Wednesday, August 11, 2010

My Take on the Liar Paradox (Part II of IV)

In Part I, I provided what I take to be a principled, non-ad hoc motivation for what, in In Contradiction, Graham Priest refers to as the “heroic solution” to the Liar Paradox. I claim that sentences of the type “this sentence is true”, “this sentence is false” and so on are meaningless.

Many people are so initially confused by this proposal that they try to charitably interpret it away. Early last year, while I was discussing my dissertation with some faculty members in my PhD program who were not on my committee, one junior professor wrinkled his forehead and said, “wait, you don’t think that the Liar is meaningless the way that ‘colourless green ideas sleep furiously’ is meaningless, do you?”

In the same spirit, in his book Saving Truth From Paradox, Hartry Field (who, a couple of years after that book came out, was on my dissertation committee!) says that those who claim that sentences like the Liar are meaningless must be using the term meaningless “in some special technical sense” that’s distinct from ordinary use of the term, and that, as such, such talk probably amounts to a confusing way of formulating something like his own “paracomplete” approach to the paradoxes.

So, to be clear, no, I don’t mean ‘meaningless’ in some non-standard way. I take sentences which (a) seem to say of themselves or other sentences that they are ‘true’ (or ‘false’ or whatever) but which (b) can’t be paraphrased into some set of base-language sentences which don’t use ‘true’ or its negation, as (c) literally meaningless, in precisely the same way that ‘colourless green ideas sleep furiously’ is meaningless. “‘Snow is white’ is true” doesn’t mean anything above and beyond what “snow is white” means, and “this sentence is true” doesn’t mean anything at all.

A major factor that seems to drive the incredulous stares often directed at meaninglessness solutions to the paradoxes is the notion that competent speakers of natural languages are infallible about questions of meaningfulness. (We could call this the Strong Principle.)

Now, one could claim that the underlying principle behind the incredulous stares is not the Strong Principle, but just the notion that it’s very unlikely that ordinary competent speakers could be wrong about issues of meaninglessness, and that we should have a strong presumption in favor of their initial intuitions. (We can call this the Weak Principle.)

Now, I actually agree with the Weak Principle, but I don’t think it explains the strength of the incredulous stare reaction meaninglessness solutions often receive.

Anecdotally, it is often the case that people react with incredulity to the claim that Liar-like sentences could be meaningless despite not being sure where to object to the reasoning that delivered that conclusion, or even having originally assented to that reasoning. If it’s very un-common for Croatians to drink single malt whiskey, and our strong default assumption for any given Croatian is that they don’t do so, but you notice a half-empty bottle of Laphroaig on Emil’s kitchen table and you smell peat on Emil’s breath, the strong statistically-based assumption becomes irrelevant. If one acknowledges the possibility of error about meaningfulness by competent speakers but finds it unlikely and subscribes to a strong default assumption against it, and then one is confronted by a good argument that a certain category of initially meaningful-seeming sentences are meaningless, then the probability becomes irrelevant.

The Strong Principle, on the other hand, seems to be trivially easy to falsify. The philosophers of the Vienna Circle were surely all competent speakers of the German language, but they mistakenly held many perfectly meaningful German sentences about various metaphysical topics to be literally meaningless, nonsense, “like music,” etc. Linguistically competent Wittgensteinians have incorrectly held contradictions to be meaningless rather than false. I flatter myself to think that I speak, read and write reasonably passable English, but, if the majority opinion on this issue is correct, I’m mistaken about the meaningfulness of Liar-like sentences.

In fact, the fun part of all of this is that, for all of the debates just mentioned, whichever side turns out to be right, given the disagreements, some otherwise competent speaker must be making a mistake about meaningfulness!

Maybe, however, the Strong Principle is implausibly strong, but something stronger than the Weak Principle is still plausible. Given standard, orthodox views on all the subjects just mentioned, one could propose the Still Fairly Strong Principle, that, while false negatives of meaningfulness are possible, cases where ordinarily competent speakers mistakenly take meaningful statements to be meaningless, false positives are still impossible.

