Modal Glitch

See the update
It’s not mistaken claims generally that bother me—a historian of philosophy needs a high tolerance for error—but those that, being put forward with every appearance of confidence, and yet in my view visibly false, put me in a situation of doubt. Why would someone who seems to know their stuff say that?
With respect to some matters I can live with, even thrive on, doubt; in others, I want the itch scratched soon. Here’s a case in point. Writing on progress in philosophy(), Timothy Williamson says at one point that
The principle that every truth is possibly necessary can now be shown to entail that every truth is necessary by a chain of elementary inferences in a perspicuous notation unavailable to Hegel.
We certainly do have notations unavailable to Hegel. We can do many things with them that Hegel couldn’t. But proving that ◊◻p ⊦ ◻p isn’t one. Not without additional assumptions. The model at the right is a counterexample. We stipulate that the proposition p is true in the greenish worlds, false in the reddish worlds (it doesn’t matter what value p has in the actual world a). ◻p holds at v because it holds in every world accessible from v (w, or w and v itself if the accessibility relation is reflexive). ◊◻p holds at a, the actual world, because ◻p holds in at least one world accessible from a, namely v. But ◻p doesn't hold at a because p is false at x, which is also accessible from a.
If the accessibility relation is symmetric (which would make every arrow double-ended), you still can produce a counterexample: just make a greenish too. If the accessibility relation is reflexive, symmetric, and transitive, there are no counterexamples. Accessibility is then an equivalence relation, so that if ◻p holds at v, and v is accessible from a (hence conversely), p must be true not only at every world accessible from v but at every world accessible from a.
So what was Williamson thinking? Did I doze off on the wrong day when I took modal logic?
Update. My colleague Gillian Russell notes that what can be proved in elementary fashion is ◊◻p ⊦ ◻p in S5, and that if the modality in question is something like physical necessity (= true in all worlds with the same laws as ours), then that modality will satisfy the axioms of S5 (because in that case the accessibility relation is indeed an equivalence relation—as, trivially, any relation of the form ‘same x as’ will be). This may be what Williamson had in mind. —Further addition: on the question of the One True Logic, see Greg Restall & J. C. Beall, “Logical Pluralism,” Australasian Journal of Philosophy, 78 (2000) 475––493, available here. The relation of consequence is defined as follows: ‘q is a consequence of p’ iff in every case where p is true, q is true; or, equivalently, in no case is p false while q is true. The crux is the specification of cases: according to Restall and Beall, there is “no canonical way to spell out the cases” to which the definition of consequence applies, hence no One True Logic, if the job of logic is to define ‘valid consequence’. It follows that unless S5 or something as strong holds for all valid modal consequences, there is no way to prove that ◊◻p ⊦ ◻p except with respect to some modal logic or other.
() Timothy Williamson, “Must do better”. The argument of the piece is that philosophers would do better to attempt less, but with more rigor. Among other things, Williamson writes: “Would it be a good bargain to sacrifice depth for rigour? That bargain is not on offer in philosophy, any more than it is in mathematics.” (p15 of the version available online on 17 Nov).
This is one of those homilies that begs for counterexamples. What about Poincaré? Dieudonné in his history of algebraic topology (see his chapter in Jean-Paul Pier, ed. Development of mathematics, 1900–1950, Birkhäuser, 1994) complains about Poincaré’s lack of rigor even as he acknowledges the fundamental contributions of Poincaré. Was P. “trading” rigor for depth? Would he have been less deep had he been more rigorous? Merely to ask these questions suggests to me that the opposition of depth and rigor is fishy. Williamson himself goes on to argue that rigor promotes depth in various ways. We’re not, therefore, talking about contraries, but about two virtues a philosopher’s thinking may exhibit.
Some of what passes for rigor is a waste of time, both in mathematics and in philosophy. I remember reading an introductory text in philosophy, very instructive in its way, that tried to cast various great arguments (the Ontological Argument, the cogito) into first-order logic, that being, apparently, the test of meaningfulness. Naturally they came off badly. The conclusion one ought to draw is that first-order logic (or rather what can be, more or less faithfully, “translated” into it) does not exhaust the domain of meaningful discourse. Insisting that philosophy limit itself to that idiom is illusory, not genuine, rigor.
There is illusory depth also, of course. Williamson writes:
No doubt, if we aim to be rigorous, we cannot expect to sound like Heraclitus, or even Kant: we have to sacrifice the stereotype of depth. Still, it is rigour, not its absence, that prevents one from sliding over the deepest difficulties, in an agonized rhetoric of profundity.
I’m sure we can agree that an agonized rhetoric of profundity is a Bad Thing, just as a timid pedantry of rigor would be. The words indicate as much. Kant has been accused of both (though “agonized” would be a bit melodramatic in his case). It is at least possible that despite what reason permits us to conceive or suppose explicable, in some events we find ourselves at a loss, at the limits of what we can easily make sense of. Kant goes to some lengths to show how that might happen.
The genuine opponent—rather than the straw men erected by Williamson—of what he calls “rigor” is a conception of philosophy according to which it is the pursuit of wisdom, a pursuit to which the quasi-scientific pursuit of rigorously argued truth—the version of philosophy Williamson is advocating—is a needed instrument, but only that (see Pierre Hadot’s Qu’est-ce que la philosophique antique? and Exercices spirituelles for surveys of ancient and early modern conceptions of the pursuit of wisdom). The difference between the two is apparent in what each regards as legitimate forms of discourse in philosophy. Though it is granted that “philosophy cannot be reduced to mathematics” (p17), the ideal of rigorous philosophy is to approach the condition of (rigorous) mathematics. The pursuit of wisdom may include the doing of mathematics (I’ve just been reading nice work by Matt Jones that makes the argument for this in some 17th century philosophers); but what one proves mathematically will not be the content of wisdom: mathematics is rather an exercise that promotes a condition of wisdom, which is not a set of propositions but a mode of human existence. Other disciplines may be equally useful, other styles of discourse equally effective—poetic, riddling, dramatic, mythic. For that matter, music and the other arts may also serve.
I’m not advocating this conception of philosophy, though I think Hadot makes a good case for conceiving much of ancient philosophy as the pursuit of wisdom through spiritual discipline. I’m suggesting only that the interesting question is not whether genuine profundity and rigor are better than fake profundity and rigor; it is understanding the two conceptions, which in my view must include an understanding of their history. “Rigorous philosophy” would certainly be included among the disciplines of wisdom. Whether only rigorous philosophy deserves to be called philosophy seems to me a matter mostly of academic politics.

LinkNovember 20, 2004 in Philosophy of Mathematics