# Department of Philosophy

### There Are Two Errors in the the Title of This Page

Many philosophers enjoy amusing puzzles and paradoxes when not doing more serious work; we hope you will too. The puzzles here have solutions: with careful analysis you can figure them out but the answer may not be what you first expect. The paradoxes provide an amusing way to raise deep philosophical problems; you may enjoy thinking about them.

#### Puzzles

On the Island of Knights and Knaves: On the island of Knights and Knaves, every inhabitant is either a knight or a knave. Knights always tell the truth. Knaves never tell the truth; any sentence uttered by a knave is false. A stranger came to the island and encountered three inhabitants, A, B, and C. He asked A, "Are you a knight, or a knave?" A mumbled an answer that the stranger could not understand. The stranger then asked B, "What did he say?" B replied, "A said that there is exactly one knight among us." Then C burst out, "Don't believe B, he is lying!" What are B and C?

One day I went to the island of knights and knaves and encountered an inhabitant who said, "Either I am a knave or else two plus two equals five." What should you conclude?
(Knights and Knaves puzzles by Raymond Smullyan)

The Surprise Test: One day the professor came into the class and announced, "Next week I will give you a surprise test. It will be a surprise, because you won't be able to figure out on which day it will occur until the class meets on the day of the test. It could happen on Monday, Tuesday, Wednesday, Thursday, or Friday, but I won't tell you which day."

The class were clever. They reasoned as follows: "She can't give the test on Friday, because then it wouldn't be a surprise; we'd know after class on Thursday that the test hadn't yet occurred, and hence we'd figure out that it would have to be on Friday. So we know the test can't be on Friday. But then it can't be on Thursday either, because if it were, we would know after class on Wednesday that it would have to be on Thursday, since it wouldn't have happened yet, and we have already shown that it can't be on Friday." Reasoning in this manner, the students concluded that the test could not occur on Wednesday either, nor on Tuesday, nor on Monday. Having concluded that a surprise test was impossible, the students didn't study. They were very disappointed and very surprised on Wednesday when they got a test. Where did the students' reasoning go wrong?

The Problem of the Light Switch: I have an ordinary light switch connected to a light. When the switch is closed, the light is on. When the switch is open, the light is off. At two minutes to noon, the light is on. At one minute to noon I flip the switch, turning the light off. At half a minute to noon I flip it again, turning the light on. At a quarter of a minute before noon I flip it again, turning the light off. I continue in this way, cutting the time between flippings of the switch in half each time. Now this will be an infinite series of flips. The switch flippings will occur closer and closer to noon, but will all be completed before noon. Will the switch be on or off at noon?

The Super Bullet: The Acme Arms Company has invented a Super Bullet: a Super Bullet penetrates anything it hits. But the Adamantine Armor company has invented a Super Strong Armor Plate: nothing that hits a Super Strong Armor Plate penetrates it. The army is planning to shoot a Super Bullet at a Super Strong Armor Plate. What will happen?

The monkey: Hanging over a pulley there is a rope with a weight at one end; at the other end clings a monkey of equal weight. The rope weighs 4 ounces per foot. The sum of the ages of the monkey and its mother is eight years, and the weight of the monkey is as many pounds as its mother is years old. The mother is twice as old as the monkey was when the mother was half as old as the monkey will be when the monkey is three times as old as its mother was when she was three times as old as the monkey was. The weight of the rope and weight is half again as much as the difference between the weight of the weight and the weight plus the weight of the monkey. How long is the rope?

The problem of the Two Sentences: Consider the first of the following sentences -- is it true or false?

1) The sentence numbered 2) is true.

2) The sentence numbered 1) is false.