Monday, August 2, 2010

Russell's Paradox, the Liar, the Barber and the Inclosure Schema (The Russell's Paradox Series, Part III of IV)

In Parts I and II, I responded to the bullet-biting approach to Russell's Paradox--that it shows that there really are true contradictions--by appealing to some general epistemic considerations about how and when we might be justified in coming to believe in the existence of mathematical objections with inconsistent properties. I argued that, given our notorious lack of direct epistemic access to mathematical objects, if there are any such things (think of Benaceraff-type arguments here), we should be extremely cautious and conservative in our theory of them. (The default assumption should be that they exhibit the same logical structure as the parts of reality we do have direct access to.) Moreover, I argued that the best argument we have for supposing any mathematical objects to exist is one from Quinean indispensibility-to-our-best-science, I pointed out that we hardly need all of the sets in the crowded universe of naive set theory to reconstruct the fragment of mathematics we need for our best current science, and ended by leaving a fallibilistic back door open for a way that hypothetical future science could force inconsistent mathematical objects on us after all.

All well and good, someone might argue, but these sorts of considerations clearly won't help us with the Liar Paradox. Any epistemic access we might have to sets is indirect and holistic, fair enough, but sentences are clearly a different matter. Thus, any plausible consistent solution to the Liar Paradox must surely be different from the solution I've offered here for Russell's Paradox. Given the deep structural similarity between the Liar and Russell's Paradox, though, aren't they both 'of a type'? And doesn't that mean that the right solution to the two paradoxes must be 'unified'?

Graham Priest has argued in exactly this way, in his article "The Structure of the Paradoxes of Self-Reference" and many subsequent articles. He argues that the Liar Paradox, Russell's Paradox, and many other paradoxes besides, are all instances of a common pattern, the Inclosure Schema, and that the right solution to any given Inclosure Schema paradox must apply to the rest as well. The technical details of the Schema (diagnolizing functions, etc.) don't really concern us at the level of detail we're going to operate on here, but the important point is that there are three elements: Existence, Closure and Transcendence.

(He often only talks about Closure and Transcendence in the articles, but existence is obviously a necessary component--there's nothing paradoxical about saying "if there was such-and-such an object, it would have strange paradoxical properties--and in Priest's book Beyond The Limits Of Thought, which includes his longest discussion of the Schema, he emphasizes Existence as much as the other elements.)

Priest's approach is to bite the bullet of inconsistency, so in each case, he accepts that there is something (existence), that it falls into a category (closure) and falls outside of it as well (transcendence). Any consistent solution to Russell's Paradox, like mine, that starts from considerations about the limits of our epistemic access to the realm of abstract mathematical objects, considerations that will obviously be irrelevant to consideration of the semantic paradoxes, must be much less unified than this, right?

Well, first of all, not necessarily, no. If one takes Liar sentences to be meaningless, or to be meaningful but to fail to express propositions, or to be expressing different propositions than they seem to express, or anything that's remotely in that family of approaches to the Liar, then one is denying the Existence condition. The solution to Russell's Paradox that I argued for in the last two posts also amounts to denying Existence. (Indeed, almost all consistent solutions to Russell's Paradox amount to denying Existence. In fact, Penelope Maddy's "set-membership gaps" approach--which amounts to denying both Closure and Transcendence--is the only exception that I can think of right now.) Thus, if you solve the Liar Paradox in any of the ways just gestured at, and pair that solution with my view about Russell's Paradox, then, at the level of abstraction the Inclosure Schema operates at, your solution will be exactly as unified as Priest's dialetheist soltuion to all of the Schema paradoxes. In all cases, Priest affirms Existence and Closure and Transcendence. In all cases, you deny Existence. There's simply no gap whatsoever in the uniformity of your respective solutions.

