Unsolved Problems in Philosophy Part 1: The Liar's Paradox

The formalist) school of math philosophy thinks that meaningful questions have to be phrased in terms of finite computational processes. But if you try to write a program for determining the truth value of "this statement is false", you'll see it recurses and never terminates:

def f():
    return (not f())

See also Kleene-Rosser paradox. This may or may not dissolve the original question for you, but it works for me.

There's more to be said about the paradox because it keeps turning up in many contexts. For example, see Terry Tao's posts about "no self-defeating object". Also note that if we replace "truth" with "provability", the liar's paradox turns into Godel's first incompleteness theorem, and Curry's paradox turns into Löb's theorem.

ETA: see also Abram Demski's explanation of Kripke's fixed point theory here on LW, if that's your cup of tea.

s :: Bool
s=not s

S'(n) = not not ... {n times} ... S

S'(n)=not S'(n+1)

S(n)=not exists m>n such that S(m)=true

S(n)=(not S(n+1)) && (not exists m>n+1 such that S(m)=true)

forall n in N : (omega>n),
not exists n in N : (n+1=omega),
omega=omega+1

The Liar's Paradox is still considered an "unsolved problem in philosophy"? I don't see why it's considered a big problem that we're able to define things that can neither be sorted into the "true" bucket nor the "false" bucket. If you could derive a paradox from, say, the Peano axioms, then that would indeed be problematic, but as it is, why is the fact that you can say "This sentence is false" any more problematic than the fact that you can say "let X = ¬X" without all of logic imploding?

Math is the art of constructing tautologies complicated enough to be useful. I don't think it's any mark against it that you can use the same language to describe things that are neither useful nor tautologous.

What about this:

The predicate "is true" usually gets applied to a sentence with a subject and predicate. The classic example is "Snow is white". As Tarski says, "'Snow is white' is true if and only if snow is white".

English allows us to pretend we're applying the words "is true" to a noun, for example "Islam is true". But this confuses Tarski: "Islam is true if and only if Islam" is nonsense. So we should properly understand "Islam" in this sentence as a stand-in for various sentences lumped under the name Islam, for example "Allah is God", and "Mohammed is His prophet." When we do this, the statement "Islam is true" unpacks to "'Allah is God' is true, and 'Mohammed is His prophet' is true." This fits nicely in Tarski form: "Islam is true if and only if Allah is God and Mohammed is His prophet."

So the general idea is that you can't use a truth-function to evaluate the truth of a noun until you unpack the noun into a sentence.

Now consider the sentence "This sentence is true". It Tarski-izes to "This sentence is true if and only if this sentence", which doesn't work. To make it work, we have to unpack the noun "this sentence" into a sentence. "This sentence" unpacks to the sentence to which it refers: "This sentence is true". So the unpacking ends with:

"'This sentence is true' is true."

The second round of unpacking ends with:

"''This sentence is true' is true' is true."

And so on, with each unpacking just adding one more "is true" after it without making it any less packed. Trying to unpack the noun fully will lead to infinite regress; stopping at any point will mean you're trying to run a truth predicate on a noun.

What can be said about a truth predicate can also be said about a falsehood predicate, so the Liar Sentence just returns "invalid argument for function", the same as if you pointed to a dog and said "That dog is false!"

The other sentences mentioned as contrasts don't have this problem. "This sentence is in English" also requires a sentence as an argument. It gets one: "The sentence 'This sentence is in English' is in English" is a perfectly valid sentence. It's not necessary to evaluate the truth of the sentence in the middle (its English-ness isn't related to whether it's true or false), so we can leave that one unevaluated and just evaluate the frame sentence, which evaluates the inner sentence's Englishness, which comes out as true.

The Liars Paradox appears to be a special case of infinite recursion.

liar() {

NOT liar()

}

Straight forward. A debugging tool would detect an infinite recursion. An English speaking logician could call it 'meaningless'. Now consider the 'strengthened paradox':

"Everything written on the board in Room 33 is either false or meaningless."

