Published: The Logical Strength of Compositional Principles

Abstract

This paper investigates a set of issues connected with the so-called conservativeness argument against deflationism. Although I do not defend that argument, I think the discussion of it has raised some interesting questions about whether what I call compositional principles, such as "A conjunction is true iff its conjuncts are true", have substantial content or are in some sense logically trivial. The paper presents a series of results that purport to show that the compositional principles for a first-order language, taken together, have substantial logical strength, amounting to a kind of abstract consistency statement.

Find it on Project Euclid, or download the pre-publication version here.

The paper is a kind of companion to "Disquotationalism and the Compositional Principles" (PDF) and is basically the philosophical side of the paper "Consistency and the Theory of Truth" (here).

Lacuna in "Is Frege's Definition of the Ancestral Adequate?"

Ran Lanzet has pointed out a significant lacuna in the proof of the main result in my paper "Is Frege's Definition of the Ancestral Adequate?" This has been repaired in the 'pre-publication' version of the paper, which can be downloaded here. See p.21 of that document.

I had certainly thought of the missing case, and seem to recall that at some point I'd introduced a 'simplifying assumption' that allowed me to ignore it. But that assumption is not mentioned in the published version of the paper, and it isn't nearly as easy as I'd supposed to see that it's permissible (which is perhaps why I removed it, but without fixing the affected part of the proof).

Interview in 3AM Magazine

Thanks to Richard Marshall for the invitation. The interview is here.

New Paper: The Frontloading Argument

Forthcoming in Philosophical Studies.

Maybe the most important argument in David Chalmers's monumental book Constructing the World is the one he calls the 'Frontloading Argument', which is used in Chapter 4 to argue for the book's central thesis, A Priori Scrutability. And, at first blush, the Frontloading Argument looks very strong. I argue here, however, that it is incapable of securing the conclusion it is meant to establish. My interest is not in the conclusion for which Chalmers is arguing. As it happens, I am skeptical about A Priori Scrutability. Indeed, my views about the a priori are closer to Quine's than to Chalmers's. But my goal here is not to argue for any substantive conclusion but just for a dialectical one: Despite its initial appeal, the Frontloading Argument fails as an argument for A Priori Scrutability.

You can find the paper here.

New Paper: Speaker's Reference, Semantic Reference, and Intuition

Forthcoming in The Review of Philosophy and Psychology.

Some years ago, Machery, Mallon, Nichols, and Stich reported the results of experiments that reveal, they claim, cross-cultural differences in speakers' `intuitions' about Kripke's famous Gödel-Schmidt case. Several authors have suggested, however, that the question they asked they subjects is ambiguous between speaker's reference and semantic reference. Machery and colleagues have since made a number of replies. It is argued here that these are ineffective. The larger lesson, however, concerns the role that first-order philosophy should, and more importantly should not, play in the design of such experiments and in the evaluation of their results.

You can find the paper here.

New Paper: Comments on Imogen Dickie's "Fixing Reference"

The main focus of my comments is the role played in Dickie's view by the idea that "the mind has a need to represent things outside itself". But there are also some remarks about her (very interesting) suggestion that descriptive names can sometimes fail to refer to the object that satisfies the associated description.

You can find the paper here.

New Paper: Logicism, Ontology, and the Epistemology of Second-Order Logic

Forthcoming in Ivette Fred and Jessica Leech, eds, Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale (Oxford: Oxford University Press)
 
In two recent papers, Bob Hale has attempted to free second-order logic of the 'staggering existential assumptions' with which Quine famously attempted to saddle it. I argue, first, that the ontological issue is at best secondary: the crucial issue about second-order logic, at least for a neo-logicist, is epistemological. I then argue that neither Crispin Wright's attempt to characterize a `neutralist' conception of quantification that is wholly independent of existential commitment, nor Hale's attempt to characterize the second-order domain in terms of definability, can serve a neo-logicist's purposes. The problem, in both cases, is similar: neither Wright nor Hale is sufficiently sensitive to the demands that impredicativity imposes. Finally, I defend my own earlier attempt to finesse this issue, in "A Logic for Frege's Theorem", from Hale's criticisms.

For the most part, the paper is not terribly technical, but there are some (what I think are) interesting applications of technical work on models of second-order arithmetic toward the end of section 3.

Do Welders Make More Than Philosophers?

Early in the Fox Business Network debate, Marco Rubio took a strong stand in favor of vocational education. "For the life of me I don't know why we stigmatize vocational education," he said. "Welders make more money than philosophers. We need more welders than philosophers."

So begins a complete debunking of this claim by Matthew Yglesias, who just happens to have been a student of mine at Harvard. Read the rest here. It's a corrective that's especially welcome to those of us who have to have the same conversation, over and over, with budding philosophy concentrators: How do you tell your parents?

Of course, as Yglesias says, that doesn't mean we should stigmatize vocational education. For many people, it's clearly the right choice. And, to be honest, I'm not sure many people do stigmatize vocational education. I happen to live very close to Blue Hills Regional Technical School, a vocational high school that serves several of the surrounding towns. (Avon, Braintree, Canton, Dedham, Holbrook, Milton, Norwood, Randolph and Westwood.) My understanding is that there are almost always more applicants than there are slots. And the school is widely respected in the area. Indeed, Blue Hills students scored higher than the state average on all three recent MCAS examinations.

Reading Frege's Grundgesetze: Now in Paperback

My book Reading Frege's Grundgesetze has just been published in paperback in Europe. It is due out in paper in the US around 1 October. (Links are to OUP websites.)

