Thursday, December 08, 2005

If the title of this guest post is true, then you should read it.

[By Charles Johnson]

Here's one of the few canonical philosophical puzzles that I had learned about by the age of five. What's the truth-value of the following statement?

(L) This statement (L) is false.

The problem, of course, is that if (L) is true then it's false, and if (L) is false then it's true. Thus, any theory of truth that assigns a truth-value to (L) is internally contradictory, since the theory will (inter alia) include the contradictory truth-ascription:

(TL) L is true if and only if L is false.

Since there are no true contradictions, a theory of truth must not assign any truth-value to (L) at all. But how do you doing it? If a statement hasn't got a truth-value, then the usual take is that they are, in some respect, nonsense; that is, they fail to make an assertion -- just as "Cat mat on the sat the" fails to make an assertion. The canonical approach to (L) in the 20th century has been to try to come up with some principled means of ruling (L) out of the language by means of setting up the right structure of rules beforehand (just as you can point to the preexisting rules of syntax to show that "Cat mat on the sat the" doesn't amount to a complete sentence). The most famous attempt, and the inspiration of many of the subsequent attempts, has been Tarski's attempt to sidestep the Liar Paradox by means of segmenting language into object-language and meta-language layers. The idea being that, if you do this assiduously, you can avoid self-referential paradoxes because self-reference won't be possible in languages whose sentences can be ascribed truth-values; because they can only be ascribed truth-values within a meta-language that contains the names of the object language's sentences and truth-predicates for those sentences. I have a lot of problems with this approach; a full explanation of them is something that I ought to spell out (indeed, have spelled out) elsewhere. But here's a quick gloss of one of the reasons: Tarski and the people inspired by him started setting up ex ante rules to try to rule out self-referential sentences because it's self-reference that makes the Liar Paradox paradoxical (and that makes for similar paradoxes in similar sentences; exercise for the reader: show how "If this sentence is true, then God exists" is both necessarily true and strictly entails the existence of God). But there's an obvious and general problem for the method: there are self-referential sentences which are unparadoxical, and indeed self-referential sentences which are true. Here's an example which may or may not cause trouble for Tarskian theories, depending on the details:

(E) This sentence (E) is in English.

(E) is truth-valuable; and in fact it is true. (If, on the other hand, it had said "This sentence is in French," it would have been false.) Now, this may cause trouble for the Tarskian method and it may not, depending on the details of a particular account. (Sometimes people want to ban all self-referential sentences; sometimes they are more careful and claim that object languages might be able to name their own sentences but only so long as they don't contain the truth-predicates for their own language.) But even if (E) is allowed, you haven't solved the problem. There are plenty of self-referential truth-ascribing sentences that aren't paradoxical, too. Here's one:

(EM) Either this sentence (EM) is true, or this sentence (EM) is false.

Unlike (L), this causes no logical paradoxes. If you suppose that it's false, that means that it turns out to be true -- since the second disjunct, "this sentence (EM) is false" turns out to be true; meaning that it cannot be false. But it *can* be true, without contradiction. So it has to be true, if it has any truth-value at all. That shouldn't be surprising; it's an instance of the law of the excluded middle, and all instances of the law of the excluded middle are true.

Now, you might think that (EM)'s relationship to ordinary talk is attenuated enough, and the reasons for thinking it unparadoxical are technical enough, that it might be an acceptable loss if some other technical stuff that saves us from (L) happens to rule out (EM) too. I'd be inclined to agree, except that (EM) isn't the only example I had up my sleeve, either. Here's another. In the Prologue to the Travels, Marco Polo wrote,

We will set down things seen as seen, things heard as heard, so that our book may be an accurate record, free from any sort of fabrication. And all who read this book or hear it may do so with full confidence, because it contains nothing but the truth.

Let M be the conjunction of all the assertions that Marco Polo makes in his book. The book contains nothing but the truth if and only if M is true, but that the book contains nothing but the truth is one of the many assertions in the book, so "M is true" is one of the conjuncts of M. Thus:

(M) This conjunction (M) is true, and Marco Polo traveled the Silk Road to Cathay, and served in the court of the Great Khan, and observed the barbarous customs of lesser Armenia, and ... and ... and ....

