Wednesday, April 27, 2011


I just gave a talk at the National University of Singapore, entitled "Liar Paradox II: Revenge of the Liar Paradox." Singapore, by the way, is lovely, and I'm going to be sorry to leave on Thursday. On the surface it feels enough like Miami to make me feel nostalgic--humid, windy, full of palm trees and outdoor bars and a mishmash of different languages and cultures--while having a lot of appealing un-Miami-ish traits, like being chock-full of restaurants serving delicious Indian food. (The strangest thing I've seen here to date has been Haw Par Villa, a "moral instruction" theme park based on Confucianism and Chinese mythology put up by some early-twentieth-century Chinese millionaires who'd made a killing in the tiger balm trade. The main attraction is the Ten Courts of Hell, where lurid statues and signs depict the punishments sinners are sentenced to by the Emperors of Hell. For example, "cheating on examinations" gets you your intestines ripped out by demons.) While I've been in Singapore, I've been staying with NUS prof Neil Sinhababu, of Possible Girls fame, who will henceforth always have a special place in my heart for saying, when I came in on Saturday night, "I made sure to save some Laphroaig for your visit."

As far as the talk itself, here's the abstract:

Dialetheists like Graham Priest and JC Beall conclude from the Liar Paradox that sentences like “This sentence is not true” are fact both true and untrue, and that we must therefore revise our logic to accommodate the existence of true contradictions. Similarly, “paracomplete” theorists like Hartry Field avoid the contradiction posed by the Liar Paradox by rejecting one of the central elements of classical logic, the Law of the Excluded Middle. A more conservative solution starts from the claim that sentences that attempt to attribute truth or untruth to themselves are meaningless, and therefore simply not the kinds of things we can logically symbolize or apply truth talk to without committing a nonsensical category mistake. The most common objections to this move are (1) that the “meaninglessness solution” is refuted by the existence of “revenge paradoxes” like the one revolving around the sentence “This sentence is either false or meaningless”, and that (2) the sentences involved are so obviously meaningful that it’s just not possible to take seriously the claim that they’re literally meaningless in any ordinary sense, like “Blorks geblork” or “Colorless green ideas sleep furiously,” whereas the dialetheist and paracomplete approaches have the advantages that they (1*) make room for the perfectly obvious fact that, in any language with normal expressive resources, we can construct perfectly meaningful sentences that attribute untruth to themselves, and (2*) are immune to refutation by means of “revenge paradoxes.” I will argue that (1), (2), (1*) and (2*) are all completely wrong.

In terms of the talk itself, I'm never 100% sure what to think about the ethics of blogging in-person discussions, given that I'm sure I wouldn't want to be represented by someone else's half-clear recollection of what I said on the spur of the moment, so I'll pretty well stick to representing what I said myself, with one exception (one hopes, a benign one): In the talk, I spent a few minutes hammering the standard Priest/Beall/Field sort line on Curry's Paradox. In the Q&A, NUS prof Ben Blumson took issue with some of that, and later in the day I ended up spending a couple of hours in his office genially arguing about Curry and related issues, and if I'm still not utterly convinced, I will definitely say that he did a better job of presenting a fairly plausible defense of the approach to Curry I was criticizing than any other defense of that approach I've seen or read before, and in future I will be scaling back at least some of my initial objections in light of some of the points he made.*

In any case, Curry aside, a lot of the ground I covered should be familiar to regular readers here. While I briefly presented my disquotationalist argument for the claim that ungrounded truth talk is literally meaningless, including the Greenness Paradox as a way of defusing the worry that, since we are able to reason about the Liar, we know what follows from it, thus what it means and thus that it means something, my main focus was on revenge paradoxes. (A paper I'm working on making some of these points is tentatively entitled 'Who Among Us Is Safest From The Liar's Revenge?') Conventional wisdom says that the dialetheist and paracomplete approaches to the Liar, given their willingness to engage in radical surgery to our basic logical notions, gain immunity from the revenge paradoxes that typically plague classical solutions to the paradoxes, of which a particularly clear case is supposed to be the problem posed by (1) for those of us who take these sentences to be meaningless:

(1) Sentence (1) is either false or meaningless.

Since paracompletists don't assert anything about the semantic status of such sentences, but rather reject the relevant instances of Excluded Middle, reject the negation of those instances and so on 'all the way down the line,' they seem to be immune from danger from sentences that attribute to themselves the status paracompletists attribute to such sentences. Even more so, dialetheists seem to be immune from any revenge problems, because any 'revenge' liar would at worst just generate yet another true contradiction, and true contradictions don't generate triviality, given the dialetheist's claim that Disjunctive Syllogism isn't universally truth-preserving.

My claim is that (1) is not a problem for the meaninglessness solution at all. A meaningless sentence does not become meaningful once we attach the word "or" to it and paste (what would otherwise be) a meaningful sentence to its tail. Just because a meaningless sentence has the syntactic form of a disjunction and a true second disjunct does not mean that it's meaningful, much less true.

