Interesting Paradoxes and Puzzles
What is a Paradox?
Let us begin by defining what a paradox is. By definition, a paradox is a statement or a sequence of statements which may seem true at first but tends to be self-contradictory. These statements may seem to have sound reasoning but have a logically unacceptable or unexpected conclusion. Paradoxes are used to promote critical thinking. They make us think twice to draw a conclusion based on the facts provided. Here are a few interesting paradoxes which are sure to impress you.
Problem of Evil:
This is a rather controversial paradox. The problem of evil questions the very existence of God. We believe that there is an omnibenevolent, omniscient, and omnipotent God who is guarding us always. So if that's the case:
1. God needs to be everywhere and so there must be no evil in this world as he is present everywhere. But the fact is that there is evil in this world. So such an omnipotent, omniscient, and omnibenevolent God cannot exist.
2. If God is all-powerful and merciful he will prevent all evil in this world. Since he is omniscient he should know the source of evil and how it comes to existence and be able to stop it. An Omnipotent God should have the power to stop evil from existing in the first place. So if there is an omnipotent, omniscient, and omnibenevolent God then there will be no evil. But evil exists.
3. If God lets suffering in this world happen for a "greater good" then he is no longer all-powerful and wholly good.
Let us start with something simple, at least relatively simple. The barber is the "person who shaves all those, and those only, who do not shave themselves". At first glance, this statement seems simple and true. Now the question here is, does the barber shave himself? This gives rise to a contradiction. The barber cannot shave himself as he will only shave those who do not shave by themselves. If he ends up shaving himself then he ceases to be a barber as per the definition. If he does not shave himself then he is part of the group who do not shave by themselves and need to be shaved by the barber who is in fact himself! Hence we end up with a contradiction.
Hitler's Murder Paradox:
This is a variation of the grandfather paradox associated with time travel. Let us consider a scenario where someone travels back in time before world war 2 occurs and murders Hitler. Thus he would have averted a great war and the Holocaust from ever happening. However, if Hitler had indeed been murdered before the start of World War 2 then there is no reason for the time travel to happen. Moreover, if World War 2 and the Holocaust had been averted then there will be a huge change in the current universe including the birth of the person instigating the time travel. Although some people consider that a parallel universe will be created in this scenario, this will not be considered as a true time travel as their universe will remain unchanged.
The Liar Paradox:
The liar paradox is essentially a statement given by a liar stating that he is lying by declaring that, "I am lying". The most classic example of this paradox was provided by a Cretan philosopher named Epimenides. He made a statement that "All Cretans are liars". He himself was a Cretan so no matter if he had told the truth or not the statement ends in a contradiction.
Another version of this paradox is the Crocodile dilemma. A crocodile has abducted a child and promises the child's mother that her child will be returned if she guesses what he will do next. If the mother guesses that the child will be returned, then the outcome will be unpredictable but straightforward based on the crocodile's choice and there won't be any contradiction. However, if the mother says that the child will not be returned when the crocodile is faced with a dilemma. If he had thought of releasing the child, then by his condition he will have to keep the child as the guess was incorrect but contradicting his original decision to release the child. If he had decided on keeping the child, then the mother's guess would have been correct and he will now have to release the child but now he will be contradicting his initial decision to keep the child.
Those who have watched the TV series "The Big Bang Theory" will be aware of this paradox. This was a thought experiment provided by Erwin Schrödinger in 1935. Let me put it in simple terms: A cat is kept inside a closed cage. Inside this container is a flask containing a poisonous substance. The poison will be released from the flask based on a fuse present in the flask but the duration of the fuse is unknown. Since we do not know when the poison will be released, the state of the cat inside can be considered as simultaneously both alive and dead. This state is called as Quantum superposition in quantum mechanics. We will not be able to determine the state of the cat unless we open the cage after which it superimposes to a single state.
Ship of Theseus:
Let us consider a wooden ship which is made up of 1000 planks of wood. Due to repairs some of the existing planks of wood are replaced by new planks. Over the course of time, all the old 1000 planks of wood have been replaced by new pieces of wood. Now the question arises on whether this ship would be considered as the original ship that was constructed or a different one as it does not have any of the components which were initially used to build it. A variation of this is the grandfather's ax paradox. An ax consists of a handle and the ax head. Over the course of time if we replace the head as well as the handle of the ax, will the ax be considered as the original or a new one?
