Year: 2011

Under reconstruction

(Friday) I’m planning, over the next day or three, to experiment with updating Logic Matters with a classier new WordPress theme. I’m so far favouring the “Tarski” theme, not just because the name seems peculiarly appropriate for a logic blog, but because it is really clean and suitable for a text-heavy site.

(Saturday) For a while, I’m afraid some navigation may not be quite optimal. But you should now mostly be able again to find what you are looking for.

(Saturday later) Huh. Have for the moment gravitated back to the new WordPress default “Twenty Ten” theme …

(Saturday evening) Beginning to aesthetically tweak the Twenty Ten theme. Most navigation restored. Wot fun!

(After midnight) This style-tweaking malarky is mildly addictive and probably already past the point of worthwhile returns. To bed …

(Sunday morning) I’ve a stable “child” of the Twenty Ten theme working which I quite like. But I’m going to be experimenting with an alternative for an hour or so (Weaver, also based on Twenty Ten).   Things could look horrible for a bit, since Weaver’s defaults are crap. But it offers ease of customization without so much digging into css files. So here goes …

Huh. Why, Mr Weaver, write a front end for customizing all kinds of things — but not the most important? Text size, line spacing, and para spacing? Back to my hand-kludged “child” theme. Now I need to get a header sorted.

(Sunday evening) Well, I still need to design a header, but I think I’ve done enough intermittent tinkering for a while. Basically the layout is a just slightly tweaked version of the new default WordPress theme; it seems to work well enough in Safari and Firefox on Macs. So if it doesn’t play nicely with your browser, then I guess you should blame WordPress or the browser. (Reads nicely on an iPad, though I say so myself!)

(Monday) Well, maybe Bauhaus austerity, no fancy header graphic, is the way to go. Right: for the moment, redesign job largely done. And marking M. Phil. essays is over too. So it’s back to thinking about ordinals … and also, at long last, back to reading Alan Weir’s book. I’ll be taking up blogging about that again next week. Watch this space!

The MRDP theorem

I gave a rough-and-ready talk yesterday, introducing the MRDP theorem to some logic-minded philosophers (mostly postgrads). The aim was to explain what the theorem says and why it is interesting, rather than to talk about the proof in detail. Here’s a slightly tidied version of my rough-and-ready handout.

I don’t pretend to know a great deal about this stuff, so I’d like to hear about needed corrections: and suggestions for additions will be most welcome too.

(Added 4 March) Comments from David Auerbach and Warren Goldfarb have prompted some small improvements!

Imogen Cooper plays Schubert … on YouTube

Thanks to Askonas Holt, her agents, three videos of Imogen Cooper playing at a concert in 2009 have just been posted on YouTube (the video isn’t HD, but the sound is just fine). There is a nice performance of Schubert’s Hungarian Melody D817, and a lovely short piece of Janacek, ‘Good Night’ from On an Overgrown Path, which I haven’t heard for ages.

But then, on a quite different scale of length and emotional intensity, she is joined by Paul Lewis for a stunning performance of the Schubert Fantasie D940. And this is surely as good as it gets: two of the greatest Schubert pianists seemingly as one in their shared vision of the piece. Just wonderful.

Reading list on computable functions

At the moment, I’m going to Thomas Forster’s Part III maths course on computable functions. I’ve put together an introductory reading list on the elementary stuff in the opening lectures, which may be of use/interest to others.

As usual, comments/corrections/suggested additions are always welcome — especially perhaps, in this case, pointers to particularly good online lecture notes, etc.

Another “last ever …” box ticked

There’s no getting away from it: it does all feel slightly odd, as the academic year rattles on, being repeatedly struck by the thought “Well, this is the last time I’ll be doing that.” At the start of the year, the last time to be faced with another year of brand-new, eager-faced students at more or less their first university lecture; in December, my last first-year logic lecture; and now this last week, my last undergraduate class ever. (For USA readers: like all but a tiny handful, I have to retire from my post at the university’s statutory retirement age.)

Will I miss undergrad. philosophy lecturing ? Difficult to predict. But I think probably not. I might try offering a Part III maths course next year, if DPMMS will have me, but that’s a quite different kettle of fish.

Brandom, continued

Let’s add a further observation to what I was saying about Brandom in the last post. I remarked that \vdash and \vDash (as defined) coincide in a classical framework. But now let \vdash be the usual consequence relation in an intuitionistic logic, and let \vDash be the derived consequence relation defined as described.

Suppose D + p is in Inc. Then, by definition, D + p \vdash q for any q, so we have, intuitionistically, D \vdash not-p. So inituitionistically again, D + not-not-p \vdash q for any q. That is to say D + not-not-p is in Inc. Hence, again by definition, not-not-p \vDash p (while of course we don’t have not-not-p \vdash p). So here \vdash and \vDash peel apart.

I don’t know whether there are cases of more impoverished frameworks which lack classical negation but where \vdash and \vDash do coincide. That’s why I was asking about general conditions for getting the round-trip equivalence. But my conjecture is that the impoverished cases aren’t going to be very interesting. (I’m encouraged in that thought by an email from Warren Goldfarb!)

