On a certain island, each inhabitant is a truth-teller or a liar, and not both.

Question

On a certain island, each inhabitant is a truth-teller or a liar, and not both. A truth-teller always tells the truth, and a liar always lies. Arnie and Barnie both live on the island. a. Suppose Arnie says, "If I am a truth-teller, then each person living on the island is a truth-teller or a liar." Can you say whether Arnie is a truth-teller or a liar? If so, which one is he? b. Suppose Arnie says, "If I am a truth-teller, than so is Barnie." Can you say what Arnie and Barnie are? If so, what are they?

Explanation / Answer

a. Let us assume Arnie is a truth teller.Let us call this statement as p.

Also it is given that each inhabitant on the island is a truth teller or a liar. Let us call this q.This is given to be true.

Then Arnie's statement becomes p -> q. But since this is Arnie's statement, it is equivalent to p.

So p -> q = p

We know that true -> true is true. But false -> true is true.

So Arnie's statement p is true implying he is a truth teller.

b. Let p hold the same meaning as earlier. Let q be the statement Barnie is a truth teller.

q can be true or false.

Then Arnie's statement becomes p -> q. But since this is Arnie's statement, it is equivalent to p.

So p -> q = p

Now if q is true, then p is true as per earlier argument.

If q is false, we have

true -> false = false and false ->false = true

Both are contradictions.

So p and q are true. Hence Arnie and Barnie are truth tellers.

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