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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.

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    "at least one knave" means "one knave or both knaves". It is true unless neither is a knave – 2017-01-03
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    Generally in these questions you are to assume that the knights speak only truths and the knaves speak only lies. Also, here, as elsewhere, "at least one" certainly includes "more than one". – 2017-01-03
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    If you assume A is a knave, then "At least one of us is a knave is a true statement." Which a knave would never say, invalidating the assumption. – 2017-01-03
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    If I give you two dollars, and then I say later "I gave Aili at least one dollar", you may not complain that I have told a lie, or even a partial lie. I told the truth. – 2017-01-03

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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.

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    but if there would be three persons, what should I understand by 'at least one of us', then? – 2017-01-03
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    Just like it says, at least one, so it could be one, two, or three. If there were three and A says at least one of us is a knave we would know that A is a knight and at least one of the others is a knave, but it could be both. – 2017-01-03
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    Thank you! So there are no half-truths...Would you kindly answer this 'either' usage, too; if it is either of two, would it be 4 or 2 options? – 2017-01-03
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    I would read "either of two" as exactly one, so there are only two options. If I were writing puzzles like this I would avoid the phrase as it might be read to include both, but certainly would not include neither. – 2017-01-03