I just finished quite a streak of formal talks in philosophy. From Thursday night until Sunday, I (like Marc Moffett) was in Vancouver for the Society for Exact Philosophy, which was quite a fun little conference with a lot of interesting talks. Then on Monday, those of us at Berkeley working with Branden Fitelson got together with the people at Stanford working with Johan van Benthem for an informal workshop, which like last year had a lot of talks on probability from the Berkeley side and dynamic epistemic logic from the Stanford side, and again helped reveal a lot of interesting connections between these two rather different formal approaches to epistemological questions. And then today we had our quasi-monthly formal epistemology reading group meeting at Berkeley, with Jonathan Vogel from UC Davis.

There was a lot of interesting stuff discussed at all these places, but I’m glad there’s a bit of a break before FEW 2007 in Pittsburgh. Anyway, it’s also very nice to know that there is all this work going on relating formal and traditional issues, both in epistemology and other areas of philosophy.

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Anyway, among the many interesting talks, the one that’s got me thinking the most about things I wasn’t already thinking about was the one by Mark Colyvan, on what he calls the “principle of uniform solution”. The basic idea is that if two paradoxes are “basically the same”, then the correct resolution to both should also be “basically the same”. So for instance, it would be very strange for someone to claim that the correct approach to Curry’s Paradox is that certain types of circularity make sentences ill-formed, while the correct approach to the Liar Paradox is to adopt a paraconsistent logic. Mark pointed out that there are some problems with properly formulating the principle though – do we decide when paradoxes are “basically the same” in terms of their formal properties, the sorts of solutions they respond to, or the role they’ve played in various arguments? For instance, Yablo’s Paradox was explicitly introduced in order to point out that self-reference is not the key issue in the Liar, Curry, and Russell paradoxes – which suggests either that the relevant formal property they share is something else, or that the proper way to think of paradoxes is something else.

In hearing this, I started to wonder just why we should believe anything like this principle of uniform solution anyway. The strongest cases of the relevant form of argument seem to me like the appeal in Tim Williamson’s *Knowledge and its Limits* to various different forms of the Surprise Examination Paradox – he points out that some traditional resolutions only solve the most traditional version, but that a slightly modified version gets through, and that his proposed solution to that version blocks the traditional version as well. Since both cases seem problematic, and one “covers” the other, it seems that we only need to worry about solving the covering case. I take it that something like this is at work as well when Graham Priest uses the Liar paradox to argue for dialetheism, and then suggests a return to Frege’s inconsistent axiomatization of mathematics rather than using the much more complex system of ZFC.

If this is the form of argument, then we shouldn’t always expect the principle of uniform solution to be worth following. If I (like most philosophers that don’t work directly on this sort of stuff) think that something like ZFC is the right approach to Russell’s Paradox, and something like Tarski’s syntactically typed notion of truth is the right approach to the Liar Paradox, then both get solved, but neither approach would work for both. Their formal similarities are interesting, but there’s no reason they should have the same solution, since there isn’t an obvious solution that works for both (unless you go for something as extreme as Priest’s approach). Formal or other similarities in paradoxes often help show that resolving one will automatically resolve the other, so that the above argument will work, but there’s no reason to think that this will always (or even normally) be the case.

But at the same time, something like this principle seems to work much more generally than in the case of paradoxes. There are certain similarities between the notion of objective chance, and the notion of subjective uncertainty, so it makes sense that we use a single mathematical formalism (probability theory) to address both. Alan Hájek has suggested that these analogies continue even to the case of conditionalizing on events of probability zero, though I think that this case isn’t as strong. (Though that might just be because I’m skeptical about objective chances.) There’s a general heuristic here, that similar issues should be dealt with similarly. In some sense, it seems very natural to suggest that differences in approaches to different issues should somehow line up with the differences between the issues. But we don’t expect it to always work out terribly nicely.

Anyway, there’s a lot of interesting methodological stuff here to think about, for paradoxes in particular, and for philosophy in general (as well as mathematics and the sciences).

Aidan McGlynn(15:06:40) :Well, as you say, the relevant notion of ‘basically the same’ is somewhat elusive, and is bound to be vague. But some people would hold that ZFC and the Tarskian solution to the Liar are ‘basically the same’ kind of solution – both get us to ‘go hierarchical’ in some sense. Alan Weir, for example, groups them together in this fashion in ‘Naive Set Theory is Innocent!’. (I should note that Alan is explicit that he doesn’t want to presuppose that ZFC is ‘interpreted via the “iterative” conception of set’ (768) in making this claim; that is, he thinks the conception of set embodied in ZFC is hierarchical in the relevant sense whether on not one thinks it is the iterative conception).

Shawn(17:48:35) :In an old article, Church shows an equivalence between Tarski’s solution to the Liar and Russell’s via the ramified theory of types. They are basically the same, both form hierarchies, but further Church shows that structurally they are basically the same.

Kenny(21:59:21) :That’s what I was going to say – the ramified theory of types is like Tarski’s solution, but ZFC compresses it all back down to one level. I’ll have to look at that Weir paper though to see how the notion of hierarchy shows up again in ZFC, since it certainly doesn’t show up as a syntactic restriction the way it does for Tarski.

Greg(13:49:28) :I’ve been trying to think of examples in science and (ordinary) math where the principle is violated in some way, shape, or form — without success. Any ideas?

Kenny(18:50:27) :Hmm… let’s see if any of these might count. None of them are about paradoxes, but paradoxes are hard to come up with outside of philosophy and logic.

In probability, there are various Laws of Large Numbers that show that some sequence of random variables converges either in probability, or almost surely, to a specific random variable. Although many of them are special cases of some more general results, I believe that these results overlap, and that there’s no single theorem of which they’re all special cases.

I believe some other basic theorems (perhaps the Cauchy-Schwartz Inequality?) are special cases of multiple more general theorems that don’t have any obvious connection to one another. I suppose this just points to a non-transitivity of similarity.

In biology, instances of convergent evolution might count. For instance, the questions “how do birds fly”, “how do bats fly”, and “how do insects fly” might all seem similar enough that they should get the same answer, but they don’t. Similarly for “how do vertebrates see”, “how do insects see”, and “how do cephalopods see”. (Though I’ve heard that the cephalopod and vertebrate eyes are much more similar than we should expect.)

John Ryskamp(09:56:29) :We are currently enjoying a renaissance in the historiography of set theory. I discuss some of the results below. Above all, do not write another word on Godel before you have read Garciadiego.

Cordially yours,

John Ryskamp

Ryskamp, John Henry, “Paradox, Natural Mathematics, Relativity and Twentieth-Century Ideas” (May 19, 2007). Available at SSRN: http://ssrn.com/abstract=897085