Friday, 27 March 2015

On Quine's Arguments Against QML, Part 2: The problem of "quantifying in"

Read part 1.

The first of the two problems we look at is related to the problem of ‘quantifying in’. Versions of this argument can be found in [1,2,3]. Quine points out that modal contexts are intensional, by which he means simply that they are non-truth-functional [1, p. 122]; this is why the class of analytic truths is larger than the class of merely logical truths. Intensional contexts are opaque, and they “do not admit pronouns which refer to quantifiers anterior to the context” [1, p. 123]. To illustrate this, he gives his now-famous example of 9 and the number of planets. He says: “The identity

(3) The number of planets = 9

is a truth (so far as we know at the moment) of astronomy” [1, p. 119], [*]. Yet compare (4) “Necessarily something is greater than 7” and (5) “There is something which is necessarily greater than 7”:

(4) L¬∀x¬(x > 7)

(5) ¬∀x¬L(x > 7)

(4) “still makes sense”, according to Quine [1, p. 123], and further more it is true; take, for example, the number 9. But in contrast, (5) is “nonsense” [1, p. 124]. It is nonsense because L(9 > 7) is true, but L(The number of planets > 7) is false, even though 9 and the number of planets are the same (at least at the time he was writing). It is false because there is no analytic connection between ‘the number of planets’ and ‘> 7’.

The problem with this as an objection is that synonymy — and hence analyticity itself, since it is defined in terms of synonymy — is a contingent matter; it is accidental whether two terms are synonymous or not. In fact, the falsity of “The number of planets = 9” demonstrates the contingency of the matter; the fact that the IAU was able to redefine what it meant to be a planet, and hence change the number of planets in our solar system, shows that there is no necessary connection between the concepts ‘9’ and ‘the number of planets’. Their synonymy was only accidental.

At this point, an interesting parallel can drawn between this example and one that can be found in another area of modal logic, namely, the Aristotelian modal syllogistic. One of the long-standing difficulties commentators (ancient, medieval, and modern) have had with interpreting Aristotle’s modal syllogistic was the Two Barbaras problem: his insistence that NXN Barbara was valid while XNN Barbara was invalid. NXN Barbara is the first-figure syllogism Barbara with a necessary major, assertoric minor, and necessary conclusion, while XNN Barbara has an assertoric major and necessary minor. Commentators find common ground against Aristotle in two ways: Either they believe that neither form should be valid, or that if one is valid, there is no way to distinguish which one, and hence both should be. Here is an example in the form of NXN Barbara:

Necessarily all elms are deciduous.
All the trees in my yard are elms.
Therefore, necessarily all trees in my yard are deciduous.

One standard objection to the validity of such an argument is that the connection between being a tree in my yard and an elm tree is accidental; there is no deep underlying relation between these two concepts. This contingency in a sense “spills over” into the conclusion; it would be acceptable to draw an assertoric conclusion, but a necessary one is too strong.

Let us compare NXN Barbara with the following:

L(9 > 7)
The number of planets = 9
Therefore, L(The number of planets > 7).

In both cases, the way to rehabilitate the argument would be to necessitate the second premise; but in order to retain soundness this would require that ‘The number of planets = 9’ or ‘All the trees in my yard are elms’ be analytic (for only then would the result of prefixing them with ‘L’ be considered true, on Quine's account); but there is no reason to think that these premises are analytic.

The fact that the analogous argument is invalid, is, far from being a reason to reject quantifying into intensional contexts as incoherent, actually evidence that Quine is correctly analysing necessity-as-analyticity. This is exactly the sort of behaviour that we would want to see, since it is precisely because the identity statement is a merely accidental identity — as witnessed by the fact that while it used to be true, it is now in fact false — that we should reject the conclusion. Thus the problems that Quine sees arising from this example are not actually reasons for rejecting quantified modal logic, but rather reasons for embracing it: It is an advantage of Quine’s analytic approach to modal logic, not a disadvantage, that it makes such arguments invalid. Given that synonymy, and hence analyticity, is a matter of accident, we should not expect analytic identities to result in necessary conclusions, and if they did, we would have reason to question these conclusions on the same grounds that people question the validity of NXN Barbara.

References & Notes

  • [1] Willard V. Quine. Notes on existence and necessity. Journal of Philosophy, 40(5):113–127, 1943.
  • [2] Willard Van Orman Quine. Reference and modality. In From a Logical Point of View, pages 139–159. Harvard University Press, 2nd edition, 1980.
  • [3] Willard Van Orman Quine. Word and Object. MIT Press, 1960.
  • [*] Nowadays, of course, we know differently. It is rather amusing that two of the enduring platitudes in philosophy—that all swans are white and that there are nine planets — have both turned out to be false; Australia provided us with black swans, and the International Astronomical Union deprived us of Pluto.

© 2015 Sara L. Uckelman


  1. Hi Sara,

    I'm happy to see this series of posts. I find the topic very interesting and have been posting on related matters lately myself (with more to come), from the point of view of some accounts I'm developing.

    It's interesting to see a connection with the problem of interpreting Aristotelian syllogistic.

    I am puzzled by quite a bit of what you're saying though. In particular from this post:

    'In fact, the falsity of “The number of planets = 9” demonstrates the contingency of the matter; the fact that the IAU was able to redefine what it meant to be a planet, and hence change the number of planets in our solar system, shows that there is no necessary connection between the concepts ‘9’ and ‘the number of planets’. Their synonymy was only accidental.'

    Firstly, about 'their synonymy': on what basis can it be said that '9' and 'the number of planets' were ever synonymous? Only, it seems to me, on a view according to which sameness of referent suffices for synonymy. But that seems intuitively wrong.

    Secondly, your point about the IAU seems to be doing a lot of work in your overall argument here, but it seems pretty dubious to me. Isn't it possible, indeed natural, to maintain that what the IAU did is to lay down a new standard technical meaning for a certain expression, and so now the sentence in question means something different. But that doesn't mean we can't revert to the old meaning for the sake of an example, nor does it mean that the truth-value of a proposition of astronomy has somehow been changed by the IAU.

    Thirdly, why bring in such considerations, just to show that there is no necessary or analytic connection between '9' and 'the number of planets'? Isn't this intuitively obvious, and if you want an argument, couldn't an argument based on considering counterfactual scenarios where the solar system had a different number of planets, do the job much more cleanly than the objectionable IAU argument?

    Finally, I'm not seeing clearly how synonymy being a contingent matter would be a problem for the objection you are considering.

  2. Hi Tristan,

    Thanks for the comments -- you've shown a number of places where I need to be more careful and precise with my argument. I particularly take your point about the IAU's definition -- if I understand it right, what you're saying is that there are two different concepts here, planet-pre-2006 and planet-post-2006, rather than one that has changed its meaning. While I think this is certainly a position that you could adopt, I'm not sure you _have_ to. Linguists tend to take the phenomenon of meaning change seriously, as something that actually happens, such that one and the same word can meaning different things at different times, rather than there being a multiplicity of equivocal terms. If you allow for genuine meaning change, then in most cases, it is contingent whether two words mean the same thing. The analogy with NAN Barbara is then clearer -- Quine tries to draw a necessary conclusion from one necessary and one contingent premise, and we might want to reject that for the same reasons that we'd reject NAN Barbara.