Now, since I find myself in the happy position of having a view of the Liar that falls within the camp of Existence-denying solutions, I'm in the clear here. What, however, if you don't find Existence-denying solutions to the Liar plausible? For example, what if you accept Barwise & Etchemendy's solution from The Liar: An Essay On Truth and Circularity, which (by denying that the Liar's truth-value depends on itself in the obvious way) arguably amounts to denying both Transcendence and Closure?

(I actually think that, as is the case with a great many solutions to the Liar Paradox, which element of the Schema is being denied is interestingly ambiguous. Whether this fact exposes a diagnostic flaw in the usefulness of the Schema is an interesting question I'll leave for another time. For now, let's say that the Barwise/Etchemendy solution--which proceeds from a complicated version of the correspondence theory of truth and some set-theoretic modeling to the surprising conclusion that Liar sentences are (just) false, since its truth-value is not included in the situation that determines that truth-value--amounts to denying Closure and Transcendence but not Existence.)

So, if one solves Russell's Paradox in my preferred Quine-inspired way, and solves the Liar in the Barwise/Etchemendy way (thus picking different aspects of the Schema to go after in each case), is one solving the paradoxes in a less uniform way than Priest is, and thus losing plausibility points?

I don't think so. Here's why not:

Imagine that someone tells you and a group of your friends that they heard that, in the village of Alcala, the barber shaves all the men in the village who don't shave themselves. (Subsequent questioning reveals that the barber is a man and that he lives in the village.) Inevitably, someone asks whether the barber shaves himself, and everyone quickly realizes that, given the information you have so far, he must both shave himself and not shave himself.

No one will say, "huh, I guess there are true contradictions about the observable physical world. Wow!" Even if your friends are all philosophy professors who accept dialetheism and believe that the Liar Paradox and Russell's Paradox must be solved in the same way because they are both instances of the Inclosure Schema, none of them will respond in this way. The response of any normal, reasonably bright human being over the age of six would, rather, be to demand evidence of the existence of such a barber. For a variety of reasons, different philosophers may have different views about whether any possible evidence would ever justify belief in the existence of such a barber, but it would take a lot to satisfy even the most zealous fallibilist.

Pretty clearly, evidence of the existence of a barber who's stated policy is to shave everyone who doesn't shave themselves won't cut it. Nor, given the enormity of the claim, would eyewitness evidence be sufficient. Perhaps, a la the set-of the short story Sylvan's Box, live streaming video of the barber simultaneously shaving and failing to shave himself, might satisfy such a demand. Certainly, one couldn't imagine much less than that convincing someone like Graham Priest or JC Beall--much less those of us who don't start out by believing in true contradictions--of the barber's existence.

But wait! Structurally, the Barber Paradox surely has at least as much in common with Russell's Paradox as the Liar Paradox does. In fact, no less an authority on Russell's Paradox than Bertrand Russell introduced the Barber Paradox as an illustration of the structure of Russell's Paradox. If the Barber Paradox turned out that it didn't fit the Inclosure Schema, this would be fairly damning evidence that the Schema had failed to capture the pattern of Russell's Paradox. Isn't someone who solves the Barber Paradox by denying existence and solves Russell's Paradox by embracing Existence, Closure and Transcendence violating the principle that all paradoxes of the same structural 'type' should be solved in precisely the same way?

In Graham Priest's article The Import of Inclosure: Some Comments on Grattan-Guiness (from the October 1998 issue of Mind and available on Jstor), Priest acknowledges that "the Barber certainly can be put into the form of the Schema" and that he (like everyone else) denies that there ever has been or ever will be a barber of the relevant kind, and that he couldn't be convinced otherwise by anything so straightforward as, for example, the testimony of people who claim this about their village barber.

