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).

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.

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.

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.