Five philosophical paradoxes

Bertrand Russell once remarked that “the point of philosophy is to start with something so simple as not to seem worth stating, and to end with something so paradoxical that no one will believe it.”

Few philosophers would wish to suggest with all seriousness that this is the foremost ambition of their subject, but it is undeniably one of its most fascinating aspects. Taking a handful of seemingly indisputable truths, philosophers throughout history have been able to show that counterintuitive or straight-up contradictory consequences follow. In fact, whilst much progress has been made, there is still little agreement over the best way to handle many of these cases.

Here are five notoriously impenetrable paradoxes:

The Unexpected Hanging

A prisoner is to be hanged. To make things worse, he is told that the execution will be a complete surprise to him: it will happen at some point over the next week, but he won’t know which day until it comes. Strangely, upon hearing this news, the prisoner reacts with glee. He surmises that, were he to be hanged on Friday, it wouldn’t come as a surprise; after all, he knows he must be hanged by the end of the week and, by Thursday night, Friday will be the only remaining possibility. Confident that he will not be hanged on Friday, the prisoner concludes that he will not be executed on Thursday either (as he would expect it by Wednesday night). In fact, by the same reasoning, he is able to rule out Wednesday, and then Tuesday, and Monday too. The prisoner comes to believe that he will never be executed.

As it turns out, he is hanged on Wednesday. This, of course, is a complete surprise to him.

There is little agreement over how this paradox arises or how it is to be solved. The prisoner appears to have ruled out his unexpected execution in a perfectly reasonable and logically acceptable way. And yet, he is still surprised when the executioner comes knocking. Why?

The Liar

This sentence is false. Is it?

Well, if it’s false, then it’s true. And if it’s true, then it’s false. It isn’t clear what to do.

The problem was first presented in or around the fourth century BC by the Greek philosopher Eubulides, of whom little is known. Nevertheless, it has become one of the most famous paradoxes in modern logic. Whilst there have been many attempts at a solution, most seem to run into problems of their own. Perhaps the most interesting of these tactics (but far from the most popular) is simply to accept that some sentences are both true and false.

The Bootstrap Paradox 

A time traveller goes back in time to meet his idol, Charles Darwin. He arrives in 19th-century England and excitedly tells Darwin about the theory of evolution; in fact, to better make his point, the traveller hands Darwin a copy of On the Origin of Species and then heads back to his own time. Darwin, taken in by the idea, simply copies the book and publishes it; many years later, a ‘first edition’ of On the Origin of Species is acquired by the time traveller, just before he journeys back and meets with Darwin.

Who, then, first wrote On the Origin of Species?

The bootstrap paradox is not as famous as its predecessor, in which a time traveller kills their own grandfather before his children are born. However, unlike this ‘grandfather paradox’, there is nothing inherently inconsistent about the bootstrap paradox – nothing about the situation itself suggests that it’s impossible. This makes the problem uniquely intriguing.

Certainly, the prospect of time travel has become an increasingly serious talking point in physics and the philosophy of science ever since Einstein’s theory of general relativity. We might, however, question whether to dismiss the possibility of time travel-conducive spacetime geometries based on the threat of paradoxes like this one.

The Raven Paradox

Consider the statement below:

1. If something is a raven, then it is black

This is logically equivalent to:

2. If something is not black, then it is not a raven

Next, consider this:

3. Cherries aren’t black and they also aren’t ravens

Now, if you were to see a black raven, and then another black raven, and then a hundred more, you may start to think that (1) (or the equivalent statement ‘all ravens are black’) is true. At least, you’d be likely to say that there was more evidence to support the claim after seeing a hundred and two black ravens than after seeing just one. Similarly, proposition (3) seems to go some way toward justifying our belief in proposition (2). However, as (2) and (1) are logically equivalent, evidence for (2) is evidence for (1).

But this is strange. Why should we think that sitting inside and observing non-black objects will be of any help when it comes to proving that all ravens are black? Surely this ‘indoor ornithology’, as it has been called, is pointless? So what went wrong?

This paradox was suggested by Carl Hempel in the mid-20th century; many answers have since been proposed, but the paradox is still thought to cause problems.

The Lottery Paradox

You buy a national lottery ticket. You know that someone must win the lottery, but you also know that your chances are slim and can conclude with near certainty that you won’t win. However, it also seems to be the case that your friend’s ticket won’t win either; in fact, extending this reasoning, it seems rational to maintain that all tickets will lose. And yet, you also wish to say with certainty that one ticket will win. How do you square these two seemingly rational beliefs?

This paradox has become hugely significant in epistemology (the branch of philosophy concerned with knowledge) as well as the study of probability. As of yet, there is no consensus on a solution.


The paradoxes discussed above are just a few of many; philosophers, scientists, and mathematicians have come up with hundreds over the years. See Wikipedia’s list of paradoxes for a good compilation.


Image: 愚木混株cdd20 on Unsplash 

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