Philosophy 100 Week #2, Logic Craig Fox

KNIGHTS AND KNAVES

“There is a wide variety of puzzles about an island in which certain inhabitants called ‘knights’ always tell the truth, and others called ‘knaves’ always lie. It is assumed that every inhabitant of the island is either a knight or a knave.”

Problems: (Working with another person, explain your answers to the following.)

1. According to this old problem, three of the inhabitants—A, B, and C—were standing together in a garden. A stranger passed by and asked A, “Are you a knight or a knave?” A answered, but rather indistinctly, so the stranger could not make out what he said. The stranger then asked B, “What did A say?” B replied, “A said that he is a knave.” At this point the third man, C, said, “Don’t believe B; he is lying!”

The question is, what are B and C?

2. Suppose that there are two people, A and B. A says, “I am a knave, but B isn’t.”

What are they?

3. Suppose there are two people, A and B. Suppose A says, “Either I am a knave or B is a knight.”

What are A and B?

4. Suppose A says, “Either I am a knave, or else two plus two equals five.”

What would you conclude?

Hints: [(i) will certainly be of help; (ii) may or may not, depending on your approach.]

(i) Use your knowledge of truth-tables for the logical connectives.

(ii) The following principle of reasoning is one that we use quite often, though you may have never explicitly thought about it. It’s called reductio ad absurdum. Loosely stated, it says that if I make some assumption and then show that something impossible would have to follow from that assumption, then I’m entitled to say that the proposed assumption cannot be true. In other words: if I assume p and show that something impossible follows, then I can conclude that ~p. (Often, the “something impossible” is a contradiction: something and its negation.)