π, τ, and Quine's pragmatism

Some readers may already be familiar with the π vs. τ debate. If not, I recommend checking out Michael Hartl's τ manifesto and this fantastic short video by Vi Hart. An attempt at a balanced evaluation of π vs. τ can be found here.

For those who don't know, τ is just 2π. The defenders of τ argue (as seen in the above links) that using it instead of π makes many things much clearer and simpler/ more elegant.

Let's assume for present purposes that the τ-proponents turn out, in the end, to be right. I want to ask a further question: what would this then say about Quine's denial of the analytic-synthetic distinction? Quine's denial, virtually all agree, is the claim that all rational belief change is pragmatic, i.e. there is no principled difference between questions of evidence/justification on the one hand, and questions of efficiency and expedience on the other (= between external and internal questions, i.e. between practical questions of which language-form to adopt, and questions of whether a particular empirical claim is supported by the available evidence).

So here's my question: if Quine is right, then is our old friend C=2πr simply wrong (and C=τr right)? If not, how can a Quinean wiggle out of that consequence? And if so (i.e. C=2πr really is wrong), does the Quinean have any way of softening the sting of this apparently absurd consequence?

Boghossian on ('metaphysical') analyticity

I've been thinking recently about an objection Paul Boghossian (and many others) make against the Tractarian/ Carnapian conception of an analytic truth, viz. a sentence that is true solely in virtue of the meaning of the sentence. (Boghossian calls this kind of analyticity 'metaphysical analyticity,' which I think is potentially misleading, given the staunch anti-metaphysical tastes of the logical empiricists. Oh well.)

Boghossian considers the notion of metaphysical analyticity untenable. Why? He asks a rhetorical question:
"Isn't it in general true---indeed, isn't it a truism---that for any statement S,

S is true iff for some p, S means that p and p?

How could the mere fact that S means that p make it the case that S is true?" (Boghossian 1996, "Analyticity Reconsidered," Nous [p.364]

Boghossian is not alone in this view: the basic idea can be found in Quine's "Carnap and Logical Truth," and is developed by Gilbert Harman, Elliott Sober, and Margolis & Laurence. How should we interpret this rhetorical question? Boghossian appears to be claiming that the truth of a sentence of the form 'S means that p' is never a sufficient condition for the the truth of a sentence of the form 'S is true'---that appears to be intended force of the rhetorical question in the quotation immediately above. And that is certainly one reasonable way of cashing out the notion of the truth of a sentence being `fixed exclusively by its meaning.'

If we do understand Boghossian's view in this way, then I think his claim is either misleading or incorrect. Consider a standard material biconditional of the form

(1) p iff q

If such a biconditional is true, we usually say that q is a necessary and sufficient condition for p. But as we teach undergraduates in Introduction to Logic classes, if this biconditional is true, then (within the classical propositional calculus) so is

(2) p iff [q and (r only if r)]

(Any other logical truth of the classical propositional calculus could be substituted for r only if r.) If we simply read off the surface structure of sentence-schema (2), one might think that q was no longer sufficient for the truth of p--because there appears to be a second condition that has to be met in order for p to be the case, namely that r only if r. Of course, strictly speaking, this is true: every sentence of the propositional calculus presupposes the truth of all the logical truths of the propositional calculus. However, it seems seriously misleading to me to say that the truth of q is not a sufficient condition for the truth of p in our original biconditional--for that is not the way we standardly understand sufficient conditions.

Hopefully the direct parallel with Boghossian's claim is clear. I certainly agree that his 'truism' quoted above is true. However, when a logical truth--which, as Carnap and Quine agree is a paradigmatic case of analytic truth (if there are any)--is substituted for p in his schema, then that instance of the truism will have (almost) exactly the form of the second biconditional (2). Then, in the usual sense of 'sufficient condition,' we will have a case in which (contra Boghossian) an instance of 'S means that p' is sufficient for 'S is true.' To say otherwise, we would have to give up either classical logic (specifically, the idea that (2) follows from (1)) or the usual understanding of sufficient conditions.

However, one could object that neither classical logic nor our standard view of sufficient conditions is sacrosanct. I think there are reasonable replies to these objections (telegraphically: for whatever non-classical logic you choose, you can substitute some other logical truth for 'r only if r' in (2) above, and the point carries); but I'll leave matters here since this post is too long already.

Comments and criticism from any angle are very welcome, but what I personally go back adn forth on with the above argument is whether it's a 'cheap point' or not... superficial logic-chopping, or genuine insight?

Analyticity in model-theoretic languages

Part of why I am drawn to philosophy of science and logic is that I like to operate with clean and neat formulations of apparently messy concepts -- and these two sub-disciplines of philosophy embrace such tastes more than other sub-fields. Of course, I am not claiming that ethicists and metaphysicians are muddle-headed; most think about their sub-discipline's topics with far more clarity and rigor than I can. I am merely expressing a personal preference for studying deontic logic instead of the most recent form of consequentialism.

Enough autobiography -- I mention it only to explain my motivation for this post. And my point is this: if we adopt the usual formalization of an interpreted language (viz., the model-theoretic one), then we apparently cannot capture the notion of analyticity -- at least in the way Carnap, who is widely recognized as the champion of analyticity, conceives of it.

