Pinocchio’s Paradox

By Tim Ballingall

Today marks the 71^st anniversary of the premier of this Disney-animated classic, Pinocchio.

We’re all familiar with the tale—Man makes Puppet, Puppet comes alive, Puppet befriends Cat and Fox, Puppet and Boys drink and smoke cigars and transform into donkeys to work in a salt mine, Puppet gets swallowed by Whale, Puppet makes Whale sneeze, Blue Fairy turns Puppet into Real Boy. Oh, the classics.

But latently entwined in the famous scenario—Pinocchio’s nose grows when he tells a lie—is a logician’s nightmare:

Pinocchio’s nose grows if and only if he tells a lie. A lie is an untrue statement made by a speaker who knows it to be so yet speaks it with the intention to deceive others. If Pinocchio knows that his nose will not grow and says, “My nose will grow now,” his nose will, in fact grow. But if this happens, it turns out his statement was true.

Conversely, if he knows his nose will grow and says so, then either his nose will grow—which contradicts its nature of only growing if and only if Pinocchio lies—or nothing will happen, in which case, as it turns out, Pinocchio did lie, in which case his nose will grow. But doesn’t that mean he told the truth?

This paradox is a variation of the Liar Paradox: “This statement is false.” Here’s a two-sentence example of this: “The next sentence is true. The previous sentence is false.” Here’s one I came up with: “This sentence is not self-referential.”

Isn’t ignoring the hierarchy of meta-language and object language fun?

Feel free to leave a totally non-non-paradoxical comment.

This is not a sentence recommending you search “Paraconsistent Logic” in the Gale Virtual Reference Library in Articles and Databases. This sentence is not recommending you search “Liar Paradox” in EBSCOhost. This is not a sentence recommending you search “Pinocchio” in Project Muse. And if this is not the last sentence of this blog post, then Pinocchio was based on a true events.