Monday, August 16, 2010

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

In Part I, I used a disquotationalist picture of truth to motivate the claim that sentences like the Liar and the Truth-Teller, although they may seem meaningful, are in fact quite literally meaningless. If “’snow is white’ is true” means nothing above and beyond what “snow is white” means, if the former inherits 100% of its meaning from the latter, then a sentence like the Truth-Teller...

“This sentence is true.”

...means nothing at all. Ascriptions of truth must inherit their entire meaning from whatever sentence truth is being ascribed to--they have no ‘independent’ meaning—and, as an orphan with no ‘true’-free sentence from which it can inherit its meaning, the Truth-Teller means nothing at all. And, of course, as Carnap liked to point out, adding the word ‘not’ to a meaningless sentence does not magically convert it into a meaningful one, so what we’ve said about the Truth-Teller applies equally to the Liar.

In Part II, I argued against those who dismiss this sort of move as absurd on the basis of the “obvious” meaningfulness of such sentences, and argued, first, that ordinarily competent speakers of the language are in fact quite capable of mistakenly taking meaningless sentences to be meaningful and vice versa, and that, in this particular case, a plausible error theory is available to explain the widespread intuition that sentences like the Liar and the Truth-Teller are meaningful.

I’ve heard the objection, though, that even if we did have good reason to suppose such sentences to be meaningless, it wouldn’t help with the Liar Paradox.

After all, meaningless sentences aren’t true, so, if one takes Liars to be meaningless, the Strengthened Liar….

“This sentence is not true.”

...still ends up being both true and untrue as a result of standard liar reasoning.
That’s wrong. Ascriptions of truth to meaningless sentences (or to bits of melting candle wax) are themselves meaningless, and, as such, so are their negations. One is surely making a mistake of some sort when one points at a salt shaker and says “that’s true,” but it’s a nonsensical category mistake (which renders the utterance “that’s true” meaningless), not a substantive factual mistake (which renders the utterance “that’s true” false). After all, one is surely also making a mistake of some sort if one points to the same salt shaker and says “that’s false.” Given Bivalence and the claim that both mistakes are factual, we have a trivially easy argument for true contradictions, and dialetheists don’t need to get into anything as esoteric as semantic paradoxes in order to make their case. For every single item in existence other than meaningful declarative sentences (e.g. salt shakers, meaningless sentences, cats, dogs, stars, galaxies…..) there is a true contradiction about the meaning of that sentence!

I’ve also heard the objection, that while applications of the truth predicate to meaningless jumbles of words—e.g. “it’s true that colorless green ideas sleep furiously”—are meaningless, applications of the truth operator to an entire quoted jumble –e.g. “‘colorless green ideas sleep furiously’ is true” are just false. Pretty clearly, given the discussion in the last paragraph, no one who wants to maintain Bivalence without inheriting vast mountains of true contradictions can make this distinction. Moreover, recall that the disquotationalist “nothing above and beyond” principle is, in its original formulation, about truth as an operator rather than truth as a predicate. Hence the metaphor of “disquotation”, that the effect of ascribing truth to a quoted sentence is simply to cancel out the quotation marks.

Certainly, at any rate, no one who accepts the “nothing above and beyond” principle can accept that the negation of the ascription of truth to a meaningless sentence is itself true. If an ascription of truth to a sentence means nothing above and beyond what the sentence means, the ascription of truth to a meaningless sentence can’t be meaningful, and, again, the negation of nonsense is nonsense. Bivalence—the logical formula that tells us that for every claim P, either Tr(P) or F(P)—but meaningless sentences, salt shakers and the rest simply aren’t the kind of thing we can symbolize, write down in a truth table, perform logical operations on and so forth without committing a nonsensical category mistake.

Of course, some readers who are particularly skilled at finding clever new ways to formulate revenge paradoxes for consistent solutions to the paradox might think they’ve found one already, in a phrase I used in the third sentence of the paragraph before last. What, after all, can we say about the following sentence, which we can call The Mistake-Maker?

“One would be making a mistake of some sort if one said that this sentence was true.”

Hold that thought. First, let’s deal with an easier case (although an interesting important one), which we can call The Babbler.

“This sentence is meaningless.”

As I mentioned in Part II, one way of pushing the intuition that sentences like the Liar and the Truth-Teller are meaningless is by means of conversations like this:

“This statement is false.”

