i myself failed this week as a perfect reasoner, but both succeeded and failed as a philosopher -- i made myself think twice, but not, i think, others. consider, if you will, the following:
you're a perfect reasoner. you've gathered with eight of your perfect reasoner friends for your weekly perfect reasoner poker rendezvous. it's the first week of a new year, and the buy-in is $1,000 from your $52,000 annual contribution fund. this week you're playing blind man's bluff. discussing any other player's card is strictly forbidden, and no one would do so. the first hand of the night is dealt. you scan clockwise around the room and see the 9 of hearts, the 6 of diamonds, the 8 of clubs, the jack of spades, the queen of hearts, the 3 of clubs, the ten of spades, and the 4 of diamonds.
before any betting has commenced, your completely trustworthy friend, c.t., enters the room. he hands each player a pencil and a clipboard with a piece of paper (each identical) on which is written several questions, and instructions to write in the correct answers to each (no player may see another's answers). please (correctly, of course) answer the questions:
1. are there at least two players with red cards?
2. how many of the players will have answered "yes" to question number 1?
3. how many of the players will have answered "9" to question number 2?
4. is the statement "there are at least two players with red cards." common knowledge?
all the players all write their answers and c.t. collects them and gives them a fresh piece of paper. c.t., who wanted to be in the poker ring, but couldn't afford it because he only had $45,000, then tells the players:
"if you know the color of your card, write that color on the piece of paper. otherwise write a question mark. if you wrote a color and you were correct, i will give you $5000, and you may go sit in the lounge until i'm done here. if you wrote a color and you were incorrect, you forfeit your $52,000 annual contribution to me, as well as your membership in the perfect reasoner poker ring for 2010." since all the players are perfect reasoners, they will not write a color unless they are absolutely certain.
5. do you write "red", "black", or "?"?
6. how many of the players will have answered "?" to question number 5?
c.t. collects all the papers, then reveals that all nine players wrote "?".
disappointed that he still has no seat at the table, but flush with cash, he passes out fresh sheets of paper and asks that the players answer once again. c.t. brought ten sheaves of nine sheets and intends to repeat this process until either he has a seat at the table, he has given away all $45,000 of his money, or he has run out of paper.
7. what will c.t. end up doing?
perfectly reasonable answers.