On Paradox, Liars, and Revenge

Posted: October 22, 2009 in Uncategorized
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I am looking at three articles right now. The first two both come from The Revenge of the Liar: New Essays on the Paradox, edited by JC Beall. The third is from an upcoming issue of Studia Logica. I’ll get to that. First, though, I want to talk about “Embracing Revenge: On the Indefinite Extendability of Language” by Roy T. Cook.

All three of these articles about the Liar and the Revenge problems. In case anyone is unclear, the Liar is a famous paradox in semantics. You’ve probably heard it before: “This sentence is false.” If it’s true, then it’s false. But if it’s false, then it’s true. So it’s a contradiction however you look at it; a paradox. The Revenge is a response to an attempt to solve this paradox. Revenge, as Cook writes it, works like this: “Given any account that purports to deal adequately with a particular paradox, that account will rely on concepts… which, if allowed into the object language, generate new paradoxes that cannot be dissolved by the account in question” (33). So basically, whatever you do to solve the Liar, there is another, stronger Liar that your solution can’t defeat.

This would be pretty frustrating for most people. The idea that every attempt to solve it just pushes it one step further leads most people to throw up their hands in frustration and look for another solution. But Cook suggests that this is the wrong way to go. In fact, he even goes so far as to say that “The Revenge Problem, it turns out, is not a problem at all, but instead affords the crucial insight that allows for a truly satisfactory solution to the semantic paradoxes” (34). In order to feel satisfied by these solutions, all we have to do is accept that “we can never speak a ‘universal language’, quantify over all sentences, or speak of all truth values at once” (35).

This is a concept I can get behind. It’s hard to wrap my mind around the idea of talking about everything. As soon as you try to get outside of everything in order to talk about it, there is something else. It’s like the idea of going beyond the universe. By definition, the universe is everything; there is no outside. If you were outside, you wouldn’t be outside, because you’d still be in the universe. So you can’t ever look at the universe from the outside. Similarly, you can’t talk about everything from the outside, because the terms you use to talk about everything would be something not included in that set of everything.

It’s like set theory; there are an infinite number of sets, because if you think of the set containing all sets, it can’t contain itself, so there is at least one set that is not included in that set (namely, itself). Anyway, the point is, I’m willing to go along for the ride with Cook.

Cook’s next step is to talk about the concept of truth values, of falsity, and of truth in and of itself. This, I think, is going a bit too basic, so for now, I’m just going to skip this stuff (he does a better job of it in his other article anyway). The point is that “more expressively powerful languages (in particular, those that contain more semantic vocabulary) require more truth-values” and that “The concept of truth-value is indefinitely extensible” (38). This makes sense. If you can’t speak of all truth values at once, then truth-values must be universally extensible; there’s always at least one more that you can’t speak of. As for more expressive languages needing more truth values, this seems pretty obvious. When all you can say is that something is either true or false, you lose any chance of going half way. Someone is either bald or hairy. They can’t have a receding hairline or be balding. But when you add a halfway value, then you’ll want values to cut that half in half again. Cook says this gives us a potential Sorites paradox (though he doesn’t see that as a problem), but I think it’s more like a Zeno paradox. Like Achilles and the Tortoise. In order to get somewhere, you first have to cross the halfway point. But to do that, you first have to cross the halfway point to that halfway point. Etc, ad infinitum.

Anyway. Cook at this point goes through the formal theory, which is complex enough that I need a bit of refreshing in modal logic to really follow all that well. But eventually, he goes back to text and makes an interesting point: “Given a language L, if we can completely describe the semantics of L, then we have (knowingly or not) extended our language to a new language L’” which means, for those who prefer computer metaphors, “the Revenge Problem is no longer a bug—it is now a feature, exemplifying the indefinite extensibility of the concepts language, statement, and truth-value” (45, his italics). So Revenge actually helps; the more we think about it, the better our language gets.

This seems like it might be a bit on the ridiculous side. If he’s right, then there are an infinite number of languages, and we’re constantly moving to a better one. But we have to remember that the way he’s using “language” isn’t the same as the way we use it in every day life. By the definition Cook is using, as soon as there is a single new word, we have a new language. So English before the word “Robot” was coined is a different language than English after robot. In other words, the Revenge feature just makes our language stronger, more expressive. That’s good.

I’ll now move on to Matti Eklund’s article, “The Liar Paradox, Expressibility, Possible Languages.” Eklund starts off by talking about ways out of the Revenge problem. He says that “One common way to avoid a threatening revenge problem is to deny semantic self-sufficiency” (57); that is to say, if you are willing to say that there aren’t ways to describe a language within the language, then you’re okay. This would also mean that “This sentence is not true.” Wouldn’t make sense, because it can’t refer to itself this way.

Eklund does give a nice approach to that problem of the word “language” I was just talking about. He says that “In logic, languages are individuated very finely: add another constant and you have a new language. Outside logic, however, natural languages are individuated more liberally: a language can undergo significant changes in its vocabulary and its syntax and still we happily speak of it as the same language after these changes” (60). As any linguist would say, languages are living things, evolving all the time. Want an example? Some people are predicting that “they” will become a singular neuter predicate. So if you don’t know someone’s gender, you can say “What gender are they?” pointing at just one person. Or look at how rarely the semicolon is used; regardless, language does evolve, and it’s good that Eklund acknowledges that he’s talking about a different definition of the word.