But it's either true or false that Marco Polo's book contains nothing but the truth; that assertion is a standard bit of understood language (passages just like it are a near-universal feature of traveler's tales, or other extraordinary stories where the author feels the need to reassure you that she's not making things up). If your theory of language throws it out as nonsense, then your "theory of language" needs to be thrown out, on the grounds that it's not semantically serious. (Whatever it's a theory of may be interesting, but it's something other than language as it actually exists.)

Now, like Polo, I may have been fudging just a bit in what I said. I suggested that M isn't paradoxical; I don't think it is, but there is a way to make it seem paradoxical. Lots of readers have doubted that Polo was telling the truth; some of them, for example, were unimpressed by the evidence that he had ever served in the court of the Great Khan; others weren't so sure about the tales of dog-headed men or giant birds that consumed elephants. Whatever the case, they believed that Marco Polo made at least two false assertions in his book: (1) the claim about his journeys that they doubted, and (2) the claim that his book contained nothing but the truth. Call these the normal skeptics. I'm sure there were also a few readers (however credulous they would have to have been), who believed that the Travels really did contain nothing but the truth; that is, that there were no false assertions in the book, including the assertion that the book contained no false assertions. Call these the normal believers. But now imagine a third kind of reader, a perverse skeptic -- a philosopher, of course -- who noticed that you could gloss the contents of the book as M, and who decided that she believed everything that Polo said in the book about his journeys, from the customs of lesser Armenia to the domains of the Great Khan to the giant birds. She believes everything in the book, except ... there is one assertion that she thinks is false -- that is, (1) the assertion that everything in the book is true, and nothing else.

There are a couple of different ways that you could approach the difficulty. One way is to point out that the perverse skeptic really is being perverse. That's just not how you can sensibly read the book. Either you think that nothing in the book is false, or you think that at least two things are; the assurance of truthfulness just can't be a candidate for falseness until something else has been shown false. But if you list the truth-conditions of M, then "M is true" is one among them, and it's hard to see how you could stop the perverse skeptic from going down the list and picking that one as the only one to be false. Certainly Polo doesn't say "The rest of the book besides this sentence contains nothing but the truth." And given that he did say what he did, I'd be hard put to say that "this book contains nothing but the truth" isn't one of the untruths denied by the sentence, if something else in the book is false.

Another way to approach it is this: you can imagine an argument between the normal skeptic and the normal believer; whether or not one ever managed to convince the other in the end, you can in principle identify the sorts of reasons that they might offer to try to determine whether Polo really did tell the truth about the birds, or about the Khan, or..., and you can say what things would be like if one or the other is true. But what kind of argument could the normal believer and the perverse skeptic have? How would one convince the other? Or, to take it beyond the merely psychological point to the epistemological point, what kind of reasons could the normal believer possibly give to the perverse skeptic to give up the belief that "this book contains nothing but the truth" is false? (She can't point to all the true statements about his journey; the perverse skeptic already believes in those.) Or, to take it beyond the epistemological point to the ontological point: what sort of truth-makers could even in principle determine whether the normal believer or the perverse skeptic is in the right?

So there is a problem with M, to be sure. But the problem is not the same as the problem posed by L: there's no logical contradiction involved, so its self-referentiality sets off no logical explosions. And the solution can't be the same either: the radical move of abandoning the sentence as meaningless works with (L), where there's just no right way to take it, but it doesn't help us out with (M), where there obviously is a right way to take it (i.e., as the normal readers take it, and not as the perverse reader takes it).

So there has to be some right way to go about ascribing a truth-value to (M) (and also (E)). Whatever it is, it may very well also explain how we can ascribe a truth-value to (EM). But it certainly cannot also mean that we try to ascribe a truth-value to (L). What is it? Is there some kind of principled and motivated general rule that we can add to our logical grammar, so as to get M and E and maybe EM but no L? If so, what in the world would it be? If not, then what do we do?

(I have my own answers; for the details, you can look up "Sentences That Can't Be Said" in the upcoming issue of Southwest Philosophy Review. Or contact me if you're interested enough to want a copy of the essay. But I want to pose the puzzle and see what y'all think about it as it stands.)

Update 2005-12-08: I fixed a minor error in phrasing. Thanks to Blar for pointing it out in comments.


  1. Shouldn't the sentence at the end of your fourth-from-last paragraph be "Or, to take it beyond the epistemological point to the ontological point: what sort of truth-makers could even in principle determine whether the normal believer or the perverse skeptic is in the right?"