Where does this leave us? It looks as though \vdash and \vDash  typically peel apart, unless we are already assuming a classical framework (or something hobbled and uninteresting). So how could an inferentialist justify the claim that \vDash is the relation that matters (the one to feature in inferentialist definitions of connectives, etc.)? The suspicion must be that Brandom’s very idea of starting from Inc and defining a consequence relation \vDash from it will just beg the question against e.g. the intuitionist. But is that right?

Brandom’s incompatibility semantics, and other distractions

First, apologies to Alan Weir and all his fans who are impatiently awaiting the next episode of my stalled discussion of his book. I will get back to it, but I have been distracted over the last couple of weeks by various events, both in good ways and in less enlivening ways.

On the debit side, it now seems definite that I’m going to get replaced on retirement by someone remote from anything to do with logic, even very broadly construed (that’s assuming I eventually get replaced at all). Difficult not to get a bit depressed by such developments, about which no doubt more anon. Let’s just say it’s a very great pity, given the flourishing of logicky enterprises here, that we aren’t able to back success.

On the plus side, one distraction has been starting a close-reading of Kaye’s classic Models of Peano Arithmetic for our math logic reading group (I know the book a bit, of course, but it is different when you have to give presentations to a seminar on what exactly is going on and why). Another distraction — more time consuming but still enjoyable — was the recent annual grad conference here in the philosophy of logic and mathematics. The format is that, apart from the keynote speakers, grads from various places give papers, and locals give responses. I was replying to a nice talk by Giacomo Turbanti on modality in Brandom’s semantics, which forced me to do an amount of reading up, because I wanted to give a scene-setting talk to help those who hadn’t come across Brandom’s project before. In retrospect, having had the chance to think a bit more, I realize what I said wasn’t spot on in certain respects, so I won’t post my talk here as I was planning. But here’s one general comment I stand by, and one techie query for anyone who knows about this stuff.

The comment is this. The familiar inferentialist approach to the logical operators takes as its setting a standard consequence relation, and then adds introduction rules for the operator O which tell us when we are canonically entitled to assert a sentence with O dominant. Then we are supposed to locate the harmonious elimination rule which enables us get from a sentence with O dominant to wherever we could have got from its canonical grounds. Now, this kind of inferentialist approach to characterizing the logical operators by their introduction and harmonious elimination rules delivers intuitionistic logic but not full classical negation. Or so the usual story goes, as worked out in the hands of Prawitz, Dummett and Tennant. Brandom, though, claims that his brand of incoherence-based inferentialism does deliver classical logic. How does he pull a classically shaped rabbit out of an inferentialist hat?

Part of the story is that he in effect gives rejection rules rather than assertion rules for connectives. What Brandom’s rule for negation (for example) does is, in effect, tell us when to reject a proposition with negation its main operator, because adding it to some other stuff would lead to incoherence. But why privilege rules of rejection over rules for assertion? Why is this any better than privileging rules of assertion over rules for rejection? I would have thought that if — like Brandom — we see sapient enquiries responding to challenges and developing arguments in response, we should at most be giving equal weight  to the rejections and inconsistencies that prompt a reasoned response as to the assertions and deductions they elicit. But developing that line of thought would take us in the direction of Timothy Smiley and Ian Rumfitt’s bilateralism which puts assertion and denial on a par. Then we do arguably get an inferentialist framework which is genuinely friendly to classical logic. My comment, then, is that I don’t see why Brandom doesn’t go down this route, given his starting point.

The techie question is this. Suppose we start out with a standard single-conclusion consequence relation S |- A between finite sets of propositions S and propositions A, where “standard” means we have reflexivity, i.e. {A} |- A, dilution on the left, and cut.

Now we can define the extensional property Inc of being a (Post)-inconsistent finite set of sentences by saying that finite S is in Inc just if S |- A for all A (of the relevant language). It is immediate that Inc satisfies the filter condition that Brandom calls persistence, meaning that if S is in Inc, and S* is a superset of S, then S* is in Inc.

We can also go the other way about. We can start (as indeed Brandom wants to) with the idea of an upwardly closed set of sets of sentences Inc, and define a relation S |= A to hold just in case, for every set of sentences D, if D + A is in Inc so is D + S. It is easy to check that, so defined,  |= is a standard consequence relation.

In the general case, however, if we start from a consequence relation |-, define Inc as suggested (using the idea of Post inconsistency), and then define |= from Inc, we don’t get back to where we started. We will have S |- A entails S |= A, but not always vice versa. So here’s the techie query (and someone out there must know this!): under what general conditions does the round trip take us in a closed circle, so  S |- A iff S |= A? The relevant language having a classical negation will suffice, but what is the weakest condition? (Maybe Brandom himself tells us, and I should have been more patient hacking through his stuff: but the mode of presentation in the appendix to the fifth Locke Lecture is pretty much a paradigm of how not to present logical results in a helpful way.)

GWT updated

The first two episodes of Gödel Without Tears have been corrected — catching a few typos but mainly to correct the silly thinko that David Makinson caught. And there are new versions of episodes 7 (Arithmetization of Syntax) and 8 (The First Incompleteness Theorem), though these aren’t much changed from last year’s NZ version. Go to the GWT page.

Later: Oops, the corrected version of Episode 2 is now properly linked.

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