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  • If his nose is not growing, he is telling a lie and his nose will grow but then he is telling the truth and it can't happen.
  • If his nose is growing, he is telling the truth, so it can't happen.
  • If his nose will grow, he will be telling the truth, but his nose grows if he lies so it can't happen.
  • If his nose will not grow, he is lying and it will grow but then he would be telling the truth so it can't happen.

What will happen? The universe will explode?

Community
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Rodrigo
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    Be careful. Saying that a falsehood is true does not a liar make. If you don't know what you are saying is false, then not a liar. You're just wrong. – David H Aug 16 '13 at 01:42
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    Well, he could get plastic surgery to enlarge his nose... I mean, wooden surgery. – commando Aug 16 '13 at 02:22
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    What happens next is up to the story-teller, because Pinocchio is a fictional character whose world is not constrained to any particular laws of physics (hence the quite literally fabled properties of his nose) nor logical consistency (which must perforce go out the window if Pinocchio succeeds in implementing the Liar's paradox). Or more precisely: the logic which is best suited for effective reasoning in Pinocchio's world depends on what the story-teller decides happens, and that logic needn't be classical logic. – Niel de Beaudrap Aug 16 '13 at 07:27
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    @NieldeBeaudrap: Well played, like the guy, who, told to measure the height of skyscraper with a barometer, dropped it off the top and timed it. – Mitch Aug 16 '13 at 12:41
  • @Mitch: the analogy would perhaps be complete if the barometer had no markings on it. One might as well ask a physics forum, how Superman avoids getting third degree burns from air friction. – Niel de Beaudrap Aug 16 '13 at 12:44
  • @NieldeBeaudrap: it's all just playing the game. – Mitch Aug 16 '13 at 12:45
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    If he is a little boy than chances is that as he grows older his nose will indeed grow. – Neil Meyer Aug 16 '13 at 14:55
  • Pinocchio's nose grows when he lies. Lying and saying something that isn't true is not the same thing. – gnasher729 Jan 03 '16 at 23:49

4 Answers4

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That looks like a paradox, but it isn't.

The assertion is so vague that it is true even if Pinocchio's nose doesn't grow immediately.

Even if he says: "My nose will grow immediately" that's a vague assertion, when does "immediately" start and end as a time lapse?

If he is more specific and states: "My nose will grow in the next five minutes" then the nose can make a suspension of disbelief, if he tells a lie in those five minutes then the nose will grow and the first assertion will be true, if he doesn't then the first assertion will be proved as false and if falsehood is considered as a lie then the nose will grow after the five minutes.

Everything that is false is a lie? That depends on some definitions, but if Pinocchio says something that is false and a mistake then the nose may (or may not) consider that Pinocchio is implicitly stating to be sure about something that he is not sure and actually wrong, hence a lie. That may be a question for Geppetto.

Nothing is true or false until it is.

If Pinocchio succeeds in creating a paradox (either the liar's paradox or some other paradox) then the nose will not react to that. A paradox is something that cannot be true and cannot be false and the nose grows on falsehood, right?

No matter which logic and assumptions (OWA and CWA for instance) that nose follows, in any logic where there can be paradoxes it is clear that the middle cannot be excluded.

Trylks
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  • While I want to give you credit for this answer, I still have to give you a –1, because your treatment of the paradox that the author was really getting at seems quite weak. The paradox is the crux, and you just hand-wave it away. – labreuer Feb 26 '14 at 22:32
  • no worries about the -1. WRT "the paradox is the crux", there is no paradox, maybe the question could be reworded or formalized so that there is some paradox, but there is no paradox so far. Anyway, if Pinocchio succeeded in creating a paradox: "then the nose will not react to that." Therefore, if the question was fixed and a paradox included, it would be already be answered. – Trylks Feb 27 '14 at 11:07
  • Actually, I thought you did a good job clarifying such that it would become a paradox? One can call it nit-picking, but in philosophy discussions, nitpicking is often a good thing! :-) I just think you ended up ensuring there was a paradox, and then left me hanging! – labreuer Feb 27 '14 at 16:59
  • There is no paradox, but if there ever was one, the nose would not react to that, because it does not react to paradoxes, it reacts to lies (as I pointed in the original answer). It's a tale to tell children not to lie, but nobody cared to say not to troll or not to hack. – Trylks Feb 28 '14 at 09:27
  • I suppose you could make a play on 'lie' vs. 'utter falsehood', but that seems to escape on a technicality. It just seems like you're taking the easy way out, and not addressing the issue head-on. But I could be wrong. – labreuer Feb 28 '14 at 09:50
  • What's "the issue" in your opinion? – Trylks Feb 28 '14 at 09:55
  • PD: I've just seen your answer. You can analyse it like that, which means the nose will crash with a "not enough heap space" exception and it will not grow. I'd hope that someone with enough knowledge to make a nose that grows (creating matter out of nothing, big deal) would be able to detect and prevent this before raising an exception. In any case, the nose doesn't grow. – Trylks Feb 28 '14 at 10:03
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The premise that Pinocchio's nose grows if and only if he claims any falsehood is given as:

