What is a paradox?
A paradox is a sentence that at first seems to be true, but actually has a major logic problem.
Types of paradox
1. Self-reference: a sentence that refers to itself, but the reference seems to make the sentence illogical
2. A vicious circle: a sentence, or collection of sentences, that go in a circle that will never end.
Using paradoxes
Paradoxes can be used in writing to make the reader think.
They do not need to have an answer (as in a half-paradox): the idea is purely to make the reader examine the words and look for an answer.
They are often used to make writing seem more interesting (and the reader to enjoy feeling clever spotting the paradox)
Examples
- Self-reference
| Paradox | Logic problem |
|---|---|
| This sentence is false. | If the sentence is false, then its meaning (that it is false) must be true. |
| A town’s only barber only shaves people who do not shave themselves. | Who shaves the barber? If the barber is shaved by someone else, then he is not the town’s only barber. If the barber shaves himself, then he does not only shave people who do not shave themselves. |
| I know that I know nothing. | To know that you know nothing means you do know something. |
| Spies do not look like spies. | A spy must look like a spy, because he/she is a spy. |
| This is madness, but it has method in it. | Madness has no method, so something with method cannot be madness. |
2. Vicious circles
| Paradox | Logic problem |
|---|---|
| The following sentence is true. The previous sentence is false. | 1. If the following sentence is true, it says this sentence must be false. 2. If this sentence is false, then the following sentence is false. 3. If the following sentence is false, then this sentence must be true. 4. If the following sentence is true, it says this sentence must be false. |
| Everything I say is a lie. | 1. If everything I say is a lie, this must be a lie. 2. If this is a lie, then not everything I say is a lie. 3. If not everything I say is a lie, then this is a lie. |
| What happens to Pinocchio’s nose when he says ‘My nose will grow now’? | 1. Pinocchio’s nose grows when he lies. If it does grow, then he was lying. 2. If he was lying, then ‘my nose will grow now’ was actually true. 3. If ‘my nose will grow now’ was true, then his nose should not have grown. 4. If his nose does not grow, then he was lying, so his nose will grow. |
| Pursuing happiness will make you miserable. Pursuing other things will give you happiness. If I want to be happy, what should I pursue? | 1. You cannot gain happiness by pursuing happiness. 2. So you pursue other things in order to be happy. 3. Pursuing other things to make you happy is pursuing happiness. 4. You cannot gain happiness by pursuing happiness. |
| Only crazy people would fly these missions. And these missions are no place for crazy people. If you know the missions are crazy, then you are not crazy and you should be flying these missions. – from Catch-22 by Joseph Heller | 1. They do not let crazy people fly the mission. 2. Sane people recognise that only a crazy person would fly the mission. 3. If you can see the mission is only for crazy people, you are not crazy. So you can fly the mission. 4. If you fly the mission, you are crazy. 5. They do not let crazy people fly the mission. |
Other paradoxes
The half-paradox is not a real paradox, but can look like one if people do not think very carefully.
| Half-paradox | Logical solution |
|---|---|
| A doctor and his son are in a car. The car crashes and the father dies. The boy is taken to hospital, where the doctor says ‘I can’t operate on this boy. This is my son.’ | The second doctor is the boy’s mother. |
| A 21-year old man is celebrating his 5th birthday today. | The man was born on 29 February. |
Thought experiments create interesting hypothetical situations that could create paradoxes.
| Thought experiment | Possible paradox |
|---|---|
| A man makes a time-travelling machine. He goes back in time and kills his grandfather before his mother and father are born. What happens to the time-traveller? – ‘The Grandfather Paradox’ | In theory, the killing of ones grandfather means one’s parents aren’t born, which means you aren’t born, which means you don’t kill your grandfather. However, as we do not know how time travel would affect timelines, this is just a thought experiment. |
| Which came first: the chicken or the egg? | A chicken needs to come from an egg. An egg needs to come from a chicken. In theory, one could neveer exist before the other. However, since we have chickens and eggs, one must have come first. And the theory of evolution suggests this is not a genuine problem. |
| You can never go from one place to another, because first you must go half way. After that you must go half way again. Then another half way. And so on. | A distance can always be halved. However, real life tells us that people can travel from one place to another, so this is again just a thought experiment. |