# About Knights and Knaves and their consistency

On an Island of Knights and Knaves (where Knights always tell the truth and Knaves always lie), a Knight will never contradict himself unless some true fact changes. I was wondering: can a Knave contradict himself (regarding presently unchanging facts)? If he did wouldn't that mean he told the truth about 'something'?

• We're going to need to know more about the problem you're referring to. How is a 'knave' supposed to respond in this context?
– user20
Sep 28, 2014 at 6:44
• Can a Knave say some statement at say ,12 noon today and then some 'time' later say something that contradicts what he just said at 12 noon 'earlier' today? Sep 29, 2014 at 1:23
• This is only interesting if your knave is restricted to yes and no. Asked what is 2+2, he could say 3 one minute and 5 the next, which contradict while not requiring any truth telling. Oct 8, 2014 at 13:47

A knave can contradict himself easily.

- Is the sky blue?
- No.
- Did you just answer no?
- No.

For example, let's take a set of facts:
Fact A, which is FALSE
Fact B, which is TRUE

If you ask a knave "what is A?" he would answer TRUE. Then "what is B?" - he would answer FALSE.
Then ask what A AND B is. He would answer TRUE (FALSE AND TRUE = FALSE).
But combining his previous answers logician would conclude that A AND B = FALSE (TRUE AND FALSE = FALSE).

• If you asked a Knave for two statements that contradict each other how might he answer? Sep 29, 2014 at 1:27
• @user128932, I do not know. Why would I ask him about statements? Usually you ask knaves about facts, which is not defined (can be true or false) and therefore can not contradict each other. Sep 29, 2014 at 6:18
• I mean if you ask a Knave to make two statement ( about info. he is aware of) and the two statements have to contradict each other could he do this? Sep 30, 2014 at 3:12
• @user128932, it depends on model of Knaves. In standard model they do not follow your wishes, just answer questions. Sep 30, 2014 at 6:22

The 'sky is red' and the 'sky is green' are contradictory, but don't require the speaker to be truthful at any point.

I think I can do it in one: "Will you answer "yes" to this question?"

If the knave will not, he would say "no", but he has to lie and say "yes".

If he will, then he would say "yes", but has to lie and say "no".

The added benefit of this question is that it does not force knights into a contradictory loop (they will truthfully say "yes"/"no").

My friend suggested another interesting idea to answer this question.
Contradict himself means to say something illogical, independently of the perception of the reality. For example, "sky is not blue" would not be contradiction, may be our Knave is colorblind, or just crazy. But must leave here a criteria for what is logical and what is not anyway. So the logic provides us a facts, which are postulated to be true for all people.

Therefore one question is enough, just ask something like "Does A is true if and only if A is true?". Since the Knave should use the same logic as we do (otherwise it would not be a Knave in our definition), he will answer "No", and this way we would understand immediately that it is a Knave.

If we take general definition of the word: "contradiction is a situation in which inconsistent elements are present.". We can see that to contradict himself one need to say a set of statement, which can not be all true simultaneously (because of logic). Sometimes to create such a conditions one statement is enough.

• Do you mean by saying 'no' to this question the knave is contadicting himself? Oct 6, 2014 at 20:27
• @user128932, yes. Oct 6, 2014 at 20:54
• But wouldn't that imply he answered something similar to that question before in a different way? How could someone contradict themselves about 'something' if they only ever said one statement about that 'something'? Oct 8, 2014 at 6:17
• @user128932, in general sence "contradiction is a situation in which inconsistent elements are present.". To contradict himself one need to say a set of statement, which can not be all true simultaneously (because of logic). Sometimes to create such a conditions one statement is enough. Oct 8, 2014 at 6:27
• Does contradict 'mean' to say something ('dict' from a similar root to 'diction') that is 'against' ( or 'contra-') something? How can you say one statement that is against itself; isn't that just saying a paradoxical statement? If a set of statements that are not all true simutaneously is a set with just one statement then that one statement would be false ; is that right? Oct 8, 2014 at 6:39