∀x: G ↔ C(x) ∧ ¬x

If Pinocchio's claim is that his nose will grow then this is given as:

G ↔ C(G) ∧ ¬G

This is a contradiction. Therefore the premise that Pinocchio's nose grows if and only if he claims any falsehood is necessarily false:

∃x: ¬(G ↔ C(x) ∧ ¬x)
Michael
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  • Once you accept a contradiction in bivalent logic, anything can be proven true. – labreuer Feb 26 '14 at 22:33
  • Then don't accept it. – Michael Feb 27 '14 at 00:57
  • If anything can be proven true, anything can be proven false. The conclusion of your argument is not the only logical conclusion. – labreuer Feb 27 '14 at 01:02
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    I don't understand how "If anything can be proven true, anything can be proven false." relates to my answer. Also, re. your first comment, are you claiming that I am accepting a contradiction? Because I'm not. I'm arguing that the premises lead to a contradiction and so one of the premises is false. Reductio ad absurdum; proof by contradiction. – Michael Feb 27 '14 at 08:57
  • If a set of axioms leads to contradiction, we can say that one of those axioms is 'bad', but why did you single out a specific axiom? – labreuer Feb 27 '14 at 17:00
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    ∀x: C(x) ∧ ¬x ↔ G isn't an axiom. It's a premise which leads to a contradiction. Therefore it is false. – Michael Feb 27 '14 at 17:12
  • If the logical combination of two axioms produces a contradiction, at least one of those axioms is generally considered 'bad'. – labreuer Feb 27 '14 at 17:58
  • Right. But again, the premise that Pinocchio's nose grows if and only if he claims any falsehood is hardly an axiom. Although as with your comment regarding axioms, that it is shown to lead to a contradiction is that it is proven to be false, à la the Exception paradox. – Michael Feb 27 '14 at 19:49
  • Why is it "hardly an axiom"? – labreuer Feb 27 '14 at 20:42
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    "An axiom, or postulate, is a premise or starting point of reasoning. A self-evident principle or one that is accepted as true without proof as the basis for argument; a postulate. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy." Pinocchio's nose growing when he claims any falsehood doesn't quite meet the criteria of being self-evident. It's not an axiom. It's just a necessarily false premise. Recognizing it as false is the solution to the paradox it gives rise to. – Michael Feb 28 '14 at 07:28
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Bad question. Stated differently, you have constructed a paradoxical world. Check out self-reference. I will take your question to be equivalent to the liar paradox:

This sentence is not true.

Something tricky is going on here: recursion:

This sentence is not true.

Let me do a substitution, "This sentence" → "This sentence is not true.":

"This sentence is not true." is not true.

But wait, what is being pointed to by "This sentence"? The substitution did not bring any clarity! The problem here is that there is an infinite regress, caused by self-reference. This kind of paradox shows up elsewhere, like Russell's paradox, which attempts to construct the set R, with membership criterion: "all sets which do not contain themselves as a member".

  1. If R contains itself, it is not a member of R.
  2. If R does not contain itself, it is a member of R.

This might take a while to completely wrap your head around, but it is a deep result of what is now called naive set theory. In order to circumvent this problem, axiomatic set theories where developed which could not produce such paradoxes. However, they lose something: they are axiomatic, and thus not ambiguous like natural language. And yet, there seems to be something deep to this ambiguity. I won't go into that now, but it would make a good separate question.

There is a theory of computation aspect to self-reference, which shows up as Turing machines being able to print out their own description. This gets at the idea of self knowledge. And yet, Thomas Breuer's The Impossibility of Accurate State Self-Measurements questions this whole perfect self-knowledge enterprise. This Turing machine self-reference thing is very important; it shows up in the Halting Problem, which presents a huge obstacle to provability of Turing complete systems, which means we cannot guarantee properties we'd like to guarantee (like that your phone won't crash).

Douglas Hofstadter introduced the idea of strange loops in his Gödel, Escher Bach. The book is a layman's introduction to some neat theory of computation issues. I do not pretend to understand this 'strange loop' idea, but it definitely has to do with self-reference. It may be that consciousness itself has to do with self-reference; indeed, it is hard not to. So there's a lot to this liar paradox!

labreuer
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your question is the question of is the statement

this statement is false

true or false?

however there are more things to consider now that we are discussing Pinocchio none that i see include the world exploding but here are some facts to consider

  1. the ambiguity of my nose will grow as trylks pointed out
  2. the fact that this statement must have more context an "if... then my nose will grow" is obviously case-dependent each with a yes/no answer (unless it is "if i say my nose will grow then my nose will grow" - which is the proper paradoxical statement)
  3. does his nose grow if he is unaware that he is lying (and if no what about if he is unsure)
Math chiller
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