6 Amazing Paradoxes That'll Leave You Scratching Your Head

October 15, 2018

|Oliver G. Alvar
paradox examples

These problems have captivated the minds of the world’s smartest people for a long time. See if you can give an answer.

Broadly speaking, a paradox is a situation or statement that leads to contradiction (even if at first it seemed true). We'll constantly find the world to be full of impenetrable puzzles that keep brilliant minds forever working on how to solve them, if we care enough to look. From linguistic idiosyncrasies to mathematical mysteries, paradoxes help us advance knowledge by checking and correcting for mistakes of all kinds in our way of thinking. But some paradoxes are so embedded in the very fabric of logic that they've remained unsolved for thousands of years —and we’ll probably never find a satisfying answer. But not all is lost: unresolved paradoxes can point to important features of language that allow us to understand its inner workings much better. 

Here are some some examples of paradoxes that will put your intelligence to the test. Bear in mind there’s no easy solution for any of them, in spite of what you might initially think. Still, challenge yourself and take a shot at solving them!

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1) Barber Paradox

Imagine you enter a small town one day while traveling. As you walk around, you notice everyone is perfectly shaved —almost obsessively so. When you come across the only barber in town, you ask him about it. He tells you that, by law, everyone living there must be cleanly shaven. “I’m required to shave all the residents who do not shave themselves, and only the residents who do not shave themselves” he says. “If a resident shaves themselves, I cannot shave them. But if anyone doesn’t shave, then I must shave them. It’s the town’s law.” What a strange notion, you think. Accordingly, clever as you are, you ask “you’re a resident too, right?,” to which the barber replies that he is.

Should the barber shave himself?

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Why it’s a paradox

Per the town’s law, the barber cannot shave himself, as he only shaves residents who do not shave themselves, but if he doesn’t actually shave himself, then he is one of the residents who do not shave themselves —so he’d be required to shave. If he shaves, he shouldn’t. If he doesn’t, he should. So, the law is paradoxical. There’s no way the barber can follow it without breaking it.

This paradox is important insofar as it represents (or is derived from) a logical conundrum called Russell's Paradox (which, briefly put, asks whether the set of all sets which do not contain themselves contains itself). The discovery of this paradox changed our way of thinking about several of the most important branches of mathematics, which in turn led to baffling scientific advances over the course of the past century.

2) The Liar Paradox

This sentence is false. And so is this sentence. An odd thing to say, for sure; yet this is the core of what has been known for centuries as the paradox of the liar. To illustrate the problem, imagine Pinocchio is real: his nose actually grows when he lies. What do you think would happen if he said "my nose grows now"?

Let's put it another way. Imagine someone came up to you and said only “I’m telling you a lie just now.” You would surely think that person is strange, but look beyond that fact. Were they telling something true or something false (or neither)?

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Why it's a paradox

If this sentence is false, then it's telling the truth. But if it's true, then it's false. The same goes for Pinocchio's nose: if he says "my nose grows now," it should grow only if it doesn’t grow, and it shouldn’t grow only if it grows. One last version. Ed claims that what John says is true. John says that what Ed claims is false. If Ed’s right, then he’s wrong. And if John’s wrong, then he (John) must be right. So, you see... That's a paradox!

This puzzle is important because it creates several problems for logical systems of truth. A good answer eluded thinkers for millennia, until it was (somewhat) addressed during the 20th century. It's still problematic for the typical way we think about truth.

3) Surprise Test Paradox

Imagine that a teacher announces there will be a surprise test the following week. He says that while students know a test is coming, the specific day will be the surprise. A clever student strongly objects to the announcement. There’s no way, she claims, that the teacher can fulfill the promise, since there’s no way that the test could be a surprise now. 

The class meets every Monday, Wednesday, and Friday. The student reasons that Friday is not an option. If the test were to fall on that day, come Tuesday the class would know about it (since it didn’t happen neither on Monday nor Wednesday); so they wouldn’t be surprised. What about Wednesday, then? Same reasoning applies: now that Friday is off the table, if the test doesn’t happen on Monday, then the students would know it’s coming on Wednesday. So, Wednesday is also off the table. By process of elimination, the students would now know that the test is on Monday, so it wouldn’t be a surprise either. 

Can the teacher actually fulfill his promise?

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Why it’s a paradox

There are two very strong points in this case. On the one hand, the process of elimination is technically correct. In order for the date to be a surprise, the students cannot know when it will happen, but by process of elimination, they will always know the date in advance. 