On the other hand, I argue that the paracompletist has a real problem about sentence (2) and that the dialetheist has a real problem about sentence (3).

(2) An ideally rational being who did not lack any relevant information would not accept sentence (2).

(3) It is not the case that sentence (3) is related to truth.

Regular readers will recognize that (2) is the latest form of a revenge paradox for paracompletism I've been tinkering with for some time. The problem, as I see it, is that, if (2) is true, we have the starkly counter-intuitive result that an ideally rational being would not accept a sentence it knew to be true, if (2) is false, we have the equally counter-intuitive result that an ideally rational being would accept a sentence it knew to be false, and if (2) is one of the sentences about which the most rational option is to 'go paracomplete' and reject both the sentence and its negation, then its a sentence that any ideally rational being would not accept (it would reject the sentence instead of accepting it!) and the sentence is true, and, once again, we have the conclusion that an ideally rational being would fail to accept a sentence it knew to be true.

(3) is a familiar problem, but as I argued here, the exact nature of the biggest problem it poses doesn't seem to be widely realized. If being-related-to-truth and not-being-related-to-truth overlap, just as being-related-to-truth and being-related-to-falsehood overlap, then when the dialetheist shows that (given the assumption of dialetheism) there are cases in which all the premises of Disjunctive Syllogism are related to truth and the conclusion is not, they have no more shown that DS is not universally truth-preserving than they would if they'd 'just' showed that all the premises of DS were true and the conclusion was false. No one thinks that "all the premises of argument A are true and the conclusion is false" is a dialetheistically-acceptable way of establishing a failure of truth-preservation. Why should "all the premises of A are related to truth and the conclusion is not" be even a little bit different? Without a better answer to that question, the dialetheist claim that Liars can be both true and false without triviality following simply doesn't hold up.

*On a similar note, I should include a quick shout-out to Brandon Watson for giving me a hard time recently about my error theory about the mistakenness of ordinary competent speakers who take Liars to be meaningful. My preferred way of putting the point now, which I used in the talk, goes about like this: Self-referential sentences are often meaningful--e.g. 'This sentence has seven words in it'--and sentences with precisely the same wording as the Liar sentence are meaningful in other context--'This sentence is false' said while pointing at a sentence about some substantive subject written on a chalkboard. To realize that it's meaningless when the intended reference of the 'this' is that very sentence is a conclusion that takes careful philosophical argumentation. Given those two facts, its quite natural that most people don't realize that it's true. By analogy, if an object is far away and looks a certain (misleading) way from a great distance, and out of a whole crowd of people watching the object, only Bob has a telescope, it's utterly unsurprising that most competent-users-of-functioning-human-eyes end up having a mistaken impression about the object. All Bob needs to do by way of explanation of the disconnect is to say, "yeah, you don't have a telescope. But, hey, look through it yourselves and you'll see where you went wrong here."


L33tminion said...

(2) An ideally rational being who did not lack any relevant information would not accept sentence (2).

Isn't (2) just the Godel-string for some set of axioms and rules of inference used by "ideally rational" beings?

(2) is one of the sentences about which the most rational option is to 'go paracomplete' and reject both the sentence and its negation

Don't you mean accept both the sentence and its negation? (I took "reject" as a synonym for "accept the negation" because in formal systems (which I'd hope is the sort of thing implied by "ideally rational") you accept propositions or fail to accept them within bounded time.)

If by "reject both the sentence and its negation" you intended to mean something distinct from "accept both the sentence and its negation", I'd say that doesn't make sense, they're equivalent.

If you meant the same thing, I'd agree that doesn't get you out of a jam in this case: If accepting (2) is self-contradictory, accepting (2) and ~(2) doesn't avoid a contradiction.

Ben said...

"Don't you mean accept both the sentence and its negation?"

Absolutely not. I meant reject both the sentence and its negation. That's the whole point of paracompleteness--they don't accept that the Law of the Excluded Middle applies to every case. They think that sometimes, as with paradoxical sentences, it 'breaks down' and there's simply no fact of the matter about whether some sentence or its negation (or both or neither) is true, so we should reject all the options.

"I took 'reject' as a synonym for 'accept the negation'...."

Again, no.

I'm not a paracompletist, but to critique the view we have to be clear on what they're saying, and their position is that those sometimes rationally come apart--their are situations in which the rational thing to do is to neither accept the sentence nor its negation, but to reject both. Part of the motivation is that if you try to reject some instance of Excluded Middle by accepting its negation, that DeMorgans into accepting an outright contradiction, and the whole point of the exercise is to prevent that. (Hartry Field's book advocating the view is called "Saving Truth From Paradox.") Of course, I don't think the attempt succeeds, given the sort of revenge paradox I discuss here, but that's a different kettle of fish.

"...because in formal systems (which I'd hope is the sort of thing implied by 'ideally rational') you accept propositions or fail to accept them within bounded time."

Not in paracomplete systems! Again, see Field's book, or any number of his articles on the subject, to see how this sort of thing works.