday …

Before I retired it was on my most frequent route to walk or cycle to the Faculty … and the view has been so familiar for half a century. But it still lifts the heart.

The post Category theory page updated appeared first on Logic Matters.

PHQ play Shostakovich

To the Wigmore Hall, to hear the Pavel Haas Quartet playing Shostakovich’s second and seventh string quartets in a lunchtime concert. PHQ are recording these plus the wonderful eighth quartet next month for Supraphon, with a planned October release. So they have been playing Shostakovich a lot in their recent concerts, and oh heavens it showed. These were stupendous performances with all PHQ’s usual commitment and passion but so much fine detail and immense control. The audience gave them a terrific reception — so a lot of smiles after all the tension in the music.

More than worth the journey to London to be there — for isn’t this music where you really want to be in the audience for the drama? But the concert was broadcast on BBC Radio 3. You can catch up on BBC Sounds here (from 2.20 in). And the concert will be broadcast again on Sunday on Radio 3 at 13.00.

The post PHQ play Shostakovich appeared first on Logic Matters.

The fragility of beauty

Notre Dame, Paris

The post The fragility of beauty appeared first on Logic Matters.

IFL2: Chapters on QL languages

One of the parts of the first edition of IFL that I liked rather better, and which seemed to work with students, was the bunch of chapters motivating the quantifier-variable notation and introducing the standard language of quantification theory. Which isn’t to say that the chapters have survived into IFL2 without some radical rewriting. But I hope the heavily revised versions still work! I think, at any rate, they are clearer than many about what is the essential Fregean insight, and what are artefacts of supplementary and optional decisions (e.g. because we decide to keep the quantifiers unary and single-sorted).

So here for your delight and delectation are the draft Chapters 25 to 29. All comments most gratefully received — ideally in the next five weeks or so, as June is the month for getting the final corrections/revisions of the book done before sending it off to CUP. Ever the optimist, here!

The post IFL2: Chapters on QL languages appeared first on Logic Matters.