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The Pinocchio paradox arises when Pinocchio says: "My nose grows"

From Wikipedia, Pinocchio is known for having a short nose that becomes longer when he is under stress, especially while telling a lie.

From the Wikipedia article on The Pinocchio Paradox.

Assume the sentence: "My nose grows now" is true:

  • Which means that Pinocchio's nose grows now because he truthfully
    says it is, but then
  • Pinocchio's nose does not grow now because according to the novel it grows only as Pinocchio lies, but then
  • Pinocchio's nose grows now because Pinocchio's nose does not grow
    now, and Pinocchio trustfully says it grows now, and it is false,
    that makes Pinocchio's sentence to be false, but then
  • Pinocchio's nose does not grow now because Pinocchio's nose grows
    now, and Pinocchio trustfully says it grows now, and it is true that
    makes Pinocchio's sentence to be true, but then
  • And so on ad infinitum.

Assume the sentence: "My nose grows now" is false:

  • Which means that Pinocchio's nose does not grow now because he
    falsely says it is, but then
  • Pinocchio's nose grows now because according to the novel it grows only as Pinocchio lies, but then
  • Pinocchio's nose does not grow now because Pinocchio's nose grows
    now, and Pinocchio falsely says it grows now, and it is false that
    makes Pinocchio's sentence to be true, but then
  • Pinocchio's nose grows now because Pinocchio's nose does not grow
    now, and Pinocchio falsely says it grows now, and it is true, that
    makes Pinocchio's sentence to be false, but then
  • And so on ad infinitum

Dr. William F. Vallicella, while admitting that he has not read the articles published in Analysis, says that he does not see a paradox in the future tense of the sentence "My nose will grow now", or in the present tense of the sentence "My nose grows now".

Dr. Vallicella argues that the future tense sentence cannot generate the Liar paradox because this sentence cannot be ever treated as a falsity. He explains his point with this example: "Suppose I predict that tomorrow morning, at 6 AM, my blood pressure will be 125/75, but my prediction turns out false: my blood pressure the next morning is 135/85. No one who heard my prediction could claim that I lied when I made it even if I had the intention of deceiving my hearers. For although I made (what turned out to be) a false statement with the intention to deceive, I had no way of knowing exactly what my blood pressure would be the next day." The same explanation could be used to explain Pinocchio's sentence. Even if his prediction that his nose will grow turns out to be false, it is impossible to claim that he has lied.

Then Dr. Vallicella explains why he does not see the Liar paradox in Pinocchio's sentence if the present tense is used

If Pinocchio says 'My nose grows now,' he is either lying or not. If he is lying, then he is making a false statement, which implies that his nose does not grow now. If he is not lying, then his statement is either true or false, which implies that either his nose does grow now or his nose does not grow now. Therefore, either his nose does not grow now or his nose does grow now. But that is wholly unproblematic.

Is Dr Vallicella's argument valid? Further, if he is correct then does it not mean that any statement we make about the future is logically invalid?

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Dr. Vallicella is correct in regards to statements regarding the future having an indeterminate truth value, if and only if perfect prediction is impossible (specifically, in a universe that is not bound by causality (determinism), fate, predestination, has no abrahamic God (He would know), etc). If the mechanism that grew pinnochio's nose was able to tap into one of these methods, then technically future statements would be verifiable.

In regards to your second question, even without perfect prediction this does not mean "any statement we make about the future is logically invalid". Logical validity has nothing to do with truth. At best you could say that statements regarding the future are of indeterminate truth value, but that doesn't mean the liklihood of two competing statements is 50/50. Statements can still be more or less likely to be true or not.

Consider the following claims:

  • In 5 minutes, someone will be looking at the moon.
  • In 5 minutes, the moon will explode.

Both these statements regard future events, and as such could be either true or false—we just don't know yet, so the verity of these statements are indeterminate. However, given all that we know about our world and the moon, it is highly likely that someone will be looking at the moon 5m from now somewhere on the planet, and it is highly unlikely that the moon will explode. Just because we don't know for certain whether a set of statements is true or false, that doesn't mean they all share the same probability of being true or false. Virtually all inductive scientific knowledge rests on this notion.

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To answer your last question first: most statements about the future are logically indeterminate. Trivially, what is the truth-status of the statement "In 2016, the Cubs will win the World Series"? Is it True? False? This has been a much-discussed question since the time of Aristotle; for example, here is an encyclopedia article on Medieval Theories of Future Contingents

Now, as for the Pinocchio Paradox: this largely comes down to how one imagines the operation of Pinocchio's nose with regard to time (i.e., what kind of a time-lag there is between the utterance of a lie/truth and the corresponding alteration in nasal length.) As such, the philosophical issues end up being subsidiary to the stipulations of the thought experiment, which makes it profoundly uninteresting for me, at least.

Put another way: are there any relevant issues the Pinocchio Paradox brings to light that we cannot derive from the age-old Liar Paradox?

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    +1: Thanks for putting into words, "stipulations of the thought experiment", for what has long bothered me about many philosophical arguments, the unspoken rules of the game. – Mitch Aug 25 '11 at 12:47

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