That's not a paradox if you distinguish between a lie and a false statement. If Pinnochio does not expect his nose to grow, then he is lying, and his nose will grow; he has lied but in so doing made a coincidentally true statement. If he does expect his nose to grow, then he is telling the truth as far as he is able, and it will not grow, even though he would be mistaken in the statement.
If Pinnochio's nose detected absolute truth, then there really should have been a subplot in the story about people who want to kidnap him and pose him questions to determine the ultimate nature of the universe.
If Pinnochio's nose detected absolute truth, then there really should have been a subplot in the story about people who want to kidnap him and pose him questions to determine the ultimate nature of the universe.
Or just make shitloads of money by asking him about the results of upcoming football matches and betting on them.
If Pinocchio's nose detected absolute truth, then there really should have been a subplot in the story about people who want to kidnap him and pose him questions to determine the ultimate nature of the universe.
Ha ha ha. Mad Computer Scientist: "And now, Pinocchio, I would like to you to assert that 'P=NP'."
(I was just primed by reading this.)
In what language did I write the description of this video?
You seem to have written it in English, a language that offers no protections against paradox inducing self reference.
I did not create this video and I have no idea who did; someone originally named the file truefalsempeg1.mpg.
The Epimenides Paradox or Liar Paradox says "This statement fails to match reality."
Why did I post this video? Well... type hierarchies supposedly resolve the Epimenides paradox. Using an indefinitely extensible, indescribably infinite, ordinal hierarchy of meta-languages. No meta-language can contain its own truth predicate - no meta-language can talk about the "truth" or "falsity" of its own sentences - and so for every meta-language we need a meta-meta-language.
This video does a pretty good job of depicting how I feel about that: pretty much the same way I feel about the original Epimenides Paradox.
Bonus problem: In what language did I write the description of this video?
Video may be a wee bit longer than needed to get the point across. And by wee bit, I mean you could cut 7.5 minutes off of it, and it runs for 7:40.
I've always (after a fun paper on this!) thought of this in modal terms. Self-referential statements are not truths about the world; they're truths about ill-defined universes. "This statement is false" doesn't refer to anything; it's its own little world, and consequently truth has no meaning.
Similarly, "This statement is true" really doesn't provide any information. What if it's a false statement? Is there a difference? How can we tell? It's its own self-referential world, and it's unclear what truth even means in that little world, though it's very clear that it does not make the least bit of difference in any world I care about.
Video may be a wee bit longer than needed to get the point across.
I believe that's the intended point. Besides, most infinite ordinals are very much larger.
"This statement is false" doesn't refer to anything; it's its own little world, and consequently truth has no meaning.
What about statements that refer both to themselves and to things in our world, such as "either this sentence is false, or Psychohistorian owes Peter de Blanc $50"?
This is also how I've always thought about self-referential statements; is there a name for this position? Is it even a position?
Modal logic, from what I remember of it, deals with "worlds" of sorts, so it might be possible to express it with modal logic; I don't remember well enough, and I certainly don't recall this being a named position. It does seem like a position, though.
And by wee bit, I mean you could cut 7.5 minutes off of it, and it runs for 7:40.
There's a very pretty bagpipe line that kicks in towards the middle -- wouldn't want to miss that.
Is that the Mingulay Boat Song?
To what extent is the paradox really solved? Can someone who understands the proof more fully comment on this?
This solution to the paradox say that the context shifts... "this statement is false" can be true if whatever "this statement" is referring to is false; in other words you have nested meanings (type hierarchies) where the whole 4-word sentence is true (meta-meaning) because it is referring to a statement (in this case itself) that is false in a given context.
Seems a bit unsatisfying to me. They've redefined the meaning of the sentence so that it now makes sense -- but have they really addressed the original paradox or just explained it away?
By the way, Godel's incompleteness theorem does not rely on the paradoxical aspect of the Liar Paradox to be undecidable. The statement p is undecidable in the given theory it is referring to, even though it is not paradoxical:
p = "This statement cannot be proven in the given formal theory"
FYI: the middle 6-7 minutes can be safely skipped (unless I'm missing the point somehow).
Most of the video either made no sense or went over my head. If it's the former, recommend against watching.
The Epimenides Paradox or Liar Paradox is "This sentence is false." Type hierarchies are supposed to resolve the Epimenides paradox... Using an indefinitely extensible, indescribably infinite, ordinal hierarchy of meta-languages. No meta-language can contain its own truth predicate - no meta-language can talk about the "truth" or "falsity" of its own sentences - and so for every meta-language we need a meta-meta-language.
I didn't create this video and I don't know who did - but it does a pretty good job of depicting how I feel about infinite type hierarchies: namely, pretty much the same way I feel about the original Epimenides Paradox.
Bonus problem: In what language did I write the description of this video?