On the other hand, we know surprise tests are possible even with advance warning (as long as the warning is vague enough). Therefore, we have a paradox. Suppose that students went home that day reassured that the surprise test couldn’t possibly happen. Then on Wednesday the teacher applies it. Now, the students are truly surprised! So, even though the elimination argument was correct, it was ultimately wrong. How so? 

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This paradox has been thoroughly discussed by many thinkers over the years, yet no satisfying answer has been offered. At some extremes, the paradox threatens to undermine our very understanding of classical logic, or even our conception of how knowledge works.

4) Paradox of the Bald

A person with a full head of hair is most certainly not bald. Now, suppose that Roger loses one hair every hour. On the first hour, he will be down by one hair, on the second hour he will be two hairs down, and so on. Furthermore, Roger never grows new hair.

The question is, on which hour will Roger turn bald?

Why it’s a paradox

If losing one hair will not turn a non-bald person into a bald one, then there cannot be bald people! It seems arbitrary to claim that, say, on hour 20,000 Roger is not yet bald, but on hour 20,001 he is. Why that hour and not the next? Or the previous? Why not 33 hours later instead? 

Since losing a single hair doesn’t turn someone who isn't bald into someone bald, there will be no point in time, no specific hour, in which someone goes bald. Yet we know plenty of people do go bald. How come? Sure, it's a gradual process, but then there's a problem when we accept that a single hair doesn't make a difference, because there's no way of building up from that to the point where baldness does occur.

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There are several paradoxes of this kind, called vagueness paradoxes. For instance, how many grains of sand does it take to make a heap? Vagueness threatens many things in our way of thinking. Philosophers, linguists, and scientists are still not sure whether the problem lies in our language, our knowledge, or reality itself.

5) Lottery Paradox

Suppose you buy a lottery ticket, then your friend comes along and buys another one. A third friend, your typical buzzkill, calls you both fools for having bought them. He argues that there’s no chance you could win. You know it’s unlikely that you will, but you also know someone must win, right?

“Wrong!,” your friend objects. After all, he argues, any individual ticket has such a low probability to win, it’s almost negligible. It’s reasonable to assume your ticket won’t win (certainly, it’s unreasonable to simply assume it will win). The same goes for your friend’s ticket. But if that’s true for both tickets, then it must be true for all of them. After all, if any ticket is as likely to win as the next one, no ticket has much of a chance of winning. But if that’s so, it’s reasonable to assume that no individual ticket will win, since none of them has a reasonable chance in the first place. “So, you see: it's more rational to believe that no one will win!”

Is your buzzkill friend right?

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Why it’s a paradox

It sounds like something went wrong with your friend’s line of reasoning. But what? It’s true that, taken individually, no ticket has a reasonable chance of winning. Does it follow that there’s no reasonable chance of any particular ticket winning? It seems so! Yet we know that a ticket must win. So we must accept two contradictory positions at the same time: that probably no ticket will win, and that at least one of them will surely win. There’s the paradox. 

It's relevant because it raises serious questions about our reasoning and the foundations of knowledge: how we reason from probability and the boundaries between induction and reasonable belief. It puts almost all of our intuitions about knowledge to the test.

6) Buridan’s Bridge

Imagine you’re walking with your prankster friend on a bridge over a river. Your friend, a much bigger person than you, suddenly has a brilliant idea to tease you. He stands by the end of the bridge and tells you that if you say something true, he’ll let you pass. But if you say something false, he’ll throw you in the water. You are the clever one and you want to tease him back. “You will throw me in the water,” you say. 

What should your friend do if he wants to keep his promise?

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Why it’s a paradox

Much like the liar’s paradox, there’s no way your friend can actually keep his promise. If he now let’s you pass, then you said something false, so he shouldn’t have let you through. Then again, if he throws you in the water, then you said something true, so he shouldn’t have thrown you. In trying to fulfill his own promise, both options will make him break it (any other option is not a part of his promise). That’s the paradox. 

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How boring the world would be without enigmas and paradoxes. If everything were straightforward, lacking difficulty or ambiguity, there would be little reward in pursuing our curiosity to the very end. Perhaps without intractable puzzles humanity would have never looked too far into the depths of the universe and its riddles. After all, science as we know it comes from philosophy, and since its beginnings philosophy has been fueled by the mysteries of language and nature. So, how would you solve them?


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Oliver G. Alvar

Oliver G. Alvar

Creative Writer