I’d argue that this proposal is falsified by what I think of as “bored dinner guest”-type examples. Imagine that two people, Jack and Jill, are at dinner. Jill goes on at length about subjects that Jack is bored to tears by, and, after a while, Jack completely tunes out. He contributes nothing to the conversation except for the occasional “yeah,” “I agree” or “that’s true,” dictated by the rising and falling of
Jill’s voice and the appropriate pauses. After a while, Jill catches on. To test him, she starts emitting a string of nonsense syllables like “glork bork de glork”, but she keeps her tone normal, and makes sure her voice rises and falls in the normal way. At the appropriate pause, Jack says “oh yeah, that’s true.”

Now, Jack has no idea whether he just “that’s true”-ed a true statement or a false one, so he doesn’t know whether his statement was true or false. He does, however, surely assume that it was one or the other, that he at least said something meaningful.

But, to paraphrase our previous President, if “it’s true that ‘glork bork de glork’” isn’t meaningless, then meaninglessness has no meaning. Jack is an ordinarily competent speaker of English who is mistaken about the meaningfulness of *his own* utterance. False positives are indeed possible.

(Moreover, remember, especially on standard deflationary stories about truth, such “blind endorsements” constitute one of the most important linguistic purposes for which the truth predicate exists.)

Now, one could water down the Still Fairly Strong Principle to the Not Terribly Weak Principle:

False negatives are possible across the board, but false positives are only possible when the competent speaker in question isn’t aware of the content of the sentence in question.

Now, to canvass a broad range of approaches here, I’ve left that word “content” intentionally ambiguous. If by “content”, one means something merely syntactic, such that “knows the content of a sentence” just means “is aware of which words appear in the sentence”, then the Still Not Terribly Weak Principle looks awfully implausible.

For one thing, I think an awful lot of linguistically-but-not-epistemically-competent people could be suckered into believing that “colorless green ideas sleep furiously” was not only meaningful but true…for example, Deepak Chopra could tell them that quantum physics had proved that colorless green ideas did that.
In response, one could interpret “knows the content” in a stronger, semantic way, where “knows the content” means “grasps the content.” It may be possible to fool people in various ways into thinking that “colorless green ideas means something” but it’s impossible to fool them into actually knowing what it means, because there’s nothing there to know.

Fair enough, but at this point deploying the (now entirely trivial) principle which we are still referring to as the Still Not Terribly Weak Principle against meaninglessness solutions to the Liar Paradox would utterly and transparently beg the question. No one would deny that, of course, if competent speakers (or, for that matter, even normally wildly incompetent speakers!) grasp the meaning of a sentence they take to be meaningful, their belief that it’s meaningful can’t be wrong. To base on objection to meaninglessness solutions on this, however, is to simply assume the precise bone of contention, which is whether Liar-like sentences have a content to grasp.

Perhaps, somewhere along this spectrum, there’s some remotely plausible principle that’s relevantly stronger than the Weak Principle, and as such justifies the “well, that’s just obviously ridiculous” unargued brush-off that some people use to refute meaninglessness solutions. If anyone has any candidates they’d like the propose in the comments section, I think that might be interesting.

Meanwhile, of course, showing that it’s possible for competent speakers of English to be mistaken about the meaningfulness of syntactically innocuous sentences formed out of English words is quite a different thing from showing that, in the case of the particular category of sentences under consideration, they actually are mistaken. In Part I, I sketched out an argument, but we need more than that. We also need an error theory to explain why many people have such a strong mistaken intuition. This obligation becomes more pressing as the number of mistaken people rises. If one ordinarily mathematically competent reasoner gets the wrong result for a simple algebra problem, then “he didn’t have enough coffee that morning” may be a plausible error theory. If, on the other hand, out of the thousands of ordinarily mathematically competent reasoners to have tried their hand at the problem, all but one have gotten the wrong answer, we need something a bit more robust to explain this away.

