Monday, February 13, 2017

The Third Trial (Some Unusual Knights and Knaves Part III)

"Don't despair," said Craig to the chief of the island police, "we may find our man yet!"

Well, a third suspect was arrested and brought to trial. He brought with him his defense attorney, and the two made the following statements in court.

DEFENSE ATTORNEY: My client is indeed a knave, but he is not Arthur York.
DEFENDANT
: My attorney always tells the truth!
Is there enough evidence either to acquit or convict the defendant?
(Source: To Mock a Mockingbird and Other Logic Puzzles: Including an Amazing Adventure in Combinatory Logic by Raymond Smullyan)

If the defense attorney is a knight, then it is the case that the defendant is a knave. Then the defendant's claim that his attorney always tell the truth (i.e. is a knight) is false, which means that the defense attorney is actually a knave! There is a contradiction here and the defense attorney cannot be a knight.

When I initially tried to solved this, my reasoning was that the defense attorney could not be a knave either, and no verdict could be attained. This conclusion was based on the belief that, for the defense attorney to be a knave, both conjuncts of the statement "My client is indeed a knave, but he is not Arthur York" would have to be false. In other words, "My client is indeed a knave" and "He is not Arthur York" would both have to be false. This assumption would lead to another contradiction: "My client is indeed a knave" would mean that the defendant is actually a knight, which would in turn mean that the defense attorney is a knight.

But in Smullyan's knights and knaves puzzles, being a knave really has the weaker meaning that at least one of the conjuncts of an "and" statement—"but" is a form of "and"—is false, though not necessarily both. In this case, the first conjunct of his statement is true (i.e., the defendant is a knave) but the second must then be false. This means the defendant is in fact Arthur York.

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