The Diagonal Lemma: An Informal Exposition

A couple years ago, I got a stream of emails from someone who shall remain nameless, who was completely convinced that there was something desperately wrong with the proof of the incompleteness theorem because of some kind of circularity in the diagonal lemma. I ended up writing him a long email explaining the diagonal lemma in terms of informal syntax, much as Quine does in Mathematical Logic.
I realized shortly thereafter that many of my students are in a position that is not so different. The diagonal lemma just is very hard to understand, in large part because its proof, in most expositions, in bound up with things like Gödel numbering, the representability of recursive functions, and the like, that really don't have very much to do with the diagonal lemma itself. The diagonal lemma is really a fact about syntax, not about arithmetic, and when one explains it in those terms it makes a whole lot more sense.
I therefore converted my original email into a short (five page) document that I've now used in a couple different courses. It can be found on my website.

The Grand Teaching Experiment (6)

Having finally dug out of the blizzard, and finally getting back to work, it's time to start thinking about teaching again. (Brown was closed on Friday, and there was still a parking ban in effect in Providence on Monday.)
Last Wednesday's class was, as I'd warned, on Dummett's paper "Truth". As I mentioned in my last post, the students in the class did an amazing job with the paper. Their responses, posted to the courses Canvas site, were all very, very good. I don't wish to take credit for that. It had, in the obvious sense, nothing to do with me. But the reading guide I posted for them, definitely seems to have helped. I've taught this paper many times, and I've never seen people make so much sense of it.
The discussion in class was at a correspondingly high level. We ended up spending the whole time talking about Dummett's two arguments against the explanatory sufficiency of Convention T.
The first, concerning non-referring names, isn't that hard to understand, but we worked through the question exactly what the argument assumes, and when one does that it becomes clear that it has a very narrow target: Frege, basically.
The second is much more interesting. The rough idea is that, if all there is to say about truth is given by Covnention T, then truth cannot play the role in logic that it is often assumed to play. In particular, the truth-tables can have no explanatory value. But what does that mean? That's the hard question.
Ultimately, I think the answer turns on the notion of truth-functionality: A deflationist cannot really make sense of the notion of truth-functionality. The usual way to try to do so is to talk about inferences like:

• A & B
• B <--> C
• So A & C

But this only works if you assume that the biconditional is itself truth-functional, and there's no reason to assume that. And the same complaint applies even if you try something like:

• (B & C) v (~B & ~C)

instead of the biconditional. But that's a larger issue.
I don't know how many people have tried giving students extensive reading notes for papers like "Truth". But I'm going to keep doing it, that's for sure, and I'd recommend trying it to everyone.

The Grand Teaching Experiment (5)

Monday's class was on Tarski's "Semantic Conception of Truth", which is what Burton Dreben used to call one of Tarski's "popularizations" and refused to take seriously. I like it for teaching, because it gives a nice introduction to Tarski's approach to the liar and to truth generally.
I felt like I fell a bit more into lecturing on this paper than on the others we've discussed. Maybe that was because it's more technical in nature, and I thought there was just stuff people needed to know. For example, Tarski indicates but does not actually argue that satisfaction of Convention T guarantees the extensional correctness of the defined truth-predicate. We needed to see why that is.
Still, the class was a lesson in how easy it is to fall back into talking a lot, something I'll try to avoid in our future sessions.
Today, then, is Dummett's "Truth". It seems, from the written responses I got from the students, as if my detailed reading notes did help. I was very impressed, in fact, with how well they'd done with this difficult paper. We'll see how the discussion goes.

The Grand Teaching Experiment (4)

I mentioned a couple posts back that we are scheduled to read Dummett's paper "Truth" this Wednesday. As many of you will know, this paper is legendarily hard and is often said to contain every major idea Dummett would spend the rest of his career developing. That is a slight overstatement, but it does indicate the rather frightening density of the paper.
Terrified by this prospect, I decided that what I needed to do was to give the students a whole lot of guidance about how to read Dummett's paper. The result can be found on the course website. We'll see how much it helps.
More generally, I realized when thinking about this that it isn't just Dummett's paper that's hard. All philosophy papers are hard. For many of my more philosophical "survey" courses, I've therefore often provided students with a handful of questions that might help structure their reading, and those sorts of questions are already on the website for this course. When I was lecturing on this material, those sorts of brief questions might have been adequate. But if I'm not lecturing, if I'm essentially expecting the students to do more of the work for themselves, then they're probably not adequate.
So, yesterday, even though we were scheduled to talk about Tarski's "Semantic Conception of Truth" today, I went to the course website and put up a similar reading guide for that paper.
To see the contrast, the original questions were:

What does Tarski think the Liar Paradox shows about our intuitive notion of truth? How is Convention T supposed to be related to our intuitive notion of truth? What are an object-language and a meta-language? How does distinguishing between them help us solve the liar paradox?