Now, many people make claims from the armchair that most speakers have the intuition that Liar-like sentences are meaningful, but I’ve never seen any concrete empirical evidence to back this up. (This has always sounded like a job for x-phi to me.) In fact, anecdotally, my impression is that, on first contact with the Liar and its ilk, some non-philosophers will respond in a way that indicates what may be precisely the opposite intuitions, having conversations like this one….

“This statement is false.”

“Wait, what statement is false?”

“Well, the one I just made.”

“Yes, but what did you say?”

“I said that what I said was false.”

“Yes, but….”

….and so on. None of this, of course, adds up to a good reason to suppose that Liars are meaningless, but it might reflect a suspicion that this is the case.

Certainly, though, whatever the proportions might turn out to be at some point in the future when we’ve collected some empirical evidence, there’s no denying that plenty people, philosophically trained and otherwise, have the meaningfulness intuition, certainly enough that “Priest and Field don’t drink enough coffee”-type explanations won’t cut it.

I think one piece of the puzzle (although, to be clear, only a secondary one) is a matter of training. Intuitions are formed, changed and molded by one’s educational experiences. The people whose intuitions about the Liar we know the most about are professional philosophers. The Liar is an ancient and venerable philosophical difficulty, and even professional philosophers who have never thought in depth about it at least have years under their belt of being vaguely aware of it in a context in which one of the fundamental background assumptions to get the difficulty off the ground is that it is meaningful. (Moreover, as good rhetoricians have known since time immemorial, it’s often easier to influence people with the opinions you never get around to explicitly stating. Think about Mark Antony’s funeral speech in Shakespeare’s “Julius Caeser.”) So people’s intuitions are shaped by the shared assumptions of the people around them. Think about the way that, in the other direction from the case at hand (non-existence solutions to Russell’s Paradox being as popular as they are unpopular when it comes to the Liar), people who work on set theory, and who initially have ‘naïve’ intuitions, often claim to have new intuitions about what sorts of things sets are and which might exist, formed as a result of the experience of exclusively working in the cumulative hierarchy of ZFC or some similar system. So that’s surely part of the story, and I think it accounts for some of the strength and self-assured fervor of meaningfulness intuitions among professional philosophers.

For many reasons, however, this can only be a small part of the overall story. After all, how do we explain how meaningfulness assumptions initially came into the picture? Besides, it’s surely plausible that a great many people have the meaningfulness intuition on first contact with the Liar “in the wild,” even in eccentric circumstances where it isn’t presented to them as a famous and interesting puzzle.

I think that the first thing to notice is that odd and unusual sentences involving self-reference are usually meaningful—indeed, they’re usually obviously and unproblematically true or obviously and unproblematically false. Consider cases like...

“This sentence has seven words in it.”

When it comes, however, to sentences whose truth-value is utterly and stubbornly inaccessible to us, we usually have no reason to doubt that they are meaningful and that they thus have truth-values. Consider “Alexander the Great’s maternal grandmother’s paternal grandfather once had a splinter in his thumb at some point during the month after his sixth birthday.” Given, among other problems, the absence of time travel, no one has the slightest idea of how to find out whether this is true or false, but this doesn’t (and shouldn’t!) make us question whether it has a semantic content and that this content either lines up with the facts or fails to.

Finally (although it would certainly be possible to go on) the Liar is a syntactically “well-formed” sentence. Now, it’s easy to show with examples that being “well-formed” is neither necessary nor sufficient for being meaningful, and I’m somewhat inclined to think that the use of the phrase “well-formed” to describe natural language sentences relies on an exaggerated, idealized and un-helpful analogy between natural language and formal logical “languages,” but let’s put that to one side and acknowledge that sentences that are “well-formed” (i.e. composed entirely of normal natural language words, arranged in a way that conforms to grammatical rules, etc.) are at least less likely to turn out to be meaningless jumbles of words than other sorts of utterances, and that most of the grammatically innocuous assertion-style combinations of words we have cause to run into in the ordinary course of things are meaningful.

