A paradox is a statement that appears self-contradictory but may contain some deeper truth. So, if you ask Rick Astley for his copy of the movie Up, he cannot give it to you as he will never give you Up. Yet, in doing so, he lets you down. Thus creating the Astley Paradox.

Mankind loves a brain-twisting paradox, and we have been puzzling them out for aeons. From old-fashioned aphorisms such as “The more you know, the more you realize; you don’t know anything.” to the almost self-help quality of “The more you try to impress people, the less impressed they’ll be.”

Divided into three categories, paradoxes can be Veridical, Falsidical and Antinomy. Veridical paradoxes are logical fallacies that illustrate the limits of reasoning and logic, such as ‘The more things change, the more they remain the same.’ Falsidical paradoxes are questions that start with an appearance of validity, but upon closer inspection lead to an impossibility of answering, such as ‘Which came first, the chicken or the egg?’ Last of all is a contradiction between two statements that both seem right, like ‘You can’t lose what you never had.’ In the sphere of philosophy, this kind of paradox is called an antinomy.

The Ship of Theseus

When he wasn’t bringing the smackdown to Minotaurs, breaking the hearts of ancient Grecian women or committing filicide, Theseus had quite a case of wanderlust. Much is written of his exploits, but I would like to draw attention to one of his ships in particular.

Let’s say Theseus goes down and buys a brand-new ship from ‘Honest Ajax’ – he even pays extra for bucket seats. Theseus starts sailing around the Greek islands and having a good time. Over time, bit by bit, the ship has to dock for repairs, and since most of these parts have a bit of life left in them, he gives the old parts to Honest Ajax.

As Theseus replaces a timber here and a plank there, Ajax has enough to start to fabricate his own ship. After a while, every part gets replaced, so, given that not one single timber of the original remains, is it still the same ship? Moreover, what of the reconstituted ship made up of the discarded parts? Is that ship also the ship of Theseus?

The question has been debated ever since it was first put forward by Plutarch in the first century AD. In later years, the problem was considered by the likes of John Locke and Thomas Hobbes. Locke believes that the ship would remain the same as its form and function remain unchanged. It is still definitely a ship, owned by Theseus for sailing around and having adventures.

Hobbes argued that there would be a threshold where the ship would become a new object.

The debate rages on.

The Raven Paradox

How could going to the reptile house at your local zoo and feeding geckos help further the study of ornithology? Not by much, you might think, and you would be right – Well, rightish. In the 1940s, a German philosopher of science, Carl Hempel, put forward the following thought experiment to underline the question of what evidence you would need to prove a hypothesis.

Consider “All ravens are black” as a hypothesis that would mean that every raven you see that is black strengthens the theory. This is the same as saying, “If something is a raven, it is black”, so it follows then that “If something is not black, then it is not a raven.”

Taken to its logical extreme, you could say that all objects that are not black and not a raven count towards your hypothesis.

Going on cold, hard logic alone, this argument completely stacks up: if A = B and B = C, then A = C. Intuitively, however, it makes little sense.

Nelson Goodman pointed out that before we all declare ourselves ornithologists for being able to point at the contents of your living room, and declaring that your TV, armchair and coffee table, while all neither black nor ravens, proves therefore all ravens are black, we should consider that the same results could also be taken as proof that all ravens are, in fact, white. If you changed the wording on the statement.

 Karl Popper, a philosopher of science, believed the whole purpose of the scientific method was to show where a hypothesis was wrong, so in Popper’s mind, the existence of green lizards didn’t support the theory that all ravens are black, but neither did a black raven.

The Crocodile’s Dilemma

The crocodile dilemma is a logic problem and paradox that dates back to ancient Greece. Over the years, it has been attributed to many philosophers, but the first recorded reference I could find was made by the Greek philosopher Chrysippus around 280-207 BC in order to explain the difficulty of self-referential scenarios.