What's there now is:

• Tarksi insists that a definition of truth must be "materially adequate" and "formally correct". What are these two notions?
• If a definition of truth satisfies convention T, that is supposed to imply that it is in some sense correct. In what sense? and why? Note here the difference between extensional and intensional correctness that Tarski himself discusses.
• What does Tarski mean by saying that truth is a "semantic" concept?
• What is Tarski's diagnosis of the Liar Paradox? That is: To what exactly do his conditions (I), (II), and (III) come? To answer this question, analyze the informal presentation of the Liar on pp. 347-8. Where exactly do the three conditions play a role? Is there anything else that plays a role that Tarski is not mentioning?
• Why exactly does Tarski mean when he says that he will not "use any language which is semantically closed"? Why, if we do that, are we then forced to distingish object-language from meta-language?
• Tarski says that the meta-language must be "essentially richer" than the object-language if we are going to be able to define truth for the object-language. How exactly must the meta-language differ from the object-language?

And I will plan to structure today's class around these same questions.

The Grand Teaching Experiment (3)

Friday, we had our first "discussion" class. I still call them that, although now every class is a discussion class. But these ones are meant to be more wide-ranging and, more specifically, devoted more to criticism and evaluation than to exposition and understanding.
When I've taught courses with this same sort of structure before—lectures Monday and Wednesday, discussion on Friday—the discussion sessions have rarely flowed well. They had a tendency to turn into question and answer sessions, with not a whole lot of actual discussion. This one was much better, amazingly better given that it was the first one, so early in the semester. So perhaps that is a good sign.
It doesn't hurt, of course, that the class is full of students with a lot of philosophical experience, and that many of them have taken at least one course with me before, some of them many more than that. But still.

The Grand Teaching Experiment (2)

Shortly after I made my first post about how I'm trying something different with teaching this year, I went off to my Wednesday class. The paper for the day was Ayer's "Truth", published in Revue Internationale de Philosophie in 1953.
This particular class seemed to go pretty well. I laid out the topics I thought we should cover: Ayer's discussions of whether "true" is eliminable, of whether convention (T) can be understood as a definition of truth, and of the metaphysical and epistemological status of instances of (T).
Most students seemed as if they'd understood the main outlines of Ayer's discussion, so it was easy to get the class to put Ayer's basic claims on the table. We were then able to work through some questions about them pretty effectively. At certain points, I felt it worthwhile to jump in and talk for a bit, but it seems unsurprising that I should need to do that from time to time, especially at the beginning of the semester.
It helps, of course, that Ayer's paper is, as one would expect with him, extremely clear. Indeed, several students remarked how much they'd appreciated Ayer's straightforward prose after having slogged through Austin and Strawson.
One lesson here may well turn out to be, then, that if you're going to try to teach a course without lecturing, you have (for the most part, at least) to choose papers that are relatively easy to understand. The ultimate test of that theory will come next Wednesday, when we read Dummett's paper "Truth".

The Grand Teaching Experiment

Over the last couple years, I've become very dissatisfied with the way I've always taught my non-logic classes, e.g., classes on philosophy of language. These tend to be fairly small classes, with enrolment in the range of 10-15, and the basic model has been this: I've lectured on Mondays and Wednesdays, and we have had discussion on Fridays, led by me.
But I've read several things recently suggesting that lecturing is not a very effective way to get students to learn things. So this semester, in my course on Theories of Truth (Phil 1890D), I'm trying something different. I propose to blog about it from time to time.
The first thing I'm doing is trying to make use of Brown's new online teaching framework, called "Canvas". It does quite a lot. For example, there is an integrated conferencing system that I may try to use later. And it has simpler stuff, like the ability to schedule assignments, which are then automatically entered into a "grade book". But the main thing I'm using is the discussion board. For each of the readings, I've set up a discussion thread, and I'm requiring everyone in the class to post to it prior to class.
Obviously, this is just taking the place of the "response papers" that lots of people use, anyway, but it has a few advantages.

1. It's easy for me to comment on people's responses simply by replying in the discussion thread. As a result, they can get feedback before we meet for class. 
2. The students' responses, and my comments, are visible to the other students, so there is some opportunity for them to learn from each other. So far, there has only been a little discussion among the students, but I'm hopeful that, as we get into the semester, and as we all adjust to this new system, there will be more. (I've set it up so they have to post before they can see what other people wrote, for the obvious sort of reason.)
3. Since contributing to discussion is an "assignment", it is linked to the grade book, and I can enter grades (not much more than "did" or "didn't") very easily.

Much of this, of course, could be done with a Google Group, but the way Canvas automatically generates a syllabus from my discussion assignments is very nice. And I can add the other course assignments, too, so a calendar for the semester is automatically created.
The second thing I'm doing differently is I'm not lecturing. At all. I told the students this at the first meeting, and when I walked into the first "real" class, I had no lecture notes. I'm trying to run the entire class as discussion.
The days that would previously have been devoted to lecture are now devoted to discussion that is aimed at understanding the readings. The day that was previously devoted to discussion is now devoted to discussion aimed at evaluating and criticising the readings. We've had two of the former so far (on Austin's and Strawson's famous papers on truth), and I'm not sure yet how they are going. My strategy has been to identify topics from the papers that we should talk about. So, in the case of Strawson's paper, for example, these were: His criticisms of Austin's account of (i) statements, (ii) facts, and (iii) correspondence, and (iv) his own positive account of the use of "true". The first class seemed to go pretty well. The second one, a bit less so, and I ended up talking more. But that may simply have been because Strawson's paper is quite hard, and maybe that is a sign that I should do something else.