But that’s about where my agreement with Eklund falls apart. He makes a claim that it is possible to talk about predicates and words that don’t yet exist. By saying that something ‘falls under the predicate that someone will speak at a specific date,’ he claims that we have expressed that predicate. But I can’t agree with that. It doesn’t follow. Words that did not exist in, say, the nineteenth century English could not be expressed. If they could, it wouldn’t be a language in which those words don’t exist. In other words, if you can express a concept, that concept exists. If you can express a word, that word exists. What Eklund is doing here, I think, is equating “refer to” with “expresses.” I can, after all, refer to predicate that the world’s shortest man will speak on January 21, 2112. But I haven’t expressed that predicate; I still have no idea what would or would not fall under that predicate, I can’t use it. It doesn’t exist. But I can refer to things that don’t exist. Any atheist is able to refer to god, even though she doesn’t believe that god exists. But god cannot be expressed without being able to examine all properties. (As a side note, I think this is the same problem Descartes and others made; just because you can refer to god and all his qualities, including existence, does not mean that god exists. You didn’t express god, you just referred to him.)

He makes a fairly convoluted argument about negation, then suggests four possibilities for determining which of two types of negation (Boolean and Priest) is correct: (a)one of them is right, one wrong; (b)it can’t be determined; (c)both exist, they’re just expressed in different languages; (d)the underlying metaphysical assumptions might be wrong (66). Then he says that “Alternatives (a)-(d) appear to be exhaustive. But they all seem objectionable. This is a paradox” (69). This I cannot agree with.

First off, even if the list appears to be exhaustive, if none of them are correct, then the list by definition is NOT exhaustive. As Sherlock Holmes said, “Eliminate the impossible. Whatever remains, however unlikely, must be the truth.” In other words, if you’re looking for a solution, and none of your attempts to solve the problem work, you haven’t found the solution yet. This does not imply that there IS no solution, only that you haven’t found it yet.

Secondly, he used a word that a philosophy professor once told me never to use: seem. This professor told me that “’seem’ is what someone says when they can’t prove something. If they say ‘it seems’ what they mean is ‘I can’t prove’.” Anecdote aside, I still don’t follow his reasoning. Even if (a)-(d) seem to be objectionable, it doesn’t mean that they are. It just means that they appear to be. That they seem objectionable. A man driving an expensive car and wearing an Armani suit might seem to be wealthy. But that doesn’t mean he is; he could be a thief.

Finally, this isn’t a paradox. A paradox is when something contradicts itself, but still seems (there it is) to be true. Saying that a list MIGHT be exhaustive and that the items on the list MAY all be false isn’t a paradox; if anything, it’s unfortunate or the solution is just unintuitive. If a list was inclusive, and all the members of the list were false, it still wouldn’t be a paradox. All it would be is proof that there was not a solution to the problem. In order to even reach towards paradox, we’d have to first assume that (1) there IS a solution and (2) the solution MUST come from the list. Even then, I’m not sure it would qualify.

Okay, let me go back to Roy Cook. His second article, “What is a Truth Value And How Many Are There?” is more or less the same idea as his other article, but explained less in formal terms and more in plain language.

Cook gives a nice explanation of how truth values work. He says “a statement receives a particular truth value because of a particular relationship which holds between the state of affairs/situation/structure/etc. described by or appropriately associated with the statement (i.e. what the statement says) and the structure of the world (i.e. what is the case)” (185). So something is true if and only if what it says matches up with the world, and false if and only if it does not. Makes sense to me.

He then goes on to explain the idea of vagueness requiring more truth values, and how more than two truth values immediately shows that truth is not based on whether something matches up with the world or doesn’t; in other words, it shows that there’s more than just the two states. It has to (188).

He also says that “the Liar sentence forces us to give up classical semantics” (189); it forces us to use more than the two values of true and false.

Then Cook moves through his earlier argument about making more and more languages. He writes that “Any language that we speak can be extended by adding the resources for describing semantics of that language, and in addition, the semantics for a particular language will always involve resources beyond those that can be described in that very language. As a result, we are (often unknowingly) extending our language all the time by the very act of attempting to describe the semantics for that language” (193-4). I think this makes intuitive sense, if we’re willing to follow along with Cook’s idea that you can’t talk about everything in a language without making a bigger language. And I am willing to follow.

What follows from this is that “We can never reach a point…where we have all of the truth values, or where extensions to our language are no longer required in order to describe things that we might wish to describe (such as the semantics of a particular language),” which means that “there is an infinite hierarchy of truth values” (199). But that’s not a bad thing. Because of this, in fact, we know that we can say very nearly everything we think we can say, and can very nearly talk about any language, so long as we know that sooner or later, we’ll run out of time to get higher up in the hierarchy.

As a side note, Roy Cook showed me a particularly cool thing the other day. He called it the Super Duper Liar. It basically looks like this: “This sentence has a truth value (i.e., one of all of them) other than true.”

He also referred to this as his version of the Revenge problem. I like that, mostly because “Roy’s Revenge” has a nice sound to it; though on reflection it should probably be the name of a band, not a philosophical argument.

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