It runs thusly: Suppose that a crocodile has kidnapped a child. Distraught, the child’s parent begs with the reptile and the crocodile backs down a little. He tells the parent that he will let the child go if the parent can guess if the crocodile intended to let the child go anyway.

There are two ways this can go: The parent can say, “Yes, you will return my child,” which is what most people’s kneejerk reaction would be. Or they can opine that the crafty croc will keep the kid. If the parent calls the Croc’s bluff and says that they will return the child, and they are correct, no problem. The paradox comes into play in the second instance. If the parent guesses you will not return my child, the crocodile is honour-bound to return the child.

Because they deduced the croc’s intentions, and in returning the child, he breaks his own word and contradicts the parent’s answer. But wait, what if the croc intended to give the child back and the parent guessed wrong? Well, the croc would have to keep the child, thus making the parent right after all, and the child would, therefore, have to be given back, crocodiles being famed for their strict adherence to contract law.

This self-referential paradox is from the same school as Pinocchio saying, “And now my nose will grow,” or its slightly older cousin, “This sentence is false.” There is no definitive answer to this paradox.

The Unexpected Hanging

A judge finds a particularly nasty man guilty of a heinous crime. The criminal is sentenced to death. The judge announces his sentence and tells the prisoner he will be hung one workday in the upcoming week, sometime before noon. The judge wants to make the criminal squirm a little bit and tells him that he won’t let him know what day it will be. The first he will know is when there is a knock on their cell door.

Thinking about their sentence – and being the type of person who thinks they can ‘well actually…’ their way out of a death sentence – the prisoner reasons that this ‘surprise’ can’t happen on a Friday. Because if it gets to a minute past noon on Thursday, and they aren’t hung, then the only logical answer is that it must be tomorrow (Friday) and therefore, they would not be surprised. This means that per the judge’s stipulation that the hanging be a surprise, so – since there is only one day left – it can’t be a surprise if it happens on Friday.

Now that the criminal has reasoned that it won’t be happening on a Friday, they can eliminate Thursday as well. Because Friday has already been eliminated, if it hasn’t happened by Wednesday at noon, the hanging must happen on Thursday. This means that it would not be a surprise, and so on and so on. Feeling smug that they have found a loophole, when the executioner knocks on the door at noon on Wednesday – he was quite surprised.

Catch 22

You lock your keys in your car, and now you need the keys to unlock your car – which you need your keys to unlock. Or, you have completed a course and set out into the world to start a professional life, only all the jobs you apply for require experience, experience you can’t get without a job, a job you need experience for. These are examples of a ‘Catch 22’ situation.

 A Catch-22 is an inescapable situation that you cannot get away from because of contradictory logical rules. In his 1961 book, Joseph Heller describes the day-to-day life of John Yossarian, a bombardier making bombing runs in the late stages of World War 2. Because of his Commanding Officer’s pettiness and desire to get ‘a feather in his cap,’ the number of runs he has to make is ever-increasing. The only way out for him is to say that he is insane. The problem is, one of the rules (the eponymous catch-22) states that only a sane man will recognise his own insanity, so if he says he is crazy, he is sane enough to fly. Also, if he is too insane to fly, he won’t mention it so both insane and sane he must be in his B-25.

Catch-22 is absurd and confusing because war is absurd and confusing. So, if you feel like none of it makes any sense or that you are the only person who can see how ridiculous the whole thing is, then you are feeling exactly like Yossarian felt seeing all his friends die.
Sometimes, Catch-22 is used, incorrectly, to describe a situation where both choices are as bad as each other.

Although these are dilemmas, this doesn’t have the circular logic of a true catch-22.

The Potato Paradox

Now for something completely different, a paradox that has an answer and is, therefore, not a paradox (so a paradoxical paradox, if you will). Imagine that you have a sack of potatoes. Now, as these are a special kind that only exist in mathematics problems, they consist of exactly 99% water and 1% solid potato goodness. (Yes, I know it’s closer to 79% for a real spud, but this makes the math easier.) You leave your 100kg bag of spuds out in the sun, and 1% of the water evaporates. What is the new weight? If you said 99kg, then congratulations, like most people, you have given a logical – albeit completely wrong – answer.