At this point, we can see that the Liar and its ilk sit at the intersection of several categories of sentences such that the overwhelming majority of members of each category are meaningful. It is, then, fantastically unsurprising that most people’s initial intuitive reaction is that it is meaningful. (If, indeed, this turns out to be the case.) Moreover, even when subjecting that intuition to critical scrutiny, it’s perfectly rational to apply a bit of simple probabilistic inference here and argue that, all else being equal, it’s reasonable to infer that a sentence that participates in a lot of categories of typically-meaningful sentences is itself meaningful.

All else is, in this particular case, simply not meaningful, as established by the argument from the disquotationalist principle that quotation marks and ‘is true’ don’t jointly add anything to the content of a sentence to the conclusion that a sentence that attempts to apply truth talk to itself won’t be meaningful. This is a substantive discovery in the course of reasoning about difficult problems, not something that one would expect to be immediately pre-reflectively obvious.

Coming up next time:

Revenge Paradoxes!


Jason Streitfeld said...

I'm going to refer to the sentence "this sentence is false" as P.

Consider pointing to a sentence which reads, "the earth is five days old," and saying P to a colleague. That would be a meaningful statement. So P can be used to make meaningful statements.

Perhaps some particular uses of P are meaningless, but then we need to figure out what makes those cases different. And we need some reason to think that they are meaningless in the relevant cases. We might thus suppose that a sentence cannot be used to assert its own falsity, or perhaps sentences cannot be used to make statements about themselves. However, these solutions are too simplistic.

We could think of the Liar's paradox as a variety of a trick children sometimes play on each other. They give a friend a piece of paper with the words "turn over" written on both sides.

We could write the Liar's paradox this way: On one side of a piece of paper, write "the sentence on the other side of this paper is true," and write "the sentence on the other side of this paper is false." People would react to this game much the way children react to the "turn over" game.

For a more interesting variation, we could write "the sentence on the other side of this paper is false" on both sides. There wouldn't obviously be a paradox at first, because we would think that the first side we looked at was true and the second side was false. (We would negate the second side which told us that the first side was false, and we would thereby affirm what we originally thought, which is what is stated on the first side.) But then we realize that we could have looked at the other side first and come to the opposite conclusion about which was true and which was false. So we have a paradox.

These are games in which sentences are used to undermine their own functionality--they are used to make assertions which, because of the game in which they are involved, cannot be made later in the game. Depending on where we are in the game, the sentence can either be true or false. At no point are they meaningless, however, because we understand them just as the game requires.

We might say that it is the game, and not the sentences themselves, which is meaningless, but that doesn't work, because the game works. People can play it with predictable results. In the games, the sentences direct our behavior in meaningful ways and lead us in predictable ways towards notions of truth and falsity.

I therefore reject the claim that these sentences are meaningless, nor do I think that they are being used to play a meaningless game. Rather, I think the problem comes from failing to recognize that the truth or falsity of a sentence depends on the game we are playing, and that the truth or falsity of a sentence depends on where you are in the game when you are using it.

Unknown said...

Hi Ben,

I can see how the solution you've worked out might apply to liar sentences which (seem to) "directly" attribute falsity to themselves. But what about those that do so "indirectly"? For example, suppose I walk into Humanities 105 and see the following sentence written on the blackboard:

(*) At least one sentence written on the blackboard in Humanities 105 is false.

If (*) is the only sentence written on the blackboard, we have a paradox, but if at least one false sentence distinct from (*) is written on the blackboard, we do not. Surely (*) cannot switch from being meaningful to being meaningless (or conversely) depending on what other sentences we write on (or erase from) the blackboard in Humanities 105?

Ben said...

Jason S.,

"Perhaps some particular uses of P are meaningless, but then we need to figure out what makes those cases different."

