Recent Paper: Intuition and the Substitution Argument

I'm not sure why it never occurred to me to post announcements of new papers here, but, well, better late than never.

This paper, "Intuition and the Substitution Argument" (PDF here), was delivered at the Analytic Philosophy symposium at the University of Texas in early December, and before that at Duke University, in October. It will appear in a special issue of Analytic Philosophy also containing the other papers from the symposium, by Mike Martin, Tamar Shapiro, and Ralph Wedgwood.

Abstract:

The 'substitution argument' purports to demonstrate the falsity of Russellian accounts of belief-ascription by observing that, e.g., these two sentences:

(LC) Lois believes that Clark can fly.
(LS) Lois believes that Superman can fly.

could have different truth-values. But what is the basis for that claim? It seems widely to be supposed, especially by Russellians, that it is simply an 'intuition', one that could then be 'explained away'. And this supposition plays an especially important role in Jennifer Saul's defense of Russellianism, based upon the existence of an allegedly similar contrast between these two sentences:

(PC) Superman is more popular than Clark.
(PS) Superman is more popular than Superman.

The latter contrast looks pragmatic. But then, Saul asks, why shouldn't we then say the same about the former?

The answer to this question is that the two cases simply are not similar. In the case of (PC) and (PS), we have only the facts that these strike us differently, and that people will sometimes say things like (PC), whereas they will never say things like (PS). By contrast, there is an argument to be given that (LS) can be true even if (LC) is false, and this argument does not appeal to anyone's 'intuitions'.

The main goal of the paper is to present such a version of the substitution argument, building upon the treatment of the Fregean argument against Russellian accounts of belief itself in "Solving Frege's Puzzle". A subsidiary goal is to contribute to the growing literature arguing that 'intuitions' simply do not play the sort of role in philosophical inquiry that so-called 'experimental philosophers' have supposed they do.

Thanks a ton to David Sosa for inviting me to the symposium, and to everyone there for showing me way too good a time.

New Paper: Is Frege's Definition of the Ancestral Correct?

I've posted a new paper to my website, titled "Is Frege's Definition of the Ancestral Correct?" (PDF here) The paper is scheduled to appear in a special issue of Philosophia Mathematica edited by Roy Cook and Erick Rech.
Abstract:

Why should one think that Frege's definition of the ancestral is correct? It can be proven to be extensionally correct, but the argument uses arithmetical induction, and that fact might seem to undermine Frege's claim to have justified induction in purely logical terms—a worry that goes back to Bruno Kerry and Henri Poincaré. In this paper, I discuss such circularity objections and then offer a new definition of the ancestral, one that is intended to be intensionally correct; its extensional correctness then follows without proof. It can then be proven to be equivalent to Frege's definition, without any use of arithmetical induction. This constitutes a proof that Frege's definition is extensionally correct that does not make any use of arithmetical induction, thus answering the circularity objections.

In the general case, the new definition is fairly complicated. But in the special case of the concept of natural number, it reduces to:

n is a natural number iff there exists a Dedekind finite concept (or set) F such that F0, Fn, and ∀x∀y[Fx & Pxy & x≠n → Fy]

The last condition says that F is closed under successors, except that it need not be true of the successor of n. The point, which has also been noted (independently) by Aldo Antonelli and Albert Visser, is that Dedekind finitude can be used here, so the definition is non-circular. The intuitive idea is just that, if n is not finite, then the other three conditions force every natural number to be F, in which case of course F is Dedekind infinite. If n is finite, by contrast, then F is just [x: 0 ≤ x ≤ n].
One can then go onto prove induction from this definition, using many of the results Frege proves for his own definition of the ancestral.

Creating PDFs for EBrary Reader

Turns out that Brown has a subscription to at least part of the Ebrary archive of online books. I found this out because I was looking for a book in our library, using the online catalog, and it turned out that said book was available online. Very cool!

Well, kind of cool, until I found out what this involved. By default, you basically have to read the document online. You can download the "EBrary Reader", which is a Java application, and read documents using it. But it's kind of clunky, to say the least. What I wanted was a PDF that I could then use as I wanted. How to get one?

I noticed that the EBrary Reader would allow me to print, so I thought maybe I could print the file as a PDF, since the default print dialog for Fedora lets you do that. Unfortunately, the Java application was not using the system dialog, but a Java dialog, so that didn't work.

A little googling led me to the cups-pdf package, which installs a system-wide PDF printer for the Common Unix Printing System. A quick "sudo yum install cups-pdf" was enough to give me access to that.

The next step was to convert this file to DjVu, which tends to be much smaller than the corresponding PDF. I've done this a million times before, so figured it would be pretty easy. Unfortunately, it was not.

The first step was to run the pdfimages command (from the poppler-utils package):

pdfimages -p file.pdf p

to extract the page images. Imagine my surprise when I got 420 images from a 20 page paper! It turned out that each page was constructed from 21 different images, stacked on top of each other. (To help with download times?)

Fortunately, I've had enough experience with ImageMagick to know this was not a problem that could not be solved. It took a little more googling, and a little experimentation, but eventually I found out that:

for i in 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20; do
  montage p-0${i}*.ppm -geometry +0+0 -background none -tile 1x21 page-$i.tiff;
done

would stack all the images back on top of each other.

So now I had 20 page images, all as TIFFs, and those could then be fed to ScanTailor for processing on the way to creating a DjVu.

I Love Perl

I posted about how to write simple Perl filters before, but I have to say that I just love doing this:

perl -ibak -pe 's/plaintext\(odocstream &(.*?), OutputParams const &(.*?)\)/plaintext\(odocstringstream &\1, OutputParams const &\2, int max_length\)/' *.h *.cpp

How much time did that save from doing it manually?

The Grand Teaching Experiment (7)

I haven't posted for a while about the teaching experiment, because it was basically on hiatus. For the last few weeks, we have been doing technical material, and it does seem worth lecturing about that. Now, though, we are back to more philosophical material, so we are back to having discussions.
Up this week were Field's paper "Tarski's Theory of Truth" and Etchemendy's paper "Tarski on Truth and Logical Consequence". Both of them are pretty clear, though the dialectical structure of Etchemendy's paper is complicated (as I argue in my own paper on the topic).
The discussions seemed to go pretty well, perhaps better on Etchemendy than on Field, and that perhaps because I have a deeper understanding myself of that paper. As previously, the students' written responses to the papers were very good. Everyone seemed to have a solid understanding of the basic outlines of the arguments, with some students (unsurprisingly) a little ahead of others. But what I'm really coming to appreciate about this way of doing things is that, as we start class, I already have a pretty good idea what people understand and what they do not, and we can focus our attention either on filling in the gaps or else, even better, digging more deeply. And because everything is based on discussion, we dig in the direction the students find interesting, or where their questions naturally lead.
It's definitely clear, as I mentioned in a previous post, that the sorts of detailed reading notes I've been giving the students recently are important to this kind of approach. I've had more than one student remark on this.

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.