This section introduces the idea of a particular logical system (with given "rules"), and shows how difficulties can arise.
In a far-off country there are two sorts of town: in one, everyone always tells the truth; in the other, everyone always lies.
Suppose you were lost in a town in that country and wanted to know whether you were heading for the road out. If you didn't know whether it was a truth-telling town or a lying town, but could ask someone two questions, what would you do?
Charlotte: I'd ask something like "Is 1+1=2?", then I'd ask whether I was heading for the road out.
Good. Now suppose you could only ask one question.
Owen: That's not possible. You can't do it with one.
Actually, you can. Imagine you're a liar, and that I am heading out of town. How would you answer if I asked "What would you answer if I asked you whether I was heading out of town?"
Owen: You're right. I'd say "yes".
Of course, we're assuming the person you ask knows the correct answer, and that he must answer yes or no. What if we allow for the possibility that he may not know the answer, and that he may respond "I don't know"? Let's go back to the beginning, where we could ask two questions.
Charlotte: I'd start with the same question: "Is 1+1=2?".
Good. Then if he answered "no" or "I don't know", you'd know he was a liar. Now can you get more information than that with just one question?
Owen: Try the same as before: "What would you answer if I asked you whether I was heading out of town?".
The truth-teller would either put you on the right road or say he didn't know (if that was the case). A liar who knew that you were heading the right way would first have to consider how he would have responded if you had simply asked him: "Am I heading for the road out?" If you were heading out of town, his answer to this question would be either "no" or "I don't know". So his answer to the question Owen suggested could be any of "yes", "no" or "I don't know".
It's interesting to consider how the liar would answer if you asked a question he didn't know the correct answer to. Suppose you said to him "Is it true that the coin hidden under my hand is heads up?" How would he answer?
Owen: He can't say "yes" or "no" because either answer might be true.
Charlotte: And he can't say "I don't know" because that would definitely be true.
Exactly. He can't answer.
Allowing propositions to have truth values other than TRUE or FALSE (for example, MAYBE) is the realm of modal logic. A modern textbook on this subject is Boulos' The Logic of Provability. This book is not accessible to the general reader, but an idea of the field can be gained from its introduction, and its bibliography is a useful guide.