  2. Yes, it should. It's been fixed in the post now, with credit given where credit is due. Thanks!

  3. What do you think of this solution?

    Simple terms, like "house" and "this chair" are neither true or false, even though they are certainly english words.

    But "this statement" is a simple term, and it identical with a statement when I say "this statement is false". So the paradoxical statement you give is identical to a simple term. So it can be neither true nor false.

    I think this solution avoids your objection "this statement is in English", for there is no problem with having a simple term be in a particular language, but only with it being true or false.

    In a word, I think asking "is this statement true or false"? is like asking "is 'blue chair' true or false"?

  4. But there isn't generally anything wrong with saying "that statement is false" (pointing to, say, a claim that grass is blue). And I assume "that statement" has the same grammatical status as "this statement".

    (I also wonder if you're confusing use and mention. Clearly blue chairs can't be in English. Statements can. But statements can also be true or false, even if the term 'this statement' cannot be. For note that in this context, our use of the term 'this statement' refers to the whole statement, and not just the term. Just like 'blue chair' refers to a blue chair, and not the describing words.)

  5. If "this statement is false" refers to some other statement (like "paradoxes are fun"), then the whole paradox disappears from the very beginning, and there is simply nothing to explain or even puzzle about. The paradox will never come up at all. I'm not totally sure I see what you mean by "grammatical status", there is difference between self reference and other reference, right? Only self refence gives us the paradox, after all.

    I think that the explanation I gave can account for this, for if one is refering to some other statement, and the other statement is not a simple term, then the subject "this statement" is not identical to a simple term. Therefore there is no impediment to the thing being true or false.

  6. in a word, I read your objection as requiring that it be the same thing to refer to self and to refer to other. If this is right, I think its reasonable to deny the objection.

  7. Shulamite, "This sentence is false" does not attempt to ascribe falsity to the words "This sentence," any more than "The first sentence written by Plato was false" attempts to ascribe falsity to the words "The first sentence written by Plato." Both of them attempt to ascribe falsity to the sentence picked out by the denoting phrase.

    Even setting that issue to one side, though, I don't think your solution is even materially adequate. Among other things, it would require us to dismiss statements such as (M) for precisely the same reasons that we dismiss (L). But (M) is a perfectly ordinary bit of understood language. I think any theory that discards it is, for exactly that reason, not a good theory.

    Shulamite: If "this statement is false" refers to some other statement (like "paradoxes are fun"), then the whole paradox disappears from the very beginning, and there is simply nothing to explain or even puzzle about.

    This is not so. There are what are called "looped liar" paradoxes. Consider:

    (P1) P2 is true.

    (P2) P1 is false.

    If P1 is true, then it follows that P2 is true; thus that what P2 says obtains; thus that P1 is false. But if P1 is false, then it follows that P2 is not true; thus what P2 says does not obtain; thus P1 is true. Similarly, if P2 is true then P2 is provably false, and if P2 is false then P2 is provably true. Any theory of truth that ascribes either truth or falsity to both P1 and P2 is therefore false, because internally contradictory.

    There are also cases where we simply don't know the contents of the sentence to which we are referring. For example, you might say, "the first assertive sentence Plato ever wrote was true," or "the first assertive sentence Plato ever wrote was false." Provided that Plato existed and did write one or more assertive sentences, one of these is true (although we will probably never know which one of them is). But allowing these kind of descriptions can be risky. For example, suppose that the first thing I say on Tuesday was, "The first thing Shulamite said today is false." And the first thing you said on Tuesday -- not knowing that I had said this -- was "The first thing Rad Geek said today is true."

  8. When I said "if it refers to something else" I was responding to Richard, who made the paradoxical statement refer to the sentence "grass is blue". In his context, I stand by what I said. I did not mean to address the looped liar paradox, only Richard's objection.

    To speak to your objection, however, I have to be more exact about what I mean. My claim is that no simple term (like "chair" or "this man" or "blue grass") is true or false. But in the self referential proposition, the whole proposition is identical to a simple term- this is simply what self referential means.

    No simple term, or what is identical to it, can be true or false. I don't see how this commits me to denying anything that Marco Polo might have said in his book. So long as "this statement" in your paradox, or "this conjuction" refers to a proposition that is not identical to a simple term, then I have no problem with it yet. I might want to argue against it for other reasons, but there is no need to now. Not all liar paradoxes are false for the same reason. The looped liar is false for other reasons- but still very clever...

