On Graham Priest's 'An Introduction to Non-Classical Logic'

Through Greg Restall's blog I was just made aware of a problem that has been discovered in Graham Priest's An Introduction to Non-Classical Logic. In particular, there is a problem with the simplified semantics for relevant logic. Since I guess many people - like me - have greatly enjoyed, and still enjoy, this book, I thought it worthy of mention.

If you want to know more, check it out here. The paper which isolates the problem is available here.

An aside: Restall alludes to the fact that we rarely see philosophers apologizing for their mistakes. Logicians on the other hand, are frequently forced to admit mistakes of various kinds. Now, it would be interesting to see if any contemporary philosophers have published clearly apologetic texts. I can't think of any while writing, but perhaps someone has a suggestion. For surely, we don't want to claim infallibility; although it is tempting.

Report from Truth and Proof: Kurt Gödel and Foundations of Mathematics

On Saturday I had the pleasure of hearing three magnificent talks at the Gödel workshop in Edinburgh. First out was John Dawson (Pennsylvania State University) with the talk "Taking Truth Seriously". Then Hannes Leitgeb (University of Bristol) with "Type-Free Necessity, Truth, and Informal Provability", followed by Richard Zach (University of Calgary) on "Gödel's First Incompleteness Theorem and Mathematical Instrumentalism". Unfortunately, I didn't have time to catch the talks on Sunday, but I hope they lived up to the standards of the first day. The abstracts from the talks are here. For a comment on Leitgeb's talk see here.

On Dawson's "Taking Truth Seriously"
According to Dawson, the logicians of the first half of the 20th century distrusted semantical methods and especially the notion of truth. In particular, when Gödel was working with his incompleteness theorems, Tarski had yet to give the notion of truth its canonical characterisation. Dawson's topic, then, was the use of pre-Tarskian notions of truth in the logical work of this period. He goes on to argue that logicians was confused about the syntax/semantics distinction, even after Tarski's 1933 paper and as late as in the early 1960s.

A point Dawson kept emphasizing with different examples is how the paramount discoveries in semantics are typically accompanied by syntactical prejudices. In fact, in many cases semantic results were in effect delayed several because they were frequently interchanged with syntactical counterparts. Dawson's first example was the Wittgenstein-Post discovery of truth tables. Although this today has a clearly semantical dimension, Wittgenstein and Post simply considered this to be a decision method for tautologies. The interpretational aspect, however, seems to lost on them. Wittgenstein seems to have maintained that his Boolean values, A and B, were primitives, that nothing interesting could be added about these values. Similarly, Post's '+' and '-' also lacked the clear relationship to truth and falsity; Post was only willing to say that '+' was a convenient "guide to thought".

Another result which today is considered to be unequivocally semantic is the Löwenheim-Skolem theorem. In 1915, Löwenheim tried to show that every satisfiable formula is satisfiable in a denumerable model. However, his "proof" turned out to be flawed and Skolem later fixed the problem, thus appending his name to the theorem's name. Dawson added that, interestingly, Skolem's work on the theorem anticipated Gödel's famous results some years later. Skolem formulated an equivalence which is quite similar to that which Gödel later used to prove completeness: if a theory is consistent, then it is satisfiable. Now, in light of Skolem's formulation many scholars have been investigating whether Gödel infact knew Skolem's work before he made his discoveries. Dawson made two comments regarding this: (i) Mark van Atten (2005) seems to give good historical evidence to the effect that Gödel never read the relevant work by Skolem. (ii) Even if Gödel did actually know Skolem's work there is an important improvement in Gödel's formulation. In contrast to Skolem's semantic understanding of consistency, Gödel used consistency as a syntactical notion and thus achieved a clear distinction with the semantic notion of satisfiability.

Yet another important semantic result with an interesting history is the compactness theorem. Dawson stressed that although Gödel proved the theorem while working with completeness, the result was not fully appreciated before it was rediscovered by Mal'cev (1941). Indeed, Dawson pointed out that text books in logic all the way up to the early 1960s presented the compactness theorem but did not apply it in semantic methods. Actually, well-known logic classics such as Kleene (1952) and Church (1956) preferred giving compactness in syntactic terms.

