[Math] Knights & Knaves question 28 -seemingly another possibility

puzzle

I am not familiar with those puzzles and I would like to find out do I miss some rules which are necessary for solving those puzzles?

Here is an example – 28 (What is the name of this book? R.M.Smullyan)

In this problem, there are only two people, A and B, each of whom is either a knight or a knave. A makes the following statement: "At least one of us is a knave."
What are A and B?<

Let's suppose A's statement is true – then, of course, A is knight, B is knave.(This is the right answer in the book)

But let's suppose A's statement is false –

then 1) A is knave, as he is making false statement, as implying that one (B in that case) is knave but not saying anything about himself – so the answer would be A – knave, B- knight;

or 2) A's statement is still false, when saying that "at least one of us is a knave" when the truth is, BOTH of them are knaves?

So my question is, can knaves make part-truth/part-lies statements? Another confusing detail is this 'either' usage in the question – when it is said "'either' of whom", does that mean a total 4 possibilities or 2 :

  1. A & B both knaves
  2. A & B both knights (not in this puzzle)
  3. A -knave, B – knight
  4. A – knight, B -knave

Thank you.

Best Answer

The negation of "At least one of us is a knave" is "Neither of us is a knave" or equivalently "We are both knights" If A speaks falsely, he must be a knave, but then the falsity of the statement requires him to be a knight. This is the contradiction that the book answer relies on.

Related Question