My other book, Frege's Theorem, has been out in paper for some time now. These were never terribly expensive (as such things go), but they are cheaper now.

Reviews of Reading Frege's Grundgesetze

With the publication of Philip Ebert's review in Phlosophia Mathematica, there are now four reviews out of my last book, Reading Frege's Grundgesetze:

• Philip Ebert, Phlosophia Mathematica 23 (2015), pp. 289-93
  (Oxford Journals)
• Gregory Landini, European Journal of Philosophy 22 (2014), pp. 159-172 (Wiley Online)
• Oran Magal, Philosophical Quarterly 64 (2014), pp. 526-9 (Oxford Journals)
• Marcus Rossberg, Notre Dame Philosophical Reviews (2014),
  http://ndpr.nd.edu/news/53842-reading-frege-s-grundgesetze/

Thanks to Philip, Gregory, Oran, and Marcus for doing these.

If anyone knows of one I have missed, please do let me know.

Thought Wins PROSE Award for Best New Journal in the Humanities and Social Sciences

The journal Thought, which was founded a few years ago by Crispin Wright, and for which I am the Associate Editor for Philosophy of Language and Philosophy of Mathematics, has been recognized as the Best New Journal in the Humanities and Social Sciences at the PROSE Awards.

I had not previously heard of these awards, but apparently they are the American Publishers Awards for Professional and Scholarly Excellence. They describe themselves this way:

The PROSE Awards annually recognize the very best in professional and scholarly publishing by bringing attention to distinguished books, journals, and electronic content in over 40 categories. Judged by peer publishers, librarians, and medical professionals since 1976, the PROSE Awards are extraordinary for their breadth and depth.

This is a wonderful recognition for what has quickly become, IMHO, an excellent journal. Congratulations are due especially to Crispin and to the other to editors: Carrie Jenkins and John Divers.

For those who have not previously heard of Thought, we specialize in publishing shorter pieces (under 4000 words), kind of like Analysis. We try our best to have a fast turn-around time, though it doesn't always work out that way.

PDF of "From a Logical Point of View"

More and more cool stuff appears in electronic form, this time a PDF of Quine's From a Logical Point of View, free on archive.org.

While we're at it, how about Russell's Inquiry Into Meaning and Truth?

The Squeezing Argument in "Is Frege's Definition of the Ancestral Correct?"

I mentioned in an earlier post my paper "Is Frege's Definition of the Ancestral Correct?" It has now been refereed at Philosophia Mathematica and officially accepted for publication. One of the two reports was unusually helpful and led to some significant improvements in the final version, which is now available online.

The most helpful comment caught a nasty thinko in one of the proofs. But the most interesting observation this referee made was that perhaps the central argument of the paper is a so-called "squeezing argument". (See Robbie Williams's discussion of such arguments here, and Peter Smith's here, a version of which was eventually published in Analysis.)

The rough structure of such arguments is as follows. Suppose there is some intuitive notion I and you want to show that some rigorous notion R is co-extensive with I. Then one way to do so is as follows. Suppose that it is uncontroversial that R gives a necessary condition for I. And suppose further that we can find a different rigorous notion Q that uncontroversially gives a sufficient condition for I. So, to put it set-theoretically, we have:

Q ⊆ I ⊆ R

Then if we can show rigorously that R is sufficient for Q, i.e., that R ⊆ Q, then it will follow that both Q and R are co-extensive with I. As it's put, I has been "squeezed" between  Q and R.

The way this works in the paper is that I is the intutive notion of the ancestral; R is Frege's definition; and Q is an alternative definition that I give and claim, in fact, is intensionally correct. In response to an objection to the intensional correctness of that definition, however, I fall back on this squeezing argument.

This makes at least three instances of this sort of argument: The original, in Kreisel, which is meant to show that the model-theoretic account of validity is extensionally correct; Smith's, which is supposed to show that Turing's analysis of computability is extensionally correct; and now this one. Are there others? I'm guessing maybe there are?

"Frege Arithmetic and 'Everyday Mathematics'" Published

My paper "Frege Arithmetic and 'Everyday Mathematics'" has been published in Philosophia Mathematica 22 (2014), pp. 279-307. Abstract:

This paper shows that predicative Frege arithmetic naturally interprets some weak but non-trivial arithmetical theories. The weak theories in question are all relational versions of Tarski, Mostowski, and Robinson's R and Q, i.e., they are formulated using predicates Pxy, Axyz, and Mxyz in place of the usual function symbols Sx, x+y, and x×y. We lose the existence and uniqueness of successor, sum, and product, as generalizations, but retain these in each particular case (much as we lose the recursion clauses for addition in R, but retain them in each particular case). In saying that the interpretation is "natural", I mean that it relies only upon "definitions" of arithmetical notions that are themselves "natural", that is, that have some claim to be "definitions" in something other than a purely formal sense.

The published version can be found here; the pre-publication version, here. If you need a copy of the published version but don't have access without paying, then send me an email.

"In Defense of Formal Relationism" Published

I am pleased to report that my paper "In Defense of Formal Relationism" has been published in the latest issue of Thought. Here's the abstract:

In his paper “Flaws of Formal Relationism”, Mahrad Almotahari argues against the sort of response to Frege's Puzzle I have defended elsewhere, which he dubs ‘Formal Relationism’. Almotahari argues that, because of its specifically formal character, this view is vulnerable to objections that cannot be raised against the otherwise similar Semantic Relationism due to Kit Fine. I argue in response that Formal Relationism has neither of the flaws Almotahari claims to identify.

Links: PhilPapers, Thought, Pre-publication PDF.

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.

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.