"My nose grows now!"

Last time, we talked about a few famous examples of paradoxes, and in particular how the American philosopher and logician W.V.O. Quine distinguished between three different classes of paradoxes - the veridical paradox, the falsidical paradox, and the antimony.  I covered veridical and falsidical paradoxes last time, so today I will focus on a famous antimony that involves a famous wooden puppet who wanted to become a real boy!

The Italian author Carlo Collodi wrote the children's book The Adventures of Pinocchio in 1883.  The book is of course different from the Walt Disney animated film, which was the studio's second animated feature film and one of the greatest animated films of all time.  The character Pinocchio is a mischevious wooden marionette, at least at first, whose dream is to become a real boy.  Pinocchio's nose is perhaps his best-known characteristic, which famously grows longer whenever he tells a lie. 

There is a famous paradox that involves Pinocchio, which was apparently first described by an eleven year-old named Veronique Eldridge-Smith in 2001 (Veronique's father, Peter, specialized in formal logic).  The father and son published what is now known as "Pinocchio's Paradox" in the journal Analysis in 2010.  

Here is the essence of the paradox: Pinocchio says, "My nose grows now."  As you can see, if Pinocchio's nose only grows longer when he tells a lie, the statement "My nose grows now" can be neither true nor false.  The statement cannot be true, because Pinocchio's nose only grows when he is lying.  So, if he is telling the truth, his nose can't grow.  However, the statement cannot be false either.  Follow?  Eldridge-Smith explains, "Pinocchio's nose is growing if and only if it is not growing."

"Pinocchio's Paradox" is a variation of the classic "Liar's Paradox" (also known as the antimony of the liar), in which a liar says, "I am lying."  If the liar is indeed lying, then the liar is telling the truth, which means the liar just lied.  More generally, if the statement "This sentence is false" is indeed true, then it is false, but the sentence states that it is false, and if it is false, then it must be true, and so on.

There are two additional variations of this paradox that I want to briefly introduce.  The first is the logic puzzle " Knights and Knaves", first described by Raymond Smullyan in his 1978 book, What is the Name of this Book?  Imagine a fictional island where everyone is either a "knight" (always tells the truth) or a "knave" (always tells a lie).  There are a number of variations to this puzzle, so here is just one example (an easy one):

Imagine you are a visitor to the island of Knights and Knaves.  There are two individuals standing in front of you, Red and Blue.  Blue says, "We are both Knaves."  Who is really the Knight and who is the Knave?

Similarly, "The Hardest Logic Puzzle Ever Designed" was described by George Boolos in The Harvard Review of Philosophy in 1996.  Boolos credits Raymond Smullyan with first introducing the puzzle, and it is certainly based upon " Knights and Knaves" described above. Here is the puzzle:

Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes–no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which.

Importantly, a single god may be asked more than one question, questions are permitted to depend on the answers to earlier questions, and the nature of Random's response should be thought of as depending on the flip of a fair coin hidden in his brain: if the coin comes down heads, he speaks truly; if tails, falsely.

You can definitely find the answers to both " Knights and Knaves" and "The Hardest Logic Puzzle Ever Designed" on the Internet.  However, I would challenge you to try to work both out on your own first!  It's probably a good idea to start with " Knights and Knaves" first.  Good luck!