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.

Sex as a Jam Session

There's something extremely important happening in feminist re-thinkings of the nature of sexual consent. Namely: the idea that there is something seriously wrong with the language of "consent"; that this language embodies a certain model of sexual interaction that is itself part of the problem. The locus of this work is the book "Yes Means Yes!" and an associated blog. There's no way I can spell out all the ideas here. The rough idea is that there is something oppositional about the language of consent and the "commodity model" of sex that underlies it. Person A has something person B wants, and so B asks A's permission to take or borrow or use it. When you see sex that way, you are just asking for people to take or borrow or use when A doesn't give permission.

So, the suggestion is, we should instead see sex as collaboration. The key advantage to thinking this way is that it now follows immediately that rape isn't sex. Period. You can't force someone to collaborate with you.

Sex educator Karen K. B. Chan has produced a (non-explicit) video that promotes this way of thinking about sex. It is absolutely fantastic stuff. Everyone should see this.

Reading Guide for Dummett's "Truth"

It occurred to me that maybe a lot of people could use a guide to Dummett's paper "Truth". So here is the "reading guide" I produced for my students.
As you'll see, I didn't even try to get into the "anti-realist" stuff at the end. If anyone wants to add some material on that, I can update this for general use.

• Dummett begins the paper by expounding Frege's claim that sentences refer to their truth-values. It is easiest to understand this claim when it is put differently: that the "semantic value" of a sentence is its truth-value. And what that claim is best understood in terms of the truth-tables: that the central semantic fact about a sentence is that it is true or false.
• Dummett then rehearses an argument that a sentence cannot refer to the proposition is expresses. The argument is:
  
  1. "Mark Twain was an author" and "Samuel Clemens was an author" express different propositions.
  2. "Mark Twain" refers to the same person as "Samuel Clemens".
  3. The corresponding parts of the two sentences therefore have the same reference.
  4. Reference is "compositional", in the sense that the reference of the whole is completely determined by the references of the parts.
  5. Hence, the two sentences must have the same reference.
  6. Hence, that reference cannot be the proposition expressed.
  The point of this is really just to introduce the idea of thinking of truth from the perspective of logic.
• Dummett then suggests that, while it's reasonable to think that sentences do have "semantic values", Frege has to earn the right to say that their semantic value is their truth-value. On pp. 142-3, Dummett introduces an analogy between truth and falsity, and winning and losing, to illustrate what Frege would have to do to earn that right. What exactly does Dummett think Frege would have to do?
• Dummett then proceeds to argue, on pp. 145-6, that (the propositional version of) the T-scheme may not even be correct. The argument turns on the idea that there may be sentences that are perfectly meaningful—they express proposition—but are neither true nor false. A putative example would be something like, "The greatest prime number is one less than a perfect square". Frege would have held that this expresses a proposition, but does not have a truth-value, due to the fact that there is no greatest prime. Why, then, does Dummett think that:
  

  It is true that the greatest prime number is one less than a perfect square iff the greatest prime number is one less than a perfect square.

  is not itself true?
• Dummett then argues, on pp. 146-9, that, even if its instances are all true, the T-scheme "cannot give the whole meaning of the word 'true'". The argument turns on the assumption that the truth-tables have some explanatory value, in particular, that they embody (at least partial) explanations of the sentential connectives. How exactly is this argument supposed to go?
• On p. 149, Dummett then concludes that a theory of truth must be possible in a certain sense. In particular, he thinks that it must be possible for us to articulate the point of our characterization of assertions as true and as false. There is a sketch of what Dummett has in mind in the paragraph running from p. 149 to p. 150. Try to articulate as best you can what research program he means to be articulating.
• On pp. 150-4, Dummett then argues in support of a very general claim that he makes on p. 150: that, given the point of the characterization of assertions as true and as false, there is no need, and no room, for any finer characterization, and so that it is senseless to say that an assertion is neither true nor false. The core of the argument is on p. 153, where Dummett suggests that, although we might call both conditionals with empty antecedents and sentences containing non-referring terms "neither true nor false", there is an important asymmetry between the two cases that this common terminology lacks. What is that asymmetry?
• Finally, on p. 154, Dummett concludes that "we should abandon the notions of truth and falsity", at least in connection with the explanation of the meanings of statements. In fact, however, that isn't quite what he means. He thinks there is a particular way of using "true" and "false" that is unhelpful and another way of using them that would still be OK. What is the difference? And how is it related to Dummett's central thesis about the role of the concept of truth?
• Dummett then proceeds, on pp. 154-7, to explore whether thre might yet be a point in calling certain statements neither true nor false. He argues that there may well be, but that, if there is, it must necessarily concern the way such statements behave when they occur as parts of other statements (e.g., as antecedents of conditionals). It is morally certain that we will not get to this material, so you do not need to read beyond p. 154. But if you wish to do so, then the question to ask here is just how Dummett's arguments here are supposed to cohere with the earlier ones, and what point he thinks there could be in distinguishing among different ways a statement might be true or false.
• Finally, on pp. 157-62, Dummett introduces a set of considerations that are supposed to show that the notions of truth and falsity that are appropriate to the evalaution of assertions are not the classical notions of truth and falsity. Rather, calling an assertion "true" is like saying it is justified, and calling it "false" is like saying that it is unjustified. This sort of argument is one that became closely associated with Dummett, and he spent much of his career trying to develop it and to fill in the details. We certainly will not discuss this material.

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.

Ayer on Truth

On a more substantive note: Ayer's 1953 paper "Truth" is really under-rated, in my opinion. Did anyone make the now familiar points about the ineliminability of "true" before him? The paper is not often cited, and when it is it is almost always cited wrongly: Volume 25, 1953. It was in issue 25, but volume 7, which makes me kind of suspect that a lot of people who cite it haven't actually read it.
I'm all the more suspicious since Ayer is so often claimed for the deflationist side, when in fact his view is much more nuanced. The crucial remarks come at the transition between two very different parts of the paper:

Let it be granted then that we must forego any general definition of truth, and let it also be granted that there are certain contexts in which the words 'true' and 'false'...are ineliminable. It does not follow that there is any mystery about their meaning. On the contrary, their function is quite clear. ...To speak of a sentence, or a statement, as true is tantamount to asserting it, and to speak of it as false is tantamount to denying it. ...[I]t is hard to see what further explanation of [truth] is required.
Can we say then that we have solved the philosophical problem of truth? What is disturbing about our solution is its simplicity. If that is all there is to it, it is hard to see how anybody, even a philosopher, can ever have been supposed that the question 'What is truth?' presented any difficulty at all. ...All the same they must have known well enough how the word 'true' was actually used. Such information as that it is true that the sky is blue if and only if the sky is blue could hardly be expected to come upon them as a revelation. ...What they would say...is that the provision of these partial definitions did not meet their problem. It remains, therefore, for us to see what this problem can have been and if possible to solve it.

Ayer then goes on to do exactly that. His understanding of what the problem is, of course, is shaped by his verificationism, but he does think there is another problem that goes by the name "the problem of truth", just as Strawson does.
Ayer's paper is pretty hard to find. Feel free to ask me for a copy if you need one. Oh, and beware! Ayer also published a paper titled "Truth" in 1963, in his collection The Concept of a Person and Other Essays. There is a fair bit of overlap between the two papers, but they are different papers.

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".