    Oh, I can't resist. I think the looped liar ascribes certainty to future contingents, i.e. the sentence coming up is true...the sentence I just said is false. But this might be too much for a brief comment.

  9. "But in the self referential proposition, the whole proposition is identical to a simple term- this is simply what self referential means."

    No it isn't! In general, reference is not identity. The words "blue chair" are not identical to a blue chair, for example. Likewise, the term "this statement" is not identical to the longer statement ("this statement is false") to which it refers. Self-reference doesn't help you here, because it is the proposition, and not the term, which is self-referential. (The term "this statement" refers to the whole statement, and not to the term "this statement" itself! As an analogy, suppose I say my own name. I thereby refer to myself. But the words I speak are not refering to themselves, they're refering to me!)

    I also don't see how you hope to avoid the "this statement is in English" objection. For you must say that the entire statement is identical to the simple term "this statement" which refers to it. And simple terms have no truth value, so neither must the original statement. But of course this is mistaken -- the original statement is straightforwardly true! (But the term "this statement", by contrast, is just as straightforwardly lacking in truth value.) So you should retract your claim that the self-referential term is identical to the whole statement to which it refers. Your argument rests on a use-mention confusion.

  10. At the risk of making a naive comment on this oldest of chestnuts - I am no philosopher - isn't L roughly the equivalent of saying
    x = not x

    or possibly
    x = [x=0] where [...] is the boolean evaluation of the enclosed statement.

    in math? The statement is just plain wrong, you don't need to know the value of x - there isn't a stable value of x that satisfies x = not x, so the statement is just wrong. Now, what you may now think is... but if it is wrong then that is like saying L is false, and L says that L is false, therefore L is true etc. etc. enter the paradox.

    No! The statement is just plain wrong, therefore don't go inside the statement to see what it tells you about anything - let alone itself.

  11. Isn't M just a variation on:

    (T) This statement (T) is true.

    It's consistent to say that T is true, or that T is false, or that T is meaningless. I'm not sure what the traditional way of handling sentences like T is, but I'm fine with ruling them out as meaningless (just like L). Whatever convention we take, we can apply to M, since M can be broken up into a "looped" version of T:

    (M1) M* is true.
    (M2) Every other statement in Travels is true (i.e. Marco Polo traveled the Silk Road and ... and ... and ...).
    (M*) M1 is true and M2 is true.

    M* is just like M (we can think of M* as saying "All of the other statements in this book are true, and so is this one", which is just a rephrasing of "All of the statements in this book are true, including this one"). Here, the perverse skeptic would accept M2 and deny M1, while the normal believer accepts both and the normal skeptic rejects both. Since M* is just a looped version of T, if T is meaningless, then so is M*, and so is M. If we have some other convention for the truth-value of T, we could apply that to M* and M as well.

    As a point of pragmatics, in actual usage we might want to interpret M as if it were the most similar meaningful statement, which would be M2. In that case, there would be no disagreement between the normal believer and the perverse skeptic.

  12. Fontwell, the question here is what you mean by the "wrongness" of a statement. If you mean that the statement is ill-formed or meaningless, many if not most philosophers in the 20th and 21st centuries would agree with you. (In fact, I would too.) But the question is what sort of principled and motivated account we can give to explain why it is ill-formed or meaningless. Just pointing out a problem doesn't solve it; the question is how you can have a language that allows you to make assertions of the form "A is true," where A is a name or description that designates a particular sentence, while avoiding the awkward consequence that one of the sentences you might deny is the very sentence by which you deny it. Just banning sentences that lead to contradictions, solely on the basis of their leading to contradictions, has a couple of awkward effects: (1) it seems to be nothing more than linguistic gerrymandering; if I write "Johnson's thesis is false," then why should I be able to name it "Jackson's thesis" but not "Johnson's thesis"? What's to stop me? (2) There are in fact self-referential sentences that don't result in logical contradictions, but do cause philosophical headaches in other ways. Sentences of the form "If this sentence is true, then P" don't directly result in any contradiction, but do allow you to prove absolutly any proposition whatsoever that you care to substitute for P. M, as discussed above, seems to be a part of our ordinary language, but it also seems to allow for the possibility of the bizarre disagreement between the normal believer and the perverse skeptic. And T, as discussed by Blar, is logically completely compliant -- if it's true, it's true, and if it's false, it's false. But its semantics seem to shrink to a vanishing point; there seems to be nothing even in principle that could make it true, or make it false.