It's important to note that, although he doesn't say so, in the case of a non-paradoxical claim like "the barber in our town shaves all the men in the town except for himself," Priest (like any other normal human being operating with normal, everyday epistemic standards) would probably take a whole lot less than the combined testimony of every single resident of the town before he accepted that the claim was probably true. This is despite the fact that he starts from the assumption that true contradictions are possible. Still, reasonably enough, he finds the suggestion of contradictions in the observable physical world to be surprising and counter-intuitive enough to require more-than-usual amounts of evidence. One might think that, by analogy, those of us who don't even start out believing in the existence of true contradictions should be well within our rights to demand greater-than-usual evidence of the existence of the Russell Set once we realize that, if it existed, it would have inconsistent properties. It's not clear, then, why (even on the assumption that naive comprehension is correct and that there are true contradictions) Priest thinks that we're rationally obligated to set the epistemic bar for what should count as evidence of the existence of the Russell Set as low as the mere fact that "for every description we can come up with, there's a set with all and only the objects satisfying that description as members" is an intuitively simple comprehension axiom.

Let's put that to one side for now, and go back to the original question. Given that the Barber Paradox fits the Schema, and that Priest solves it by denying Existence rather than by embracing Existence, Transcendence and Closure (as he does for Russell's Paradox), isn't Priest guilty of disunified solution?

He argues that "there is more to the Schema than its form...An Inclosure Argument is valid. But one needs more than this for a paradoxical argument: the premises must also be true, at least prima facie--or no one would suppose the situation paradoxical. It is this fact that rules out the Barber Paradox and its ilk as inclosure paradoxes. We have no good reason to suppose that there is a barber of the required kind." (p. 386)

Now, in this passage, Priest is teetering dangerously close to abandoning the claim that the Schema has any role at all as a tool for showing us which paradoxes are 'of a type' and hence require uniform solution. Premises, after all, can fail to be true for all sorts of different reasons in different cases, and its not clear what's added by the phrase "at least prima facie." Now, I have enormous amounts of respect for Priest's philosophical abilities--if I didn't think the man was making good arguments, I wouldn't waste so much time and energy worrying about how to respond to them!) And, certainly, it doesn't seem terribly charitable to read him as claiming here that, whenever the premises of some Barber-like puzzle initially seem true, we acquire some permanent duty to solve it in precisely the same way we solve Russell's Paradox, but that we don't have such a duty if we see through the false premise right away. That said, since it's unclear what significance he does attach to the phrase, let's just move on.

Given that, if we have a good enough reason to doubt that "the premises are true," it's OK to solve a paradox with the form of the Inclosure Schema in a way that's different from the way you solve the rest of the Schema paradoxes, is there a way still available for Priest to insist that the Liar and Russell's Paradox be solved in the same way?

He thinks there is. He goes on to say that, to be a Schema Paradox in good standing to which one's uniform solution must apply, it's important "not just that the premises are prima facie true, but that they are a priori so. For exactly this is true of the standard paradoxes of self-reference: Transcendence and Closure would appear to be a priori certified. The contradictions that these give rise to seem to be inherent in thought itself, intrinsic to our conceptual structures... The premises in the case of the Barber and its ilk, were they ever to be true, could only ever be so a postereori." (p. 386)

With that firmly in mind, let's remember Stephen Yablo's summary of the recent history of philosophy, which I quoted in my last post.

"About fifty years ago, Quine convinced almost everyone who cared that the argument for abstract objects, if there was going to be one, would have to be a posteriori in nature. And it would have to be an a posteriori argument of a particular sort: an indispensability argument representing numbers, to use that example, as entities that ‘total science’ cannot do without."

In other words, at this point most people are convinced that they only good reason we could ever have to believe in the existence of any sets or any other mathematical objects--surely, a crucial premise of the dialetheist's argument from Russell's Paradox--is, in its own way, as a posteriori as any reason we could ever have to believe in the existence of the self-shaving-and-non-self-shaving barber.

Now, I don't, of course, want to engage in an appeal to popularity here. Priest may well have a knock-down argument against the prevailing consensus up his sleeve. Failing that, though, given Priest's own standards about which sorts of Schema-fitting arguments need to solved with one's favored 'uniform solution,' anyone who finds themselves in Yablo's "almost everyone" should feel free to solve Russell's Paradox and the Liar Paradox in entirely different ways.

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