The truth is that the new weight is 50kg.

This is because, in these spuds, there is a water-to-solid ratio of 99:1. The water evaporates, but the solid is not affected. So, the water content going down to 98% means the solid now accounts for 2% of the mass. The ratio has gone up to 98:2 or 49:1; the solid still weighs 1kg, but the water now accounts for 49kg.

Another way to look at it is that, to begin with, the solid matter accounted for 1% of the mass, but after evaporation, that same 1kg now accounts for 2% of its mass. So, if the total solid accounts for 2/100, then that’s the same as 1/50. if that 1kg solid is 1/50 of the total mass, the total mass must be 50kg.

Russell’s Paradox
Consider a small town with an overzealous mayor who is pogonophobic (scared of beards). Their phobia is so bad that they insist that every man be clean-shaven. On the one hand, this is good news for the town barber, who is taking in money hand over fist. However, one of the bylaws of the town states that a barber is “one who shaves all those, and those only, who do not shave themselves.”

The paradox arises when the barber comes to shave himself; he cannot shave himself as he only shaves those who do not shave themselves. So, if he shaves himself, he is not the barber laid out in the bylaws, but if he does not shave then he fits into the group of people who should be shaved by the barber, so as that barber he must shave himself… and so on.

There is no doubt that the polymath and Nobel laureate Bertrand Russell was a stone-cold genius. Yet, he is not the main player in the paradox that bears his name. The real hero of the story is Gottlob Frege. Frege argued that

“If numbers are properties of objects, then only one number should belong to any object, and it shouldn’t be influenced by matter of opinion.”

Frege put it that numbers did not apply to objects but to concepts; think of a deck of cards, is it one deck or is it 52 cards? Frege’s goal in all this was to prove that logic is the foundation of mathematics – an idea called logicism. To this end, Frege defined numbers using the idea of concepts and extensions; a concept is any idea you can think of, from transistors on a processor to clouds in the sky. An extension is a set of all things that fall under that concept.

Numbers, Frege argued, are extensions of concepts. So, the number 10 is an extension of the concept that “All things are made up of a collection of this many objects.” Pretty solid reasoning; however, Bertrand Russell spotted the fatal flaw by asking, “Consider a set of all sets that are not members of themselves, is that set a member of itself?” This leads us to the paradox: if it is a member of itself, it is not, and if it is not a member of itself, then it is.

Monty hall problem

“Let’s Make a Deal” is a popular American game show that has been broadcast nonstop since 1963. The show is synonymous with its most iconic host, Monty Hall, who was the face of the franchise for almost 30 years. The Monty Hall paradox is named after the final game in the show.
A contestant is confronted by 3 doors. Behind one of them is a desirable prize, say a car, and behind the other two are Zonks – the show’s colourful expression for an undesirable prize – like a goat. Monty has the contestant pick a door at random. Monty then (knowing which door the car was behind) opens one of the remaining two doors, always revealing one of the Zonks, and asks the player if they want to stick with their first choice or switch. You would assume that the odds of you picking the right door to begin with are 1 in 3 and, if you stick the odds are still 1 in 3 of you driving home in a new car (rather than trying to figure out how to ride a goat) do you stick, or do you switch?

The fact is that you should always switch.

The odds of picking the right door the first time are 1 in 3, which means you have a 2 in 3 chance of having to learn what a goat eats (anything, FYI). When Monty opens his door to show you a goat, the odds of the car being behind the last, unopened, door rather than going down to 1 in 3 stick at being 2 in 3. So, although it isn’t guaranteed, it is twice as likely that the car will be behind the door that you didn’t pick, rather than go down to a fifty/fifty shot.