Finally, Dawson got to my topic of choice, namely Tarski's work on truth and logical consequence. As Dawson admitted, his points here are more or less adopted from Etchemendy (1988) and (1990). Tarski, they claim, did not formulate the modern notion of truth-in-a-structure in his famous 1933 paper on truth. Of course, he gave a case of it, the calculus of classes, but nowhere does he give a general formulation á la that which is used by today's logicians. Not before Tarski & Vaught (1957) did he give the formulation of truth-in-a-structure which is nowadays used in model-theory. This goes to show, I guess Dawson would add, that the development of the technical notion of truth did not at all come to an end with Tarski's work. The reception and application of his original work has lead to the notion we now use, and especially the interplay between Gödel's theorems and Tarski's work has been important to stabilize the notion.

Though, of course, some - like Etchemendy - would probably say that the distinction between semantics and syntax is still confused. In particular, that the heavy emphasis on model-theory has been an unfortunate guide to the concepts of truth and logical consequence. This discussion, however, must wait for a later post.

News from the Mothership

Not many things worth posting about happens in Norway, at least not on the philosophy front. However, as some of my readers have noticed, I frequently post about the up-and-coming Norwegian chess master, Magnus Carlsen. Of course, this young "Mozart of Chess" has been given quite a lot of media attention back home. Nevertheless, chess not being a particularly popular sport, he rarely makes it to the front page.

Today, however, is an exception. One of these absurd coincedences - which goes to prove how small Norway is - just landed him in the headlines of one of our major newspapers. This coincedence is connected to another Norwegian phenomenon with some international importance, namely the theft of Edvard Munch's paintings (one of them was the Madonna). At the moment, Norway is witnessing a trial against a fistful of men accused of the crime. And what has this got to do with a 15-year old GM? It just happens that one of the accused claims to have been taking part in a chess simul against Magnus Carlsen on the day of the robbery.

"The Norwegians are leaving"

Maybe some of you remember that line from Twin Peaks. Anyway, today Scotland has yet again been invaded by a boatful of Norwegians. Fifty philosophy students from the University of Oslo came by St. Andrews today, as part of their trip to Scotland. As the only Norwegian member of the M.Litt class, I got the honour of welcoming them to our glorious town. I do hope that the little they saw, and, of course, the talk by Sarah Broadie, will inspire some of them to do their Master degree in Scotland.

Congratulations

Congratulations to my girlfriend, Lene Sælen, who just a few days ago finished her master dissertation in physics, called 'The N-dimensional Coulomb problem and its application to few-particle quantum theory'.

Truth and Proof: Kurt Gödel and the Foundations of Mathematics

This weekend a conference on Gödel and the foundations of mathematics will be held in Edinburgh. It is the Department of Philosophy at the University of Edinburgh who are organizing the event in celebration of the centenary of Gödel's birth. Some of my fellow students and myself will be headed down to Edinburgh to catch at least one day of the conference.

If you want to sit in on the talks, it's not too late. Find information and the programme here.

PS According to the programme, Shapiro is doing his Zermelo self-evident talk which I have posted on earlier.

Magnus Carlsen beats Anand

Okey, so his 2-0 score over world number 3 (soon to be number 1 since Kasparov will disappear from the FIDE's April list due to inactivity and Topalov lost points in Linares) was not in classical chess but in blitz. Nevertheless, out-calculating Anand, a reknowned blitz expert, is a tremendous feat. The victory came in a follow-up-tournament after his 6th place (five people shared 1st) in the Reykjavik Open. Although it was a bit of a disappointment that he didn't climb to the top in the classical tournament, it is more than compensation that he robbed Anand of the 1st place in the rapid games.

PS The link is to a Norwegian article.

Update: And here, finally, is an English article.

Formal Philosophy to the Masses: Update

A while ago, I wrote about Vincent Hendricks' and John Symons' Masses of Formal Philosophy. One of the philosophers I mentioned as a good candidate for answering the author's five questions has now posted about the project on his own blog. Stay tuned.

On the web: The Blind Spot

It just recently came to my attention that the French logician Jean-Yves Girard has recently made available an extensive talk he gave in Rome in 2004. The Blind Spot is a transcript of the talk (which according to the introduction lasted for months), and contains much of Girard's work in proof-theory. When and how I will read this monster talk, I do not know, but I can safely recommend the introduction to those who want (as me) to peak into Girard's wonderful world. (Warning: explicit content!)

Find it here.

Winner of Linares 2006: The underdog snatches the title

As you perhaps remember, Topalov, Leko, Radjabov and Aronian were all in shared 1st before the last round in Linares. Leko, the only one who had the advantage of the white pieces, played against Aronian, while the other two played against participants without a shot at the victoriy - Topalov against Spanish GM Vallejo, Radjabov against French GM Bacrot. With a scenario like this, everything was set for Topalov bringing home the trophy: not only did he play the weakest player in the tournament, he also had the advantage of winning on tie break rules if none of the four players managed a win. Of course, with Leko playing white against Aronian, a draw was already in the air.

