Question

7. (10 pts.) Amelia visits several islands. The individuals on each these islands are either truth-tellers or liars. Amelia is interested in finding out whether everyone on a given island is a truth-teller, or whether everyone is liar, or whether there are both truth-tellers and liars. (Note: it only takes one liar for an island’s inhabitants to include both truth-tellers and liars.) Amelia is also interested in finding someone from whom to bum a cigarette. (Yes, she shouldn’t smoke, but she does anyway and intends to stop soon.) In each of the following questions, use complete English sentences to present the reasoning that backs up your answer. By the way, in this problem, and indeed every problem, you should remember that any universally generalized conditional whose antecedent-predicate is true of nothing is true. Amelia has already visited two of these islands. Now she visits two more.

a. On the third island, Amelia asks whether anyone on the island smokes. Everyone answers: “If I smoke, then everyone smokes.” What can Amelia conclude about the third island? b. On the fourth island, in response to the same question, everyone replies, “Some of us smoke, but I do not.” What can Amelia conclude about the fourth island?

b. On the fourth island, in response to the same question, everyone replies, “Some of us smoke, but I do not.” What can Amelia conclude about the fourth island?

Answer 1

Case: Third island

There are 3 possible cases:

All truth tellers

If everyone smokes, this means that all of them are truth tellers since their statement is verified to be the truth

If even one person doesn't it means that the assumption that all are truth tellers are contradicted.

Hence, there is a possibility of all of them being truth tellers and all of them being smokers.

All liars

If everyone smokes, it means all are truth tellers which means that the assumption of all being liars is wrong

If some of them smoke and some don't, the peoplewho smoke are liars and the people who don't smoke tell the truth which means that the assumption of all of them being liars is wrong.

Hence, the situation of all liars is not possible

Some truth tellers, some liars

As can be seen from above if some of them smoke and some don't, the population consists of a mixture of truth tellers and liars

Hence, in island 3 either all of them are truth tellers aor it is a mix of truth tellers and liars. We need more information to determine the exact composition.

Case: Fourth Island

There are 3 possible cases:

All truth tellers

If all are truth tellers, this means that everyone tells the truth that he/she does not smoke. This means that no one on the island smokes. This contradicts the first part of the statement and hence this case is not possible

All liars

If all are liars this means everyone lies about not smoking, which means everyone smokes. This means that the first part of their statement is also a lie. Hence, everyone in this island is a liar.

Some truth tellers, some liars

The truth tellers do not smoke

The liars smoke

But then the first part of their statement is true which contradicts the fact that they are liars. Hence this is not possible

Hence in island 4, everyone is a liar.