This isn't translatable as a function. 'Meaningful' and 'meaningless' aren't values bivalent functions return so they shouldn't be values in our logic. Instead they should be thought of as flags for errors detected by our brain's debugging tool. But our debugging tool is embedded into the semantics of our language. We talk about sentences having the property of 'meaninglessness' instead of our brains not knowing what to do with the string of letters shown to it. You could probably build a language that returned a pseudo-value of "Meaningless" for infinitely recursive functions. It wouldn't "really" be a value, the program would just output a line that read "x = Meaningless" (not, and this is crucial, assign the variable the value of the string 'Meaningless') when asked to find Liar(x). That is basically what the human brain does.

When we get confused by the strengthened liar paradox we're committing a category error, thinking "meaningless" is a value when it is really an error message. In fact both versions of the paradox are meaningless (assuming 'meaningless' can be taken to mean something like 'Can't compute').

Of course it gets hairy since error messages have truth values too. But

Meaningless(Sliar()) = 1

Is not the same as

Sliar() =1

Thus there is no contradiction. Same will go for Meaningless(Meaningless(Sliar())).

Any discussion of the Liar ought to mention the books of the late Jon Barwise The Liar and Vicious Circles. Also worth mentioning is Raymond Smulyan's lighter puzzle books based on this paradox.

I like the approach of 'paraconsistency' discussed here. But there are some prominent logicians (Girard, for example) who absolutely hate it.

As for "dissolving" the Liar, I would say that it has been dissolved many times, in multiple contradictory ways. Which only goes to show that everything, even logic, can profitably be looked at from divergent viewpoints.

Well, we here agree that beliefs should pay rent. So you see the sentence "this sentence is false" or similar. What new things do you expect now?

Me, I expect to see a philosopher trying to keep his sanity. But the only thing I learned about the sentence's subject is that it is that same sentence, and that it says it's false.

So in that sense it's meaningless.

EDIT: I thought about what I said a bit, and concluded that I'm probably not entirely correct. What the above sentence predicts is that I can read this sentence a second time and see a falsehood.

I realized this after considering that "this sentence is true" is obviously true, yet very similar. And it can also be represented as an infinite recursive function.

EDIT2: Actually, no. "This sentence is true" is NOT obviously true. Infinite recursion is what should happen for it, too.

(I R confuzzled)

I like the article's approach, but it's a bit arbitrary in that "true contradiction" and "false contradiction" are equivalent. But perhaps due to bias towards the positive they get characterized as "true."

What the Liar's paradox really demonstrates is that true and false are not general enough to apply to every sentence, and so to deal with such cases satisfactorily we must generalize our logic somehow.

Then the question is - which generalization do we make? Going with the first thing that pops into our heads is probably bad. Well, let's start with some desiderata:

1) We want it to assign a definite classification to the Liar's sentence. Fairly straightforward - whether it's "option 3" or "1/2" or "0.321374..." we want our system to be able to handle the Liar's sentence without breaking.

2) It should reduce to classical logic in classical cases.

3) It should not be more complicated than necessary.

4) it should not be obviously vulnerable to a strengthened Liar's paradox.

5, etc.) Help me out here :P

Desideratum (3) suggests something along the lines of this, but that might fall prey to (4). I think it's possible that we'll need to allow a continuous truth value. But for now, sleep!

EDIT: After a little experience with this stuff, I don't like the article's approach anymore. "This sentence is not true and is not a 'true paradox.'"

This is a tangent to dialethei, but something I wanted to bring up for a while:

http://video.ias.edu/voevodsky-80th

Voevodsky asks: what if the current foundations of math are inconsistent? Answers: probably nothing so bad.

My first thought was to look for the technical version of Priest's article, which turns out to be his book "In Contradiction", which turns out not to be in my university library. The Amazon preview tells me that he discusses Gödel's theorems but not the computational models that so many comments here talk about, and he gives a formalisation of some form of paraconsistent logic. However, the preview isn't enough to answer the basic question to ask about any non-standard logic: is it intertranslatable with classical logic, such that every truth of either is mapped to a truth of the other? If it is, then there is no philosophically interesting distinction between them, any more than there is between English and French, or C++ and Perl, or standard analysis and non-standard analysis.