The first member of the leading four to call it the day was young Radjabov. His quick win against Svidler the day before must have dulled his appetite, for without leaving the opening theory, he agreed on a draw with Bacrot. On the Chessbase server, the on-lookers speculated on whether Radjabov perhaps was securing some easy cash. At any rate, the youngster's performance in the tournament has been surprisingly good. To win the hearts of the fans, however, one cannot draw the last game without even trying.

At the most, close to 6.000 people were watching the games on the Chessbase server. Without a doubt, the majority had their eyes fixed on the Vallejo vs. Topalov-board. A strange game was about to be played. Certainly, non of these players had much to lose: Topalov had already seen that on the next table, Aronian was slowly building up a winning position against Leko, grabbing a pawn for nothing. Moreover, it is not Topalov's nature to draw home a win. His opponent, young Vallejo, was not only dead last (and so could get no worse by losing), but also had the home fans to satisfy. How better to do it than to win against the Champion on the last day?

Despite all this enthusiasm, the two players followed theoretical lines into a drawn position. More precisely, Vallejo sacrificed a bishop to gain an offensive on Topalov's kingside. Quite obviously, though, the sacrifice would not win the game. But unfortunately, it did make sure that Vallejo had the resources to force a draw whenever he wanted. Topalov, with no other options than moving his king back and forth, had to watch as his opponent pondered his options. It soon became all too apparent that Vallejo had nothing to play for - all continuations would probably cost him another game due to the sacrificed bishop. Only one options was left, namely forcing the draw. By playing himself into this situation, and eventually opting for the draw, Vallejo owned himself the dubious title of most unpopular player of the tournament. 6.000 fans was roaring for blood, demanding that he continued the game, thus genlemanly preserve Topalov's winning chances. Needless to say, Vallejo wan not as concerned with Topalov's winning chances as the latter's fans.

With the two other games drawn, Aronian must have felt his chances increase. All he had to do was make the transition from a winning middle-game to a winning end-game against the slippery Leko. Never before have I seen so many spectators cheering for Leko, if only to give him half a point and Topalov the trophy. But all the cheering in the world could not have saved Leko on a day when Aronian outplayed him brutally from the opening and throughout the game. Several pawns down, Leko decided that it was time for tea, and gave Aronian the full point and the Linares 2006. Not a bad day for the 23 years old Armenian GM.


PS Aronian, now number five in the world, might climb on the FIDE lists due in early April. It is worth noting that his recent success is going to make the his candidate match against GM Magnus Carlsen even more exciting for the chess world.

Jason Stanley on The Use Theory of Meaning

A few days ago, on the Leiter Reports, guestblogger Jason Stanley posted a provocative criticism of what he calls "the use theory of meaning". There are chiefly two reasons for bringing this up here: (i) I have some investements in the "meaning as use"-idea seeing that I favour proof-theoretical semantics for logic. Moreover, (ii) even if I didn't have any particular interest in the programme, I feel obliged to object on reasons of philosophical integrity alone.

What exactly does Stanley have in mind with the phrase "the use theory of meaning"? He says: "One project in the philosophy of language centrally involves the thesis of deflationism about the semantic notions of truth and reference. According to the deflationist, truth and reference play no significant explanatory roles." As some commentators go on the point out, this fits a wide range of philosophers. Inspired by Wittgenstein, people like Grice and Dummet have started different ways of emphasizing the role of use in meaning. The latter's ideas have been picked up by for instance Wright and Prawitz, developing them for different philosophical problems. Furthermore, people like Brandom and Field represent yet another couple of positions within the broad term coined by Jason Stanley. However, he leaves us no clue as to more precisely which branch of deflationism he has in mind.

Enter Stanley's chief claim: "The use-theory of meaning will never be developed in a form that can play the role that current semantic theories do in these ongoing philosophical projects." (By these ongoing philosophical projects, Stanley is referring to tensed statements, knowledge-attributions and identity through time.) Several questions pop out of the philosophical woodwork. How are we to understand "current semantic theories"? Non-deflationist positions perhaps? Or truth-conditional semantics? Formal semantics such as model-theory? If Stanley is claiming that one of these account have made great progress on the topics he mentions, I suspect that he is getting things the wrong way.