Achilles and the Tortoise Paradox:
This is one of the paradoxes created by Zeno of Elea. A tortoise challenges Achilles to a foot race. Achilles knows that the tortoise is very slow and in order to keep things fair he gives the tortoise a head start of 100 meters. Considering that the distance for the race is significant we can confidently say that Achilles will win. Or can we? The race begins and the tortoise is given a head start of 100 meters as promised. Then by the time Achilles had reached the 100-meter mark, the tortoise had moved an additional 10 meters. By the time he reaches the 110-meter mark, the tortoise had traveled a total of 111 meters. This goes on and on infinitely and in each iteration, the tortoise is at least a minuscule distance ahead of Achilles and thus he can never overtake the tortoise.
A different version of this paradox is the ant on a rubber rope paradox. Consider an ant crawling along the length of a rubber string 1km long at the speed of 1cm per second. However, the string stretches by 1km per second continuously. So by the time the ant has traveled 1cm, the length of the string will be 2kms and after traveling 2cms it will be 3kms in length and so on. So will the ant ever be able to reach the other end of the string? Well, it turns out that it can, although it will take quite a bit of time to do it! The string is stretching at the start as well as the end and if the ant is making continuous progress at the same speed it will eventually be able to make continued progress by conserving the portion of the rope it has already crossed.
Monty Hall Problem:
This is based on the American television game show "Let's Make a Deal" which was hosted by Monty Hall. In the game show, the contestant is asked to choose one of 3 doors- A, B or C. One of these doors has a brand new car as the prize whereas the other two doors are empty. You are allowed to pick one door at the start, let us consider that you choose door B. Then the game host opens either door A or C and reveals its contents, consider that door C is opened. Note that the door which is opened will always be empty. Now you are faced with an option, either keep your option the same(B) or you can switch to door A (the unopened door of the two). So which do you think will be the better option? Or will it make no difference?
Surprisingly switching to the other door will increase your chances of getting the prize. When you initially chose one of the three doors (B), you have a 1/3 chance that you have chosen the door with the car. So this means that there is a 2/3 chance that the car might be in the other two doors (A or C). The game show host will ultimately eliminate the door which has nothing (C) and so when he eliminates one of the two doors the 2/3 probability will be present in opening the other door which you have not chosen (A). Normally we would think that we always have a 50% chance to win but in reality, if we switched we have a 66.67% chance to win and if you do not switch the probability is actually 33.34%. This is a paradox of the 'veridical' type wherein the results seem absurd initially but are nevertheless true.
Video explaining the Monty Hall problem
Paradox of Hedonism:
The paradox of hedonism or pleasure paradox explains the various difficulties faced during the search for pleasure. The paradox says that if we are constantly seeking pleasure then this will not yield the most happiness for a person since we are consciously pursuing it. Let us consider an example in which a person collects stamps. If you ask him whether he collects stamps for the pleasure he will most likely say no. He collects stamps because he likes it and not for collecting pleasure. But indirectly he finds pleasure in it. If we had heard that collecting stamps was pleasurable and so we start collecting stamps to get that pleasure, it is inevitable that we will not get the desired effect. The following quote by Politician William Bennett perfectly explains this:
Happiness is like a cat, If you try to coax it or call it, it will avoid you; it will never come. But if you pay no attention to it and go about your business, you'll find it rubbing against your legs and jumping into your lap.
Unexpected Hanging Paradox:
The unexpected hanging or hangman paradox is a paradox about a person's expectation of a future event when they know that it will happen at an unexpected time. A judge informs a prisoner on trial that he will be hanged at noon on a weekday the following week but that the day of hanging will be a surprise to the prisoner.
The prisoner is devastated at first but after reasoning the judge's statement he draws on the conclusion that he will escape from the hanging. His reasoning can be explained as follows. Since this is a surprise hanging, it cannot be Friday as if he has not been hanged till Thursday it will not be a surprise to him on the hanging day as Friday is the last day of the week. He then concludes that the hanging cannot happen on Thursday. Since Friday has already been eliminated, if he has not been hanged till Wednesday, then hanging him on Thursday will also not be a surprise (having eliminated Friday). Following the same logic, he confidently concludes that he will not be hanged on any day as it cannot be a surprise no matter what day it is. The following Wednesday the prisoner was hanged. And yes he was utterly surprised. This paradox can be applied to a prisoner's hanging or even a surprise school test. Even though this seems simple at first glance this is a complex problem for philosophy.
Which Paradox did you like the most?
© 2017 Sudhir