So summarizing the thread and the links I've read it looks like there are two basic strategies to solving this problem. One is the Dialetheist strategy used in para-consistent logics. This strategy rejects the principle of non-contradiction and one of the rules that leads to the principle of explosion (any part of the disjunction syllogism used in explosion). The other is the strategy characterized by the formalist school's approach and cousin_it's comment variations of which were given throughout the thread (I consider Yvain's Tarski sentence approach to be an instance of this strategy). The idea here is that by treating the paradox as a bivalent function we notice that it is a function which recurses infinitely. We then add the perhaps not obvious but certainly intuitive premise that sentences which suggest non-terminating functions are part of the subset of 'meaningless' sentences.

From the IEP

The Liar Sentence L is meaningless, so the Liar Argument can’t even get started because its main assumption (that the Liar Sentence exists or is meaningful) is faulty. Natural language is incoherent, and its underlying sensible structure is that of an infinite hierarchy of levels. Because the Liar Sentence would have to reside on more than one level simultaneously if it were a legitimate sentence, it’s not really a legitimate sentence. This way out of the paradox is taken by Russell in his ramified theory of types and, following Tarski, by Quine in his hierarchy of meta-languages. For Russell, the referential phrase “This sentence” in L is the culprit because the phrase is not allowed to refer to the sentence in which the phrase itself occurs. For Quine, instead, the culprit is the phrase “is false” in L because the phrase must be satisfied by sentences in a language lower in the hierarchy and not by the very sentence in which the phrase occurs.

Are what has been proposed here different from the approaches of Quine and Russell? If not which of the two is right? It isn't obviously a difference that makes no difference; Russell's approach rules out all self-reference while Quine's does not.

Now, this method (call it the non-termination approach) certainly seems to dissolve the confusion. And compared to the Dialetheist approach the non-termination approach appears much superior. The former gives up the principle of non-contradiction and a useful rule like disjunction introduction (or some other alteration to deductive logic, there appear to be a lot of alternatives and I haven't gone through them all).

So the question becomes: what advantages does the paraconsistent logic approach have? Does anyone know of examples of logic that don't have to sacrifice significant power in order to accommodate dialetheias? It doesn't seem like it would be worth it.

I remember being bothered by this problem, and feeling like I had resolved it as an undergrad. Calling it a "true contradiction" seems absurd; you've just drawn a circle around it and said, "Nothing to see here! Move along!"

I think the solution is related to modal logic. "This sentence is false" creates a self-referential universe devoid of meaning, and thus has no truth value. It refers only to the world of itself, and there are no rules that it can be evaluated against, nor are there any observations that can confirm or disconfirm it. It is, in a sense, epiphenomenal, as there is no actual thing which it corresponds, predicts, or relates to. It is, in a sense, a one-sentence universe that cannot be tied to anything in any other universe.

This concept seems more robust in my mind; I suspect I am either making a mistake or failing to explain myself. Criticism or questions would be appreciated.

There's nothing inherent in a statement that makes it true or false. It's just useful to think that way.

I'd say that it's really just somebody vibrating the air, but even that is an abstraction, and has no more real truth than anything else.

http://en.wikipedia.org/wiki/Liar%27s_paradox#Possible_resolutions

Tarski and Prior both have good approaches. I wouldn't call this problem unsolved.

I don't mean to throw away all the wonderful complexity and intricacy of the argument, but it seems like they had it just about right when they added "is meaningless".

The trick is just not to write "Everything on Board 33 is meaningless" on that same board. Honestly, that board is just a bad choice for this task.

Which is to say, we can see, from our vantage point outside the sentence, that the sentence is meaningless (in the sense of "can't tell us anything"). That seems like it ought to be enough. Why try to inject our vantage point into this whirlpool of contradiction when we can just notice, remaining outside the whirlpool, that the question is meaningless, and move on anyway?

Dissolving the Liar's Paradox seems to me like trying to falsify a proof by contradiction.