    Blar, I think you're right to show interest in T (in my essay I talk about it as an essential part of understanding what's wrong with L) but I don't think that M can be reduced to it. The simple reason being that T can't meaningfully be asserted but M can (and was, by Marco Polo). I think that part of what a theory has to do in accounting for the "data," as it were, is to account for the fact that Polo wrote M (or rather, wrote its equivalent in Italian) and we understood what he wrote.

    One way to think about this is that when we evaluate M (and so, if we try to evaluate your looped case of M1, M2, M*, in such a way as to capture what M said) there seems to be a right order to do the evaluation in. First you figure out whether the normal believer or the normal skeptic is right about all the other statements in the book, then you count the assurance as false only if that's entailed by the falsity of one of the other conjuncts. And that's how you get the truth-value of M.

    You could say, "O.K., well, that gives us a convention for calling M true or false and so also a convention for calling T true or false." But of course if that is the convention, then we don't have one for T, since T doesn't have any "conjuncts" besides itself. There doesn't seem to be any point at all at which it could be tied down to anything in logical space. So it does seem to me that there has to be an important difference between M and T; the question is how to spell out what that difference is.

    As for the suggestion that we can translate M simply as M2, and so get the truth-conditions that we want, well, I agree that we can, but I'm not at all sympathetic to the claim that that's how we should understand what Polo said. Because, well, that's not what Polo said, and there are also technical problems that surface in most of the accounts that would give you some motivation for making the translation. I don't know about you, but it certainly seems true to me that if Polo lied when he said, "This book contains nothing but the truth," then his book contains at least two counterexamples to his claim: first, whatever it was he was lying about as far as his journey is concerned, and second, the assurance that he was telling the truth.

  13. Thanks for the comments Rad Geek. You are right to say that my use of "wrong" equates to ill-formed. I also take your following points too.

    As a bit of a diversion, I'm a digital silicon chip designer and so I'm used to the idea of expressions like a = NOT(b), a = b AND c etc being turned into real hardware circuits. Now, given the handful of boolean operators it is possible to combine various signals (or variables, if you like) in any way that can be expressed in boolean logic - including self referential expressions/circuits e.g.

    z = a AND b This is a normal circuit.
    z = NOT (a AND NOT (z AND b) ) This is a 'proper' self referential circuit, where z=true or z=false are both acceptable [bi-stable] when a=b=true.
    z = NOT (z) This is a 'bad' self referential.

    In the last case, while it is obvious that there is no value of z that satisfies the equation it is interesting to note that the statement still appears to have meaning, even to the extent that one can actually build the real circuit. The hardware reality of this circuit would usually result in z oscillating as fast as the circuit can operate (in an amusing anthropomorphic way, it is just like someone is saying "If z is true then NOT(z) is false, which means that z is false, but that means NOT(z) is true, which means that z is true...). This would be classed as 'unstable' in electronics.

    So, what I'm getting towards is that in terms of boolean operators the three previous statements are all similar in as much as they are all proper boolean expressions, unlike, say, z = AND a b NOT , which is doesn't even conform to the rules of boolean syntax. However, one of the expressions is stable, one is bi-stable (sometimes) and one is unstable/ill-formed/unsolvable for z/wrong whatever you want to call it. The only way to tell if an expression is unstable is to go through the process of analysing it (obviously z = NOT (z) doesn't take much analysing, where as z = NOT (a AND NOT (NOT(z) AND b) ) needs at least a quick glance - it is unstable when a=b=true).

    Therefore for proposed logical expressions (in my world of hardware), I could define four output states (values of z)

    1) Stable values
    2) Bi-stable values
    3) Unstable values
    4) no value - the expressions that do not even conform to boolean syntax.

    Isn't this very like the situation with L, M etc? My sort of conclusion by analogy is that, yes, we are allowed at the syntactical level to form expressions that turn out to have stable/bi-stable/unstable solutions (meanings) but the only way to find out which kind of expression it is, is to poke around with the details inside the expression. But there is nothing inherently paradoxical about having a unstable (ill-formed) expression that uses proper boolean (English) to form the expression.

    Or have I still missed the point?

  14. Duh. The sentence in question is meaningless.


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