Graham Priest discusses The Liar's Paradox for a NY Times blog. It seems that one way of solving the Liar's Paradox is defining dialethei, a true contradiction. Less Wrong, can you do what modern philosophers have failed to do and solve or successfully dissolve the Liar's Paradox? This doesn't seem nearly as hard as solving free will.
This post is a practice problem for what may become a sequence on unsolved problems in philosophy.
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:
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.
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"... (read more)
The Liars Paradox appears to be a special case of infinite recursion.
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 h... (read more)
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 viewp... (read more)
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 falseho... (read more)
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.... (read more)
This is a tangent to dialethei, but something I wanted to bring up for a while:
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... (read more)
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 ... (read more)
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 disc... (read more)
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.
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 vant... (read more)
Dissolving the Liar's Paradox seems to me like trying to falsify a proof by contradiction.