Conceiving of a language in model-theoretic terms is one widely-used way of introducing precision into a philosophical endeavor. Most readers probably can recite the definition of a model-theoretically understood langauge by heart, but for the innocent:
A language L consists of a ordered triple , where
- L carries grammatical information: which symbols belong to the language, which strings of symbols count as sentences, which grammatical category each symbol belongs to, etc.;
- M is a model =< D, f >, where the domain of discourse D is a set of individuals, and f is an interepretation function, which assigns an individual in D to each proper name in L, sets in D to one-place predicates, sets of ordered pairs drawn from D to two-place predicates, and so on; and
- r specifies the truth-values of certain compound sentences, given the truth-values of their components -- in other words, r basically specifies the truth-tables.

So much for the model-theoretic conception of language; what about analyticity? Carnap, throughout his career, identifies the analytic truths as those sentences that are true merely in virtue of the language one speaks. That is, if we specify that I am speaking a particular language, in the course of that specification, I might present enough information that the truth-values of certain sentences within that language are fixed. (For an obvious example: if I specify what 'and' and 'not' mean in my language via the usual truth tables for those words, any sentence of the form 'p and not-p' comes out false merely in virtue of the rules governing the language I am using.)

Now, after all that rehearsal of material most readers probably know well, I can get to my point. In a model-theoretically characterized language, the truth-values of ALL sentences are determined by the specification of that language. For example, the truth-value of atomic sentences such as 'Fb' are true iff the individual named by 'b' is in the extension of the set associated with 'F' (i.e. 'Fb' is true iff f(b) is an element of the set f(F)). And Carnap certainly never wanted every sentence of a (non-contradictory) language to be analytic.

The problem then is: one of my favorite tools for 'precisification' in philosophy -- model-theoretic languages -- apparently affords no way to characterize one of the concepts I'm most interested in: analyticity. What to make of this? The first, obvious thing to say is: "Of course there couldn't be any explication of analyticity in such languages, because such languages are extensional, and Carnap and Quine (who represent opposing positions in debates over analyticity) both basically agree that analyticity is an intensional notion."

This is right, but I think there is something further to note: in a straightforward sense, every sentence in a (classical) model-theoretic language has its truth-value determined by the specification of the language. That is, by specifying the language, we fix the truth-values of all the sentences in such a language. That seems odd -- the model-theoretic way of specifying a language that has proved very useful in certain situations, but it likely cannot be a fundamental and/or universally applicable one.

One further point: Carnap, Quine, and the other primary antagonists in battles over analyticity all agree that if there is any such thing as analytic truth, then the (so-called) logical truths are paradigm instances of analytic truths, i.e., truth in virtue of meaning (if you are thinking of "Two Dogmas" and don't believe me, look at Word & Object, sec. 14, fn.3, p.65). But the model-theoretic conception of language characterizes the logical truths as a class of sentences that are true across a set of related langauges. That is, to know whether a sentence is a logical truth in one model-theoretic language, you have to check whether that sentence is true in a bunch of other model-theoretic languages that share certain features with the first one.

So, one might think that the way to cash out analyticity in the idiom of the philosophical logician is to use something like Kripkean possible world semantics (which are used, with some variations, in modal, deontic, epistemic, and temporal logics). But these are usually not given linguistic interpretations, and it's not clear to me that it's possible to give a decent one... though I'd love to be wrong. Any thoughts?

Galileo:Scholastic natural philosophers :: Carnap:Quine

I'm pretty sure the following has been said before, but I don't know where: Frege (and Russell, et alii) did for the study of language what Galileo (et alii) did for the study of material nature. Galileo 'mathematized' new portions of the physical world -- previous students of nature thought that (most of) nature was too messy, imprecise, or chancey to be susceptible mathematical treatment: how could the unchanging, eternal realm of mathematics model the changing and temporal material world? Analogously, Frege and other founders of modern logic turned language into a mathematical object, by treating (e.g.) subject-predicate assertions in terms of functions and their arguments.

I bring this up because I've been looking at one of Quine's arguments against Carnapian analyticity in the 1940s. This argument appears in the long 1943 letter from Quine to Carnap in Creath's Dear Carnap, Dear Van, and in print in a 1947 article in the Journal of Symbolic Logic ("The Problem of Interpreting Modal Logic"):

The class of analytic statements is broader than that of logical truths, for it contains in addition such statements as 'No bachelor is married.' This example might be assimilated to the logical truths by considering it a definitional abbreviation of 'No unmarried man is unmarried,' which is indeed a logical truth; but I should prefer not to rest analyticity thus on an unrealistic fiction of there being standard definitions of extra-logical expressions in terms of a standard set of extra-logical primitives. What is rather in point, I think, is a relation of synonymy, or sameness of meaning, which holds between expressions of real language, though there be no standard hierarchy of definitions. (p.44, italics Quine's, boldface mine)

Quine puts the point somewhat differently in different places, but the basic idea is always that the 'rational reconstruction' of language, however it is carried out, is an 'unrealistic fiction.'

Now, I can ask the question: is Quine's charge that a 'hierarchy of definitions' is an 'unrealistic fiction' any different from Galileo's scholastic critics' charge that Galileo is somehow 'falsifying' nature by rendering it thoroughly mathematically? The answer to this question will turn on what Good Things the Galilean mathematizing strategy is able to achieve (explanatory power, new predictions, etc.), and whether these Good Things (or analogues of them) also appear in the case of a Carnapian language. It would also be useful to know of other scientific cases where the analogue of the Scholastic triumphed over the analogue of Galileo, i.e., someone tries to mathematize certain phenomena, but this mathematization is rejected for good reasons by workers in the field.