If you’re having problems believing this solution, don’t worry; it’s completely unintuitive, and you’re in good company. In the 90s, a magazine in America published the solution to the problem and even PhD holders, such as Dr Paul Erdos – an emeritus professor of mathematics – got the wrong answer and only changed his mind after seeing a computer simulation of the problem in action.

Zeno’s paradoxes

Zeno of Elea was an enigmatic pre-Socratic Greek philosopher who lived around 495 – 430 BC. Little is known about his life, but his legacy in the field of logic and the very concepts of movement and infinity are profound and long-lasting. Zeno is best known for his four paradoxes, These were never meant to be taken literally, but more as thought experiments and logical exercises.

Dichotomy Paradox

This paradox states that any object in motion can never reach its destination. This is because first of all the object would have to get halfway there, that’s not a problem but then it would still need to make the other half of the journey which would necessitate it getting halfway again, and again and again to infinity, each time the half getting a smaller and smaller amount. In other words, he argued that there was an infinite amount of points between point A and Point B, so since it’s impossible for any object to move through an infinite amount of points, movement as we understand it is paradoxical.

Achilles and the Tortoise.

Zeno’s paradox of Achilles and the Tortoise suggests that it is impossible for the swiftest of runners, Achilles, to catch up with a slow-moving turtle, If the tortoise is given a head start, Achilles would forever remain behind since every time he arrived at the place where the tortoise was, it would have already inched its way ahead. No matter how fast Achilles runs, he always has to catch up with the tortoise one more time in order for him to succeed in passing it.

Arrow Paradox

This paradox states that an arrow in flight can not hit its target because it is never actually moving. At any given moment, in the smallest sliver of time conceivable, the arrow appears to be stationary. If it is stationary for this sliver of time, it must be for the next sliver of time and so on. The logical conclusion is that the arrow does not move at all. Don’t tell the person you have shot in the neck that one; they might take issue with it.

Stadium Paradox

Here, Zeno delved deeper into the Dichotomy Paradox, which suggests that a sprinter on a race track cannot reach the goal due to having to reach an endless number of halfway points. For instance, they would have to get to the precise midpoint before completing another midpoint, and that keeps reoccurring with no end in sight, thus making it a never-ending cycle that the runner cannot escape from and, therefore they cannot finish the race.

Galileo’s paradox

Galileo Galilei was a polymath, being known for his works in astronomy, physics and engineering as well as mathematics. Galileo is credited with the invention of the telescope, the thermoscope, and the pendulum clock. Often referred to as the father of the scientific method and a pioneer of observational astronomy. He was a controversial figure in his lifetime, put on trial for heresy for putting forth the theory that the Earth revolved around the sun, which ran against the church’s teaching. During his observation of the heavens, he discovered Saturn’s rings and four of the largest moons of Jupiter that now bear his name. He also had some interesting ideas about infinity and numbers.

In his book “Dialogues Concerning two New Sciences. “ published in 1632, he puts forth the following paradox: Consider two sets of numbers, in one, are all the natural numbers (1,2,3,4 and so on) and in the other set are all of the perfect Squares (Numbers that are the product of a number multiplied by itself, e.g. 1(1), 4(2), 9(3), 16(4)), you would assume that the set with all of the natural numbers would be larger because by their very nature there are more of them.

However, he proved that it was possible to create a one-to-one correspondence between them by pairing natural numbers together with their squares up to infinity. This means that there are the same number of natural numbers as there are perfect squares. The paradox had no solution, as such, but it inspired Georg Cantor’s theory of infinite sets. This late 19th-century concept allows us to comprehend and contrast infinite collections of objects. Cantor demonstrated that there are different ‘sizes’ of infinity, some of which are bigger than others – he proved the set of real numbers is larger than the set of natural numbers. This discovery was revolutionary and influenced a great deal of mathematics.

In Conclusion

Paradoxes twist our brains and challenge the boundaries of logic, language, and mathematics. They show us that reality isn’t always what it seems—and sometimes, asking the impossible question is the smartest move of all.