Only superficially, I think. Pinocchio's claim is "my nose grows now." It is only as a linguistic convention that @Michael is using the future tense to discuss that claim from a perspective outside the event of its utterance. A bit infelicitous perhaps, but nothing that cannot be solved with a dash of charity.There is a contradiction here. — Sir2u
Premise 2. Pinocchio claims "my nose grows now" — Michael
"Now" isn't a reference to time duration but reference to a lying utterance (i.e., my nose grows in response to this sentence). — Soylent
The very condition that his nose grows if and only if he claims any falsehood is logically impossible. The paradox only arises when you insist on a contradiction being true, but that's not surprising. — Michael
It is only in the timeless world of logic that such feedback becomes a contradiction demanding both states at once. — unenlightened
It is declared a contradiction only because a paradox arises. — unenlightened
"Now" is a place in time though. — ProbablyTrue
I am not sure exactly what you mean here. If you say "now" is the cause of the growth then it is obviously false because he has not said anything before this to cause it to happen. So his nose will grow for lying. — Sir2u
But there isn't a world in which Pinocchio exists. — Michael
There is a fictional world of Pinocchio whereby his nose grows if and only if he claims any falsehood. — Soylent
In replying that it is impossible by virtue of a contradiction is to import metaphysical baggage into the Pinocchio world.
The Pinocchio paradox does not shed light on how to resolve the contradiction.
Pinocchio's nose will grow and not grow simultaneously.
If helpful, it can be likened by analogy to Schrödinger's cat.
Why do we need to resolve the contradiction? Do we need to resolve the contradictions "an X is a four-sided triangle" and "I am a married bachelor"? No. We just point out that they're contradictions and so necessarily false. The same with the rule "Pinocchio's nose grows if and only if he claims any falsehood". — Michael
It has nothing to do with metaphysics. It just points out that it is logically impossible for the rule G ↔ C(x) to hold for any x. — Michael
Other contradictions (e.g., square circle) are semantic contradictions. — Soylent
Does logically impossible entail ontological impossibility? How does logical impossibility impose itself on ontological possibility?
It's declared a contradiction because where x is G the given rule ∀x: G ↔ C(x) ∧ ¬x becomes G ↔ C(G) ∧ ¬G. — Michael
Why do we need to resolve the contradiction? — Michael
Indeed. But where is x G? First, x wasn't G and everything was fine, and then becomes G. — unenlightened
We need to resolve it because we want to use expressions of the form ∀x: G ↔ C(x) ∧ ¬x all the time.
Isn't G ↔ ¬G a semantic contradiction? — Michael
How can "X exist" be true if "X" is logically-impossible? — Michael
The rule can't be followed. There is an x, G, where ¬(G ↔ ¬x). — Michael
The Pinocchio paradox is, in a way, a counter–example to solutions to the Liar that would exclude semantic predicates from an object–language, because "is growing" is not a semantic predicate. — Eldridge-Smith
If it is a true contradiction that Pinocchio's nose grows and does not grow, then such a world is metaphysically impossible, not merely semantically impossible. — Eldridge-Smith
Having one’s nose grow is a facial, not a semantic feature. Moreover, although Pinocchio’s nose grows just when he is telling an untruth, the relationship is not semantic. It might be supposed to be causal or of some other nature, but it is not semantic. If Pinocchio’s nose is growing it is because he is saying something false; otherwise, it is not growing; and ‘because’ stands here for a non-semantic relation. The facts are that ‘is growing’ is not a synonym for ‘is not true’ and the Pinocchio story is intelligible without pretending that ‘is growing’ has a different meaning." — Peter Eldrige-Smith
Eldridge-Smith used the paradox to allegedly undermine dialetheism, but he doesn't address the issues I raised here (i.e., the justification for claiming logical impossibility entails ontological impossibility). — Soylent
I assume he's saying that it's ontologically impossible for someone's nose to both grow and not grow. — Michael
The Pinocchio paradox is, in a way, a counter–example to solutions to the Liar that would exclude semantic predicates from an object–language, because "is growing" is not a semantic predicate. — Eldridge-Smith
I don't think this is correct. That liar sentences (and variations on them) contain contradictions is a diagnosis. In that sense, it's true that such sentences are declared contradictions only because an apparent paradox arises: if they didn't present any sort of problem, there would be nothing to diagnose and resolve. But plenty of paradoxes are resolved without any declarations of contradiction, and plenty of sentences are declared contradictions even when no paradox arises. It's just that in this case, the problem is—according to this solution—that we have a sentence that is a contradiction without appearing as such. Only on analysis can we see what the sentence is really saying. So again, it is true that the diagnosis says the paradox only arises in this case because someone tries to understand a necessarily false sentence as true, but that's not circular. It's just one diagnosis: liar sentences are false because they are contradictions, but they give rise to an apparent paradox because some people are fooled by their syntax and don't realize that they are contradictions (and thus think that anyone who says they are false is also committed to saying they are true, which appears paradoxical).This won't do. It is declared a contradiction only because a paradox arises. When you then explain the paradox as arising because there is a contradiction taken to be true, you have rather gone in a circle. — unenlightened
This isn't a point that you will see defended very often because it is generally thought that logical possibility is—by definition—a stricter criterion than metaphysical possibility, which is a stricter criterion than physical possibility. Therefore, that something is physically impossible does not entail that it is metaphysically impossible, and that it is metaphysically impossible does not entail that it is logically impossible. But anything that is logically impossible is metaphysically impossible, and anything that is metaphysically impossible is physically impossible. One can take issue with this, of course, but many would argue that this is just how the terms are used.This is where the justification is missing. The leap from logical impossibility (e.g., that the Pinocchio paradox demonstrates a logical contradiction in causing someone's nose to both grow and not grow) to the ontological impossibility of a world with simultaneous contradictory truth-values. — Soylent
There's another angle that can be taken on this, that is of interest to mathematicians, although perhaps to nobody else. That is that Premise 2 is a statement about the velocity of the end of the nose. Velocity is the derivative of position with respect to time.Premise 2. Pinocchio claims "my nose grows now" — Michael
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