Advocates of use-theoretical semantics (in whatever shape), typically claim that a reason why the problems mentioned by Stanley have proved difficult to solve is exactly because we tried to account for them within a classical semantic framework. The truth-conditional heritage from Frege and the model-theoretic heritage from Tarski are inadequate to deal with some of the more subtle features of our languages. As pointed out by Shapiro, this is largely due to the fact that these classical frameworks were intended for mathematical language, not for natural language.

The accounts that I believe Stanley is implicitly endorsing are not part of the solution; they are part of the problem.

Day of Defeat: 10th of March

Last year, on the 10th of March, Topalov (not yet the World Champion) played against Kasparov in the last game of Linares 2005. Kasparov had played an astonishing tournament, crushing all opponents - only Topalov was still hanging in, a mere point behind the Beast. Sensationally, Topalov won the last game, equalizing the score and securing a shared victory (strictly speaking Kasparov won on tie breaker rules). But something more happened. Topalov anticipated his own triumph in San Luis, playing the last game against Kasparov before the latter retired. Because only minutes after their game finished, Kasparov held his famous press conference, announcing that he was retiring from chess while still on top. The 10th of March proved to be a spectacular day for Topalov. Needless to say, Kasparov was not thrilled about losing his last game.

And who says there is no magic in numbers? For yesterday, the 10th of March and the next to last game of Linares 2006, Topalov dazzled the audience yet again. After playing terrible in the first half of the tournament, he has steadily played himself upwards the last week. Before yesterday's game, he was only a single point behind his opponent Peter Leko. With the black pieces against a rampant Topalov, all Leko was hoping for was a relatively easy draw, but, alas, it turned out to be a long day for the Hungarian. After over seventy moves, Leko lost in an objectivly drawn position, succumbing to Topalov's scaringly precise play. The current FIDE Champion continued his pace by playing like a computer (in the literal sense that he does exactly the moves that the analyzing computers recommend). Winning the point, he is now in shared 1st, not only with Leko himself but with Aronian and Radjabov.

Everything is decided tonight. Download software from Chessbase.com and follow the games for free!

Dinner of Champions

What is Topalov eating these days? Certainly he has shifted his diet after returning to the Old Country (Linares, Spain) for the second half of the tournament. I remember that Nigel Short complained that he had to dine with Topalov everyday at San Luis (when Topalov won his title in spectacular manner) because the latter thought it brought him good luck. I wonder who's dining with Topalov now, but whoever it is, he or she is worth keeping.

The FIDE Champion awoke from his dogmatic slumber the moment he landed on European soil. Three consecutive wins have taken him from second last to third place in manner more than sufficient to scare the always careful Leko. Svidler, Bacrot and Aronian have all been swept of the board, and only today did Topalov find reason to give away half a point, admitting a draw to Radjabov. Meanwhile, the man on top, Peter Leko, has entertained the audience with four draws in a row! In the moment of writing, he has just drained the winning chances out of a game against Ivanchuk (who by the way have not reacted as well as Topalov to the change of scenery), and is probably already calling on his muses to land him another draw against Bacrot tomorrow. But draws will only get you as far as Friday, Leko, for with one point separating you from Topalov, he will play for a brutally unbalanced game, throwing all but the title at the table to topple you over. Good luck.

Do not claim too much

This is actually the title of a Shapiro paper about second-order logic. This is not a post about second-order logic, however, but rather a post about broken promises. Before the Vagueness Workshop, I had ambitions of posting something on all of the talks, but as it turned out I managed none. Why is that? Incidentally, my presentation on, yes, exactly, vagueness in the philosophy of logic course had to be finished by Monday; thus the promise of an extensive report from the workshop proved too much.

Nevertheless, for those who want to read a bit about what happened, I can safely recommend my dear friend and fellow student Andreas' report. For Peter Pagin look here: for Timothy Williamson here. As for my presentation: when it is reworked a bit, I'll post it here. I promise.

Blogging from Sixth Arché Vagueness Workshop

Later today Timothy Williamson is starting the Sixth Arché Vagueness Workshop. All handouts and drafts are now available online here, together with the schedule. Hopefully, I'll be able to provide some modest reports from (at least) some of the talks over the weekend.

Meanwhile; here is another paper that Agustín Rayo, formerly Archè, now MIT, will be presenting on Monday in the Arché Maths seminar.

The Logic Museum

Just became aware of this through the FOM list. The Museum contains a collection of online texts in the history of logic, up to and including Frege and Russell.

The Museum also has its own weblog.