Tuesday, November 20, 2018

Kids These Days

A few weeks ago, a student at Brown, Lauren Black, interviewed me for a podcast she was doing about children's beliefs about the universe. The result is here, at Now Here This. The one I'm in is titled The Conference Room in the Sky.

Saturday, November 3, 2018

Sir Michael Anthony Eardley Dummett

A wonderful email I got from a past student reminded me of how much I miss my old BPhil supervisor, Sir Michael Dummett. I won't say much about him here. Philosophically, my work speaks to his influence, I hope, and I had my say, among many others, in the remembrances that Ernie Lepore assembled when Michael died. I also wrote about his impact on philosophy of mathematics for Philosophia Mathematica.

Michael was way ahead of his time, in so many ways. To mention just a few: His distinction between ingredient sense and assertoric content; his emphasis on indefinitely extensible concepts; his insistence, way back, that theories of meaning (for natural languages) must be theories of truth; that both must be theories of understanding; and that those in turn must be theories of what competent speakers know. Dummett, despite himself, was a neo-Davidsonian a decade or more before James Higginbotham, another of my teachers, would bring that sort of view into the center of philosophy of language and, for that matter, natural language semantics.

Tuesday, October 30, 2018

Anne Koedt, "The Myth of the Vaginal Orgasm"

Anne Koedt's essay "The Myth of the Vaginal Orgasm" (1970) is a classic of second wave feminism, an important example of the concern feminists of that era had with sexuality, but also an intellectual precursor of political lesbian and the divisive debates that surrounded that topic. One can find 'reprints' all over the web, and the essay was reprinted in a collection of essays, Radical Feminism, edited by Koedt that was published in 1973. But the version published in that book, and (from what I can tell) 'reprinted' elsewhere, is not quite the same as the original.

I know this because I was fortunate enough to find a copy of the original on abebooks.com. It was published by the New England Free Press (operating out of Boston) on two double-letter (8.5"x22") pages, printed both sides, and folded into a letter-sized (8.5"x11") pamphlet (with no staples, at least in mind). Cost: 10 cents, about 70 cents in 2018. It's the kind of thing one would have found in 'radical' bookstores back in those days.

Because of the historical importance of this work, it seems worth making a copy available online. So here's a DjVu and a PDF.

(If anyone should have good reason to object to my making this available, please let me know, and I'll be happy to remove it.)

Saturday, October 20, 2018

Corner Quotes in LaTeX

I'm posting this just because I had a hard time, today, finding the original source for a macro I've been using for a while in LaTeX. The macro in question typesets 'corner quotes', such as:
⌜A ∧ B⌝
The corners themselves are not hard to create, since LaTeX has \ulcorner and \urcorner macros. That will usually work fine, but there are issues involving the height and spacing of the corners that can arise in some cases. A macro due to Sam Buss solves these problems. (I discovered it here.) I've put the macro in a style file, godelnum.sty, which you can download here. Here's the contents:
\newbox\gnBoxA
\newdimen\gnCornerHgt
\setbox\gnBoxA=\hbox{$\ulcorner$}
\global\gnCornerHgt=\ht\gnBoxA
\newdimen\gnArgHgt


\def\Godelnum #1{%
       \setbox\gnBoxA=\hbox{$#1$}%
       \gnArgHgt=\ht\gnBoxA%
       \ifnum \gnArgHgt<\gnCornerHgt
               \gnArgHgt=0pt%
       \else
               \advance \gnArgHgt by -\gnCornerHgt%
       \fi
       \raise\gnArgHgt\hbox{$\ulcorner$} \box\gnBoxA %
               \raise\gnArgHgt\hbox{$\urcorner$}}
Usage is just: \Godelnum{A \wedge B}, and the like. Note that one can do this outside math, since the macro inserts $s around the argument. (Probably \ensuremath would be better.)

Friday, October 12, 2018

Newly Published: Logicism, Ontology, and the Epistemology of Second-Order Logic

In Ivette Fred and Jessica Leech, eds, Being Necessary: Themes of Ontology and Modality from the Work of Bob Hale (Oxford: Oxford University Press), pp. 140-69 (PDF here)

Abstract:
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.
And from the acknowledgements:
It is the peculiar tradition of our tribe to express our respect for other members by highlighting our disagreements with them. So, in case it is not clear, let me just say explicitly how much I admire Bob Hale’s work. I learned a lot from him over the years—both in conversation and from his written work—and greatly enjoyed the time we were able to spend together. Bob’s enthusiastic support for me and my work, early in my career, was particularly important to me. So I am honored to be able to contribute to this volume and thank Ivette and Jessica for the invitation.
I'm particularly sad, for myself, that Bob passed before we had a chance to discuss these issues one more time....

Thursday, September 13, 2018

Tarski: A Semantic Proof of Incompleteness, and a Possible Error

In re-reading Tarski's paper "The Semantic Conception of Truth and the Foundations of Semantics", for my course on theories of truth, I was struck by some remarks he makes in footnote 17:
...[I]n view of the elementary nature of the notion of provability, a precise definition of this notion requires only rather simple logical devices. In most cases, those logical devices which are available in the formalized discipline itself (to which the notion of provability is related) are more than sufficient for this purpose. We know, however, that as regards the definition of truth just the opposite holds. Hence, as a rule, the notions of truth and provability cannot coincide; and since every provable sentence is true, there must be true sentences which are not provable. 
It was not immediately obvious to me what argument Tarski was making here. It seems worth spelling it out in some detail.

