## Outline Mathematics

Logic Problems

# Knights and Knaves 3

Here's a problem to tackle:

On an island, the populace is of two kinds: knights and knaves. Knights always tell the truth, knaves always lie.

An islander - call him A - made a statement: "Either I am a knave or 2 + 2 = 5."

What can we conclude?

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Copyright © 1996-2018 Alexander Bogomolny

### Solution

On an island, the populace is of two kinds: knights and knaves. Knights always tell the truth, knaves always lie.

An islander - call him A - made a statement: "Either I am a knave or 2 + 2 = 5."

What can we conclude?

Recollect that disjunction (this or that) is false if and only if both this and that are false,both this and that are false,at least one of this or that is false. It is true otherwise, in particular if either this or that is true,false,true. If one of the components, say, 'that' is false, then the disjunction is false or true depending on whether the other component is false or true. Thus the statement made is equivalent to him claiming to be a knave,knave,knight. But none of the islanders could possibly make such a statment. Indeed, if a knave said 'I am a knave', he would have made a correct,wrong,correct statement, which is impossible,rare,impossible,morally correct. On the other hand, a knight can't truthfully,truthfully,deceitfully claim to be a knave,knave,knight.

We are forced to conclude that the problem was ill posed; it's entirely self-contradictory.

### References

- R. Smullyan,
*What Is the Name of This Book?*, Simon & Schuster, 1978

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