“Wait, what statement is false?”

“That one, that I just made.”

“OK, but what were you saying was false?”

...and so on. One could, of course, have the same conversation about the alleged meaninglessness of the Babbler, but, of course, for obvious reasons, if it is meaningless, it is true, and, by definition, all true statements are meaningful ones, so if it is meaningless, it is meaningful, and we have a contradiction.

The answer is that, although the same vague niggling intuitive sense that meaninglessness may be afoot might arise with both the Liar and the Babbler, there is a principled way of differentiating between the cases. In the case of the Liar, we can tell a well-motivated conceptual story to back up and justify our suspicion that the sentence may be meaningless, whereas nothing similar is available for anyone who may wish to argue that the Babbler is meaningless. The story we told about the Liar was specific to the notion of truth, and doesn’t automatically generalize to all other semantic notions. It may go for some—I’ve already indicated, in Part IV of the Russell’s Paradox series, that I think precisely the same deflationary story should be told for the meaning of the phrase “applies to itself”—but this sort of thing has to be decided on a case by case basis, and in the case of the meaningfulness predicate, we have multiple excellent reasons to rule out any similar story being told. For one thing, “‘snow is green’ is meaningful” pretty much has to mean something different than “snow is green”, since the two statements have opposite truth-values! Moreover, the whole linguistic function of the meaningfulness predicate is to differentiate meaningful statements from meaningless ones. It wouldn’t be worth anything if we couldn’t meaningfully apply it to meaningless sentences, and thus say things like “’colorless green ideas sleep furiously’ is a meaningless sentence.” Thus, for any sentence, whether it is meaningful or meaningless, the claim that it is meaningful (or the negation of that claim) is itself meaningful. As such, we have a good, principled, non-ad hoc reason to suppose that the Babbler is simply false. No contradiction follows.

All this leads nicely to the standard, throw-away revenge paradox that is standardly thought to sink meaninglessness solutions. We can call it the Strengthened Babbler.

“This sentence is either false or meaningless.”

If it’s true, then either its true and false or it’s true and meaningless, and either way a contradiction follows. If it’s false, it’s true. If it’s meaningless, and it’s true, and hence meaningful. All roads lead to contradiction.


Well, actually, I’d argue, no, no they don’t. It’s meaningless.

Wait! Doesn’t that mean that it’s true?


Think of another Disjunctive Babbler:

“Either glork blork de glork or this sentence is meaningless.”

Or, for that matter, think of cases like...

“Either glork blork de glork or the Normans conquered England in 1066.”

No one would look at that sentence and say “oh, that’s a true disjunction, since the second disjunction is true” nor should they. Putting “or” and then a combination of words that, on their own, would be a true sentence, at the end of a meaningless string of nonsense doesn’t render the whole thing meaningful, much less true. We know what it means to say that the Normans conquered England in 1066, but we don’t know what it means to say that ‘either that happened or glork blork de glork.’ No one knows what the latter combination of words means, for the simple reason that there is nothing there to know. It’s meaningless.

If someone points to a rock and says “either that, or Japan bombed Pearl Harbor in 1941”, and subsequent investigation shows that he really means for the word “that” to refer to the rock itself rather than any claim about it, the pointer has not succeeded in making a true statement. Rocks aren’t the kinds of things that can be meaningfully symbolized, that we can perform logical operations on, that are candidates to be “disjuncts” of meaningful statements and so on. The fact that the “second disjunct” would, if isolated, be a true sentence, has absolutely nothing to do with whether the “disjunction” succeeds in being meaningful, much less true. The same goes for the Disjunctive Babbler. Meaningless statements are no more candidates for being disjuncts of meaningful statements than are rocks. The fact that we can carefully formulate the Disjunctive Babbler in terms like this...

Sentence DB: “Either sentence DB is false or sentence DB is meaningless.”

...such that the words that come after that “or”, if they were carved off into a separate sentence, would constitute a true sentence of their own…..

...simply does not make the original meaningless claim meaningful, much less true.

Moreover, the sentence we called the Mistake-Maker above, which initially seemed so worrying, is on closer inspection just a disguised form of the Disjunctive Babbler. Or, to be more careful about this, the relationship between the Mistake-Maker and the Disjunctive Babbler is more or less the same as the relationship between the Simple Liar and the Strengethened Liar.