The two central premises are of course clear enough. First, we know that in any theory of sufficient strength (e.g., any theory extending Robinson arithmetic Q), the notion of provability can be formally defined. More precisely, the relation "P is a proof of S in T" can be 'represented' in Q in a familiar sense, and provability in T can then be defined as the existential quantification of this relation. Second, we have Tarski's theorem: For any sufficiently expressive language L, such as the language of arithmetic, there can be no formula Tr(x) that defines truth in L.

What's puzzling about this is what's often puzzling about Tarski's remarks on such topics: The remarks about provability are most familiarly applied to theories, such as Q; but Tarski's theorem is most familiarly applied to languages, such as the language of arithmetic. Tarski famously conflates these two notions in many of his writings on truth. And, in fact, it seems to me that Tarski's argument here is only correct if it is primarily one about languages.

Suppose we try to interpret it as one about theories. There is no formula Tr(x) such that, in PA, we can prove all instances of "Tr(S) iff S". But no matter how we interpret the remarks about provability, it will not then follow that "truth and provability cannot coincide". It's perfectly possible that Tr(x) should have the same extension as Pr(x) even though we cannot prove that fact in PA.

So Tarski must mean something like the following. Consider e.g. the language of arithmetic, or any other sufficiently expressive language. In that language, we can define the notion of provability in PA and, indeed, in any formal theory stated in that langauge: That is, there is, for each such theory, a formula PrT(x) that is true of all and only the Gödel numbers of T-provable sentences. But we know from Tarski's theorem that there is no formula Tr(x) that is true of all and only the Gödel numbers of true sentences of L. Hence, those two sets cannot coincide, no matter what formal theory T might be.

As Tarski mentions in note 18, this is no real improvement over Gödel. Most of the machinery that Gödel develops for the purposes of his proof is needed for this one: the definition of provability and the diagonal lemma. It is perhaps worth mentioning, however, that there is no need to suppose that any particular theory proves anything about T-provability, and no hypothesis of the consistency of T is needed here, either.

There's another puzzling remark that Tarski makes in this paper that I am not sure this one can be salvaged. The remark is one Tarski makes during the discussion of 'essential richness':
If the condition of “essential richness” is not satisfied, it can usually be shown that an interpretation of the meta-language in the object-language is possible; that is to say, with any given term of the meta-language a well-determined term of the object-language can be correlated in such a way that the assertible sentences of the one language turn out to be correlated with assertible sentences of the other. As a result of this interpretation, the hypothesis that a satisfactory definition of truth has been formulated in the meta-language turns out to imply the possibility of reconstructing in that language the antinomy of the liar; and this in turn forces us to reject the hypothesis in question. (pp. 351-2)
I take there to be here an assertion of the following claim. Suppose that a theory M is (relatively) interpretable in another theory O. Then truth for the language of O cannot be defined in M (since then the liar would be reproducible in M). So understood, the claim is false. Ali Enayat and Albert Visser showed that PA plus a Tarski-style truth-theory for the language of arithmetic is interpretable in PA. (A proof of the same result is given in my paper "Consistency and the Theory of Truth". I should have mentioned all this there.)

Tarski's unclarity about 'language' vs 'theory' makes it unclear, however, exactly what he meant. So, I take his claim to concern theories because I don't know of any coherent notion of interpretation for languages, and of course Tarski is largely responsible for the notion of interpretation to which he is here alluding. (The paper in which Tarski first introduces and studies this notion would not be published until 1953, however: nine years later.) Moreover, his talk of "reconstructing in that language the antinomy of the liar" certainly sounds like talk of provability. But perhaps there is something else he had in mind.

Friday, July 6, 2018

Pasta on the Grill

One of our "go to" summer dinners has become what we call "pasta on the grill". It's basically a roasted tomato sauce, which we started making when we started growing tomatos. Its a fast and easy way to use some of them, and we freeze several jars of it every year to use over the winter. It's simple and delicious and easily varied. I'll give the basic idea here and mention some variations.

Ingredients

  • 2-3 pounds of tomatos, whatever kind you have available
  • Half a large Vidalia onion, or a single large yellow onion
  • 4-5 cloves of garlic
  • A small eggplant (we usually use Asian styles)
  • A red or yellow pepper
  • 1/2 a jalepeño or other hot pepper (optional)
  • 1/2 cup chopped fresh herbs (basil, oregano, thyme, whatever)
  • Olive oil
  • One can tomato paste (optional)
Varying the types of tomatos and onions and herbs will affect the flavor a lot, as will omitting the garlic, which I do from time to time. You can also use zucchini or summer squash in addition to or in place of the eggplant. The jalepeño adds a nice bit of heat, if you like that, or you can use Thai basil or spicy oregano. And so forth.

Directions

Prepare your grill. We have a gas one, so that is easy. I tend to cook this over pretty high heat, but you'll want to experiment with that. (You can also do this in the oven, if you wish. Preheat it to 450F.)

Chop and otherwise prepare the veggies and herbs, and put them in a big bowl. Add the olive oil and stir. You want just enough to coat the veggies. Line a 13x9 disposable aluminum baking pan with foil (for ease of cleanup and re-usability of the pan) and pour the veggies into it. Cover the top with more foil, and put the whole thing onto the grill. Cook until the veggies are tender, roughly 30-45 minutes.

I often find that the sauce is a bit watery at this point. If so, then I pour it into an appropriately sized pot and add some tomato paste---half a can usually suffices, but sometimes I need a full can---and simmer for about 10 minutes, just to cook the paste.

Serve over your favorite pasta.