Actually, one interesting thing is that, while it's certainly true that sentences like "this sentence is true" and "this sentence is false" can in some contexts be meaningful and unproblematically true or unproblematically false (e.g. if one points to a historical claim written on a chalkboard and says "this sentence is false!"), it's also true that in some contexts they can be uncontroversially meaningless....or, at least, the fact that they are meaningless in these contexts is *far less* controversial than it is in paradoxical contexts. For example, if the sentence written on the chalkboard was "colorless green ideas sleep furiously", or some string of nonsense syllables, and I said "this sentence is true", surely, at least, anyone who accepts the universal intersubtitutivity of P and Tr(P) would have to grant that I've just used the combination of words "this sentence is true" as a meaningless utterance.

So given that the sentence can be meaningful (if the "this" is applied to a clearly meaningful sentence) and it can be meaningless (if the "this" is applied to a clearly meaningless sentence), the interesting remaining question is whether the sentence is meaningful when the "this" is intended to refer to the sentence itself. I've provided a reason (in Part I) why I don't think it is meaningful in those cases, and it's a reason that generalizes to cases like the turning-over-the-page example you mentioned, infinite series of sentences that each ascribe truth to the next sentence in the series but never "ground out" in a 'true'-free sentence from which the rest can inherit their meaning.

Ben said...

"And we need some reason to think that they are meaningless in the relevant cases."

Again, it might be convincing or it might be unconvincing, but I did provide and argue for such a reason, based on the disquotationalist "nothing above and beyond" principle--ascribing truth to "snow is white" is nothing above and beyond ascribing whiteness to snow. The effect of adding "is true" to a quoted sentence is simply to remove the quotation marks. Starting from this (simple, efficient, and to me at least, quite compelling) picture of truth, it seems to me that, if sentences that ascribe truth to some other sentence inherit 100% of their meaning with nothing left over from the sentence to which they ascribe truth, then "orphaned" sentences like the Liar and the Truth-Teller, which never have any 'true'-free sentence from which they can inherit their meaning, are by virtue of this fact meaningless.

"We might thus suppose that a sentence cannot be used to assert its own falsity, or perhaps sentences cannot be used to make statements about themselves. However, these solutions are too simplistic."

Did you read Part I? Because I did provide a motivation for the view that these particular sentences are meaningless there, and that motivation is one you don't bother to canvass as a possible option.

I do think that sentences can't meaningfully assert their own falsity, but this is a principled consequence of a much larger view, which is that meaningful ascriptions of truth (or falsity) must inherit their meaning from the sentences they directly or indirectly ascribe truth to (or their negations). I mostly definitely do not think that "sentences cannot be used to make statements about themselves."

In fact, in the post you're commenting on--Part II--I explicitly said that sentences can be used to make statements about themselves:

"I think that the first thing to notice is that odd and unusual sentences involving self-reference are usually meaningful—indeed, they’re usually obviously and unproblematically true or obviously and unproblematically false. Consider cases like...

"'This sentence has seven words in it.'"

Ben said...

Jason Z.,

"I can see how the solution you've worked out might apply to liar sentences which (seem to) 'directly' attribute falsity to themselves. But what about those that do so 'indirectly'?"

Well, it's certainly an unavoidable consequence of my view (and on that I embrace) that it applies to sentences (whether paradoxical or not) that seem to attribute truth or falsity to themselves indirectly, or for that matter directly or indirectly attribute truth or falsity to some other sentence that attribute it to some other sentence that go on forever without ever stopping at some 'true'-free sentence from which the rest of the sentences in the series can inherit their meaning.

"For example, suppose I walk into Humanities 105 and see the following sentence written on the blackboard:

"(*) At least one sentence written on the blackboard in Humanities 105 is false.