So far, so good, but some readers might note that I’ve used some version of the phrase “if such-and-such sentence is true, it must be meaningful” and the like on several occasions, and wonder if this sort of thing couldn’t be used to bring the Strengthened Liar back into the equation as a problem for the solution I’m defending. After all, the following statement is obviously true:

“If some sentence is true, then that sentence is meaningful.”

Now, I’ve argued that sentence like, for example, the Strengthened Liar, are not meaningful. By Modus Tollens (one of those laws of classical logic I’m so keen to defend with this solution to the Liar Paradox), doesn’t it follow that the Strengthened Liar is not true? And from there, doesn’t it follow by standard liar reasoning that the Strengthened Liar is both true and untrue?

This all sounds pretty worrying, until you realize that there’s an absolutely missing step in that little proof, which is the inference from...

“If some sentence is true, then that sentence is meaningful.”

“If the Strengthened Liar sentence is true, then the Strengthened Liar sentence is meaningful.”

....a string of words that, according to my solution, obviously comes out as just as meaningless as:

“If blork glork de blork, then the Normans conquered England in 1166.”
Assuming that any string of words with the “antecedent” “if the Strengthened Liar is true” is meaningful, much less true, utterly and transparently begs the question against the question against meaninglessness solutions to the paradox. One may find the suggestion that such ‘conditionals’ are meaningless implausible, but surely it is no more implausible than the suggestion that Liar sentences are themselves meaningless. No additional objection to the view can be launched on the basis of one’s intuition that this obvious consequence of that classification is implausible.

‘If P, then Q’ and ‘either Q or ~P’ entail each other. As such, given classical logic—and, again, it begs the question against my defense of classical logic to assume that the classical equivalencies don’t hold—and the principle that the ‘negation’ of a meaningless string of words is itself meaningless, whether strings of words with the grammatical form of disjunctions and a meaningless string of words as one ‘disjunct’ are themselves meaningful is not a separate question from whether strings of words with the grammatical form of conditionals and meaningless strings of words as ‘antecedents’ are not separate questions.

Moreover, careful consideration of this last point should lead to one clear virtue of the classical approach to the paradoxes that I’m championing here over any of the leading non-classical approaches to those paradoxes on the market today. Both the dialetheist solution propounded by Priest, Beall and others, and the Excluded Middle-denying solution offered by ‘paracomplete’ theorists like Hartry Field, are disunified in their treatment of the Liar and Curry Paradoxes. We’ll pick up on that point in Part IV.


William Moore. said...

Sorry but I'm really suspicious about your approach to the disjunctive babbler.

Here's what I have from my paper on the subject, because I rather not try to repeat myself when I think I made it clear before (hopefully my paper is clear)(Here take 'ungrounded' to mean meaningless:)

(L++) (L++) is false, or (L++) is ungrounded.

What is (L++)’s truth-value? If we assume bivalence, (L++) is either true or false. Assume that (L++) is true. Then (L++) is either false or ungrounded. Note, however, that if (L++) is true, then “(L++) is ungrounded” is false. Therefore, “(L++) is false” must be true. Therefore, (L++) is false, if it is true. Assume, instead, that (L++) is false. Then what it says is not the case, and thus it is neither false nor ungrounded, making it true. Therefore, (L++) is true, if it is false. Under the assumption of bivalence, we are back to inconsistency. We need a way to apply truth-values consistently to bits of (L++)."

What I see with this revenge problem is that it seems to push for the need to evaluate these sentences with possibly a third value. But that might not be helpful either.

Tell me what you think ;]?

All the input is helpful, specially when I'm in the refining the paper process.

Ben said...


So far, that just looks like a re-statement of the original problem, not a response to what I have to say about it.

(Moreover, if, as instructed, I read your explanation with "meaningless" substituted for "ungrounded", starting the explanation with "What is (L++)'s truth-value?" seems to sort of beg the question. The view under attack is, after all, the claim that sentences like (L++) are literally meaningless, and that, as such, they simply aren't the sorts of things to which truth talk meaningfully applies. This makes "is it true or false or some other value" a nonsense question, like "what color is Bivalence?")

So, again, what do you think about that approach?

