Logical Inconsistency

  1. Imagine an island somewhere with two large families. One family (unlike us "normals") can tell only the truth, even when they'd rather lie. This Truthteller family can't ever make false statements, even by mistake. The other family, the False-Teller family, is just as reliable but in the opposite way: they can't make true statements ever.

    You are visiting the island and meet two of its people, Adam and Eve.

    Adam says, "Eve and I are from the same family."

    1. Can you say for sure which family Eve is from?
    2. Can you say for sure which family Adam is from?
Betsy: I'm pretty sure Eve is a Truthteller, but I don't know how to prove it.
Proving that something is false by assuming that it's true and showing how that's impossible is called proof by contradiction.
Gamal: Sometimes it's easier to prove that something is false than to prove that something is true. So let's assume the opposite of what you want to prove, and see where that leads us. So we're going to assume that Eve is a False-teller.
Betsy: Okay. Now suppose Adam is a Truthteller. If Eve is a False-teller, they're from different families, and what Adam said is false, but that can't be if he's a Truthteller.
Alphie: So, Adam has to be a False-teller.
Gamal: But that won't work either! If Adam and Eve are both False-tellers, they're in the same family, and Adam would have to say the opposite of what he said.
Betsy: So either way, the assumption that Eve is a False-teller led us to a contradiction. Eve can't be a False-teller, so she's a Truthteller, and we proved it.
  1. You meet someone named Derek and ask him if he's from the Truthteller family. What does he answer?
  2. What if you ask Derek if he's from the False-Teller family?
Betsy: The statement I'm making right now is false.
Gamal thinks a moment.
Gamal: Wait! What?
  1. Betsy claims her statement is false. What do you think? Make sure you can explain your thinking clearly.

Some statements are inconsistent, self-contradictory, and can be neither True nor False. Some, like Adam's statement, cannot be determined with the information we have so far, but could be either True or False.

  1. What questions can you ask in order to determine whether a person is a Truthteller or a False-Teller?
    1. Talking with others, find at least four questions that will work reliably.
    2. If Adam were a Truthteller, how would he answer your questions? Check to make sure that if he were a False-Teller, he'd answer differently.

Problem 5 asked you to devise a "test" to determine whether the person you've met is a Truthteller or a False-Teller. Some statements, as you will see on the next page, could be true or false but we can prove that there is no way to test it.

  1. On that island of Truthtellers and False-Tellers, you meet Max and Min. Max says "Min and I are both liars!" Which kind of statement is this? Is it self-contradictory? Is it indeterminate (it could be either, but there's no way to tell)? Or is it definitely resolvable?