"If (*) is the only sentence written on the blackboard, we have a paradox, but if at least one false sentence distinct from (*) is written on the blackboard, we do not. Surely (*) cannot switch from being meaningful to being meaningless (or conversely) depending on what other sentences we write on (or erase from) the blackboard in Humanities 105?"

Three thoughts about this:

(1) If a lazy and inefficient maintenance person comes in after class and erases most of the words, but happens to leave some, and the remaining words on the chalkboard, by some weird coincidence, add up to the sentence "colorless green ideas sleep furiously", is it a problem that it's contingently meaningless?

(2) This is a nit-picky point that doesn't really address the force of the example (which I promise I will get to!), but there's actually an interesting point here about tense. Of course, if the sentence means "there's at least one sentence that has ever been written on the chalkboard that's false", that's unproblematic. The apparent paradoxicality--and, given my view about truth ascriptions, apparent meaninglessness--rests on the tensed reading, "at least one sentence that is *now* written on the chalkboard is false."

Ben said...

Now, if one is a B-Theory about time and denies that the universe includes a real, stance-independent objective property of "nowness", and thus that true tensed sentences can be tenselessly paraphrased, one might argue (using the date theory) that "at least one sentence written on the chalkboard is false", written at 3 PM, really means "at least one sentence that (is) on the chalkbaord at 3 PM (is) false." Or, if one prefers to token-reflexive theory, one could translate it as something more like "at least one sentence that (is) on the chalkboard simultaneously with this sentence being written is false." Either way, no paradoxicality.

(3) OK, that said, if one's an A-Theorist--taking sentences to change truth-value over time as the 'now' changes and all that--but accepts my view, then yes, absolutely, an obvious consequence is that the sentence on the chalkboard is now "ungrounded" and thus meaningless. I'm comfortable with that.

(Certainly, one could construct the case to get around my nit-picky point about tense in any case.)

I think my Jack & Jill case in this post provides a plausible parallel situation--when Jill tests Jack by emitting a string of nonsense syllables and Jack, not paying attention, says "that's true", it's contingent happenstance based on factors outside of his control that he happens to be making a meaningless statement, but it still seems to me that he quite clearly is doing so.

Jason Streitfeld said...

Hi Ben,

Sorry, I somehow overlooked the fact that you had presented that argument in Par I. I didn't read it before posting earlier.

I'm not sure why "This sentence is false" (P) is meaningless when said of "colorless green ideas sleep furiously." What if somebody really believed that idea made sense and was false? They might be wrong, but that doesn't make their assertion meaningless.

I think you are making a mistake in your argument. You over-generalize from Quine's disquotationalism. Quine's point is about adding "is true" to sentence. That does not mean that the meaning of all "X is true" sentences mean the same as X. For example, "What she says is true" does not simply mean "What she says." "What she says" is not a well-constructed sentence. Similarly, "This sentence is true" does not mean "This sentence," which also is not a well-constructed sentence.

As you agreed earlier, we can use "this sentence is true" in clearly meaningful ways. I think you want to distinguish these meaningful cases with the Liar Paradox by claiming that "this" in P lacks content (in the relevant contexts). But I don't see how you are establishing that.

Jonas said...

I may be missing something obvious here, but:

What about modifications of the Liar, in which it intuitively seems to become clearly true or false, but still refers to itself by means of the truth-predicate?

Examples (with my suggested truth-value in parentheses): "This sentence is obviously false" (false(but not obviously so)), "it has not been shown empirically that this sentence is false" (true).

Would you say that these sentences and other like them are meaningless?

Ben said...

Jason and Jonas,

I just wanted to pop my head in to say that I haven't forgotten about your comments. I got into Korea today (having flown in to start up a visiting professorship at the University of Ulsan), and before that I was swamped with preparations. In any case, given that there have been a couple of posts since one so some interested parties might not be following the discussion back at this post at this point, instead of trying to dash off a quick reply in the comments, I'll devote a separate post next week to following up on the points you two raised.

Ben said...

(Just updating to say that the post responding to your comments is now up.)