You say that you're suspicious, and fair enough, but let's break it down a bit. Take a string of words we'd both regard as meaningless, say "colorless green ideas sleep furiously" (just because it's the classic stock example) or, if you aren't sure about that, "blork glork de blork." Would you agree that "Either blork glork de blork or the Normans conquered England in 1066" is a meaningless string of words, nonwithstanding the fact that the last five words in the string, would, if taken on their own, be a meaningful (and, in fact, true) statement? If you accept this principle--that, if you take a meaningless sentence, append the word "or" to it and then copy and paste a meaningful sentence to the end of the sentence, the whole thing does not therefore become meaningful--then my solution to the Disjunctive Babbler follows. You may find it counter-intuitive that "this sentence is false or meaningless" is meaningless, but it's surely *no more* counter-intuitive than the claim that "this sentence is false" is meaningless. Moreover, the fact that the string of words with the grammatical place of being the "second disjunct" is true should no more make the whole thing true than "either blork glork de blork or the Normans conquered England in 1066" is made true by the true second disjunct.

I think Bivalence is a red herring here....saying that a sentence like L++ is meaningful and truth-evaluable but has a third truth-value lands you with a disjunctive revenge liar that no solution nearly this neat and straightforward will be available for. Call it L+++.

(L+++) (L+++) is false, or (L+++) has 3rd-value.

What is (L+++)'s truth-value? Well, if it's false, it's true. If it's true, it's either false or 3rd-valued. If it's true, it would be false to say that it's 3rd-valued, so if it's true, it's false. If it's 3rd-valued, it's true, so it isn't 3rd-valued!

(After all, if being 3rd-valued was compatible with being true, postulating a 3rd-value would be totally irrelevant to the solution of the paradox.)

William Moore. said...

I am actually unsure of whether or not sticking the "or" inbetween a meaningful and meaningless expression makes the whole expression meaningful. That is where my suspicion raised.

Whether or not "Either blah blah or ten grams is more than five grams" is meaningful is a perplexing issue and harder more to find its truth value. Well in this case the truth value is true because one of the disjuncts is true.

However, if you have a meaningless expession, as we have, and disjoin the expression with a statement about its meaning, this is where the problem gets expressed.

For instance,

"Either green ideas sleep... or this sentence is meaningless."

If we accept that "this expression is meaningless" allows for the whole expression to be meaningful, then the second disjunct is false and then whether the whole thing is true or false depends on the meaningless expression. This is where I sought a third truth value, for the meaningless disjunct--in which case the whole expression's truth value is probably undefined.

This is how, I feel, the babbler case is to be at least examined at first.

"Either (insert meaningless expression here) or this is meaningless" is either true, false, or undefined.

Your response to the third value brings up the point of revenge problems, which makes me worrisome about approaching the disjunctive liar in general with a third value.

But is the disjunctive liar then meaningless, and then true because the second disjunct is true? Or what, I was not clear on your point.

Ben said...

"But is the disjunctive liar then meaningless, and then true because the second disjunct is true? Or what, I was not clear on your point."

Which disjunctive liar--the meaninglessness one, or the third-value one?

My point about the third-value disjunctive liar was that, by trying to solve the paradoxes with a third value, you're already admitting that such sentences are meaningful, truth-evaluable, etc. What truth-value, then, does "this sentence has either the value 'false' or the third value" get?

Well, if it's false, it's true (since one of the disjuncts is true), so we (implicitly) have a contradiction. If it's true, it's either both true and false (contradiction) or true and third-valued (contradiction). If it's third-valued, then, since one of the disjuncts is true, the whole disjunction is true. Presuming that it's impossible to have both the first and third values simultaneously, all tree options get us contradictions. If, on the other hand, it *is* possible for a sentence to be both true and third-valued, then the introduction of the third value doesn't do any explanatory saying that a paradoxical sentence is third-valued, you haven't yet said anything about whether it's true. That was my point.

My view on "this sentence is false or meaningless" (or "either colorless green ideas sleep furiously or this sentence is meaningless") is that the sentences are both meaningless. They aren't the sort of thing to which truth talk can be meaningfully applied. If we symbolize them, put them at the top of a truth-table, try to perform logical operations on them and so on, we're committing a nonsensical category mistake....and the fact that, if some of the words were taken on their own, they would be meaningful (and true!) changes nothing.

You said that you aren't sure whether adding "or (something meaningful)" to a meaningless sentence makes it meaningful. I'd suggest that one consideration that speaks in favor of "no" is that that answer--combined with the view that Liar-like sentences are meaningless--gets you a simple, workable solution to the revenge paradoxes that's otherwise hard to come by.