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
-  Willard V. Quine. Notes on existence and necessity. Journal of Philosophy, 40(5):113–127, 1943.
-  Willard Van Orman Quine. Reference and modality. In From a Logical Point of View, pages 139–159. Harvard University Press, 2nd edition, 1980.
-  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