Steven G
Elite Member
- Joined
- Dec 30, 2014
- Messages
- 14,394
I offer you the opportunity to play a game with the following perfectly true introduction.
"I have 3 envelopes here. Two of them contain a slip of paper that says 'You win!'. One contains a slip that says 'You lose!'. You choose an envelope. If the slip in that envelope says 'You win!', I pay you $1. Otherwise, you pay me $1. Since there are 2 winning envelopes and only one losing envelope, the odds are clearly in your favor. Do you want to play?"
Most folks, knowing that something interesting must be up, will take me up on the game.
When I hand over the envelopes, it may be seen that there are 3 distinct statements, one written on each envelope:
1. At most one of the statements on these envelopes is true.
2. This envelope contains a "You win!" slip.
3. This envelope contains the "You lose!" slip.
The expected line of reasoning goes as follows: If the statement on envelope 1 is true, then the others are false and envelope 2 contains the “You lose!” slip. If the statement on envelope 1 is false, then the others are true and envelope 3 contains the “You lose!” slip. In neither case does envelope 1 contain the “You lose!” slip. Therefore, choose envelope 1.
As you might now expect, I had put the “You lose!” slip in envelope 1. What’s going on here is an apparent violation of theLaw of the Excluded Middlewhich most of us are inclined to accept as a matter of course. But here we have clearly demonstrated that the statement on envelope 1 is neither true nor false! Think of it as a proof by contradiction: Assuming it is either true or false, we arrive at a contradiction.
I get everything up to the last paragraph. That is, I agree that one should choose envelope 1 (and expect a you win slip inside it).
As you might now expect, I had put the “You lose!” slip in envelope 1. This means to me that none of the statements need to be true and that you should not choose envelope 1. I thought that one statement could be true. Does the writing on the envelopes mean anything? That is, does at most one have to be true?
"I have 3 envelopes here. Two of them contain a slip of paper that says 'You win!'. One contains a slip that says 'You lose!'. You choose an envelope. If the slip in that envelope says 'You win!', I pay you $1. Otherwise, you pay me $1. Since there are 2 winning envelopes and only one losing envelope, the odds are clearly in your favor. Do you want to play?"
Most folks, knowing that something interesting must be up, will take me up on the game.
When I hand over the envelopes, it may be seen that there are 3 distinct statements, one written on each envelope:
1. At most one of the statements on these envelopes is true.
2. This envelope contains a "You win!" slip.
3. This envelope contains the "You lose!" slip.
The expected line of reasoning goes as follows: If the statement on envelope 1 is true, then the others are false and envelope 2 contains the “You lose!” slip. If the statement on envelope 1 is false, then the others are true and envelope 3 contains the “You lose!” slip. In neither case does envelope 1 contain the “You lose!” slip. Therefore, choose envelope 1.
As you might now expect, I had put the “You lose!” slip in envelope 1. What’s going on here is an apparent violation of theLaw of the Excluded Middlewhich most of us are inclined to accept as a matter of course. But here we have clearly demonstrated that the statement on envelope 1 is neither true nor false! Think of it as a proof by contradiction: Assuming it is either true or false, we arrive at a contradiction.
I get everything up to the last paragraph. That is, I agree that one should choose envelope 1 (and expect a you win slip inside it).
As you might now expect, I had put the “You lose!” slip in envelope 1. This means to me that none of the statements need to be true and that you should not choose envelope 1. I thought that one statement could be true. Does the writing on the envelopes mean anything? That is, does at most one have to be true?
