Liar Game

While this is the name of a certain story, it is also the name I am giving to the game I am about to propose. The rules are that I propose a certain number of statements, and the player has to categorize them as true or false. The player wins if all of them are correctly categorized, and fails if he does not categorize them all, or if he mistakenly categorizes a true statement as false, or a false statement as true. It is against the rules for him to place a statement in both categories.

The statements I propose are the following:

  1. 2+2=4.
  2. 2+2=5.
  3. The player will categorize this as false.

It can easily be seen that the player is guaranteed to lose the game. If the player does not categorize the third statement, then it is false, and he has failed to categorize them all. On the other hand, if he categorizes it as true, it is false, and if he categorizes it as false, it is true. In any case either he fails to categorize it, or he categorizes it incorrectly.

It is evident that this is related to the paradox of the Liar, but there is a significant difference. The original liar statement is paradoxical, in the sense that applying the ordinary rules of logic results in a contradiction regardless of whether one considers the statement to be true or false.

This is not the case here. There is nothing paradoxical about the statement, in this sense. Given an actual player and an actual instance of playing the game, the statement will plainly be true or false in an objective sense, and without any contradiction being implied. It is just that the player cannot possibly categorize it correctly, since its truth is correlated with the player categorizing it as false.



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