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 simpler 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.

Shortcakes

We had some friends over the other night, and I made shortcakes, which we had with berries and ice cream and whip cream. Yum! I've made quite a few shortcake recipes over the years, and these were the best I've had. Very sweet and flaky. The recipe I used was one from Epicurious, which can be found here. I'll reproduce it for ease of use.

Ingredients

2 cups flour
1/4 cup sugar
1 tbsp baking powder
1/4 tsp salt

1/2 cup (1 stick) chilled unsalted butter
1/2 cup milk
1 large egg

Directions

Preheat the oven to 400F. Line a big baking sheet with parchment paper (which keeps the bottoms from getting too crunchy).

Combine the dry ingredients in a big bowl and stir to blend. Cut the butter into 1/2 inch pieces and sprinkle it into the flour, then mix it a bit so the butter gets coated. Now, work the butter into the flour with your fingers. You basically want to pick up the butter pieces and mash them up. (There are other ways to cut butter into flour, too, but that one seems to work best for this recipe.) Keep doing this until the flour looks like a coarse meal. 

Whisk the egg into the milk, and pour that into the flour mixture. Stir until a dough forms, then turn it out onto a floured work surface and knead it just a bit (maybe 6-8 turns) until its smooth. 

Finally, make eight shortcakes, however you like. You can flatten the dough and cut circles with a glass, or cut triangles with a knife, or divide the dough into eight pieces and shape the shortcakes that way.

Put the shortcakes on the baking sheet. Leave a bit of room between them, as they will rise a bit. Bake them for about 12 minutes, until a tester comes out dry.

Thursday, April 26, 2018

If sexual desire is coded as male....

If sexual desire is coded as male, women begin to wonder if they are really ever sexual. Do we distrust our passion, thinking it perhaps not our own, but the construction of patriarchal culture? Can women be sexual actors? Can we act on our own behalf? Or are we purely victims, whose efforts must be directed at resisting male depredations in a patriarchal culture? Must our passion await expression for a safer time? When will that time come? Will any of us remember what her passion was? Does exceeding the bounds of femininity – passivity, helplessness, and victimization – make us deeply uncomfortable? Do we fear that if we act on our most deeply felt sexual passion that we will no longer be women? Do we wish, instead, to bind ourselves together into a sisterhood which seeks to curb male lust but does little to promote female pleasure?
Those words were published in 1984, by Carol Vance, in "Pleasure and Danger: Towards a Politics of Sexuality". That is the introduction to the extraordinary volume Pleasure and Danger: Exploring Female Sexuality, which is basically the proceedings of the famous 1982 'Barnard Sex Conference'. But it strikes me, in the midst of #MeToo, in all its various manifestations, that those words could just as well have been written yesterday. There are all kinds of ways in which women's sexual desires are questioned, not just by slut-shamers but also by a certain breed of 'feminist'. Of course, there are all kinds of ways in which (sexual) desire is shaped by culture. But, as the women who contributed to Pleasure and Danger understood, we need to find a way to reconcile that fact with the reality of the desires we have: to allow ourselves to enjoy what we enjoy while simultaneously fighting against the culture that shapes us. There's no contradiction in that.

Friday, January 26, 2018

Fascinating Reflection on Transexuality

My friend Anne Eaton directed me to this fascinating but in some ways deeply puzzling essay by Andrea Long Chu, who is (I take it) a graduate student in Comparative Literature at NYU. It's a lengthy investigation of the fraught, and too little discussed, relationship between identity and desire in the experience of trans women, specifically. Probably, Chu rightly restricts her discussion to that case: the one she knows from the inside. But I strongly suspect that her reflections have something much broader to teach us about gender, and our experience of it. Certainly, as someone who is genderqueer, it rang a lot of bells with me.

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).