We ponder over such problems because of a worry that they give us reason to doubt our most trusted source of knowledge – reason itself. At the very least – they are a fun way to baffle ourselves and confuse others at parties.
Here are five great puzzles/paradoxes to tickle the mind:
Imagine you are playing a game – it works like this: a rich scientist has presented you with two opaque boxes labelled A and B. You can’t see what is inside of either of them. The scientist tells you that you can make one of the following two choices. You can either take box B, or you can take both boxes. These are your two options.
Now in box A he has placed a thousand dollars. But he doesn’t tell you what exactly is in box B. He tells you that he is a very great scientist that has developed a perfect model of human behaviour – and he can predict in advance any decision anyone is likely to make. He tells you that he has made a prediction as to which box you’re likely to take. And if he predicted that you would take both boxes, he left box B empty. If he predicted that you would take only box B – he placed a million dollars in that box. Of course he doesn’t tell you what prediction he has made.
He then asks you to make your choice. What do you do? Do you pick box B only? Or do you take both?
To make sure the puzzle cuts deeply enough – let’s rule one possibility of thought out. The scientist is as good as he says he is. It’s in the spirit of the puzzle to accept he can successfully make his prediction. Perhaps you have seen a hundred or so people play his game. All those that chose both boxes walked away with a thousand dollars only – but all those that chose box B only, walked away with one million dollars.
When confronted with this puzzle most people tend to say: that’s easy I’d do such and such. But the problem is that half the people you speak to will say they will take box B, and the other half say they’ll take both boxes. And each thinks the other is crazy.
There are two lines of equally compelling reasoning going on. On the one hand – you’re pretty confident that the scientist knows what you are going to do. As such, if you decide to take both boxes, the predictor would have known that and would have left box B empty. But if you decide to take only box B – then the predictor would have known that as well and would have placed a million dollars. Your decision to pick box B alone seems to count as evidence as to what the scientist has done. It seems obvious – pick box B only – and walk home with a million bucks.
But wait – says the other line of reasoning. The predictor has already made his choice. Whatever is there in those boxes is there. Irrespective of what the scientist has predicted – you stand to gain the most if you pick both boxes. Consider the two possibilities. Either he has predicted you would take both, or you wouldn’t. If he predicted you would take both, and you do, you get 1000 dollars. But in that scenario, if you pick box B only, you’ll get nothing. The other scenario, the scientist predicted that you would pick box B only. In this case, if you pick both boxes, you would get one million and one thousand dollars, but if you pick box B only, you get one million dollars only. So no matter what the scientist has predicted, you stand to gain the most by choosing both boxes.
It’s a cunning puzzle – and the philosophers are still arguing over it. But it is relatively recent. You can find the original paper by Robert Nozick in the reference below:
Nozick, Robert (1969), “Newcomb’s Problem and Two principles of Choice,” in Essays in Honor of Carl G. Hempel, ed. Nicholas Rescher, Synthese Library (Dordrecht, the Netherlands: D. Reidel), p 115.
The Liar Paradox
Consider the following sentence:
“This sentence is false.???
Now ask yourself if it is true. If it is true then what it actually says, that it is false, is true – so it’s false. If it is false, then what it actually says, that it is false, is false – so the sentence is true. So when it’s true, it’s false – and when false, true. This is a paradox of the first rank.
It’s a very old paradox, being attributed to an ancient greek fellow Eubulides of Miletus around 400 bc. Philosophers and logicians have been arguing about it ever since. Think about it – that’s thousands of years of argument.
Some obvious solutions are considered to be failures. For example – some say the problem arises because we assume that all sentences must be true or false. But that’s just not the way natural language works. There are plenty of sentences that are neither true nor false – and this sentence is one of them. But you can reframe the puzzle in the following way:
“This sentence is not true???
And the whole game starts up again.
Some have claimed the problem to be one about self reference. Sentences shouldn’t be allowed to refer to themselves. This solution is too strong – plenty of sentences refer to themselves meaningfully – without paradox. “This sentence is written in english??? – for example. But besides that – the solution is just wrong. Consider the following two sentences:
1) Sentence 2 is true
2) Sentence 1 is false
I’ll leave it to you to see why this causes all the same problems – but neither of these sentences refer to themselves – only to each other. Hence this version avoids that object.
The debate goes on and the literature is vast. They’ve been arguing over this one for centuries – so good luck.
The Unexpected Hanging
A prisoner who has commited a heinous crime is before a judge. The judge sentences the prisoner to death by hanging, but adds a cruel twist to the sentence (the prisoner’s crime is particularly heinous). The prisoner is to be hanged on one of the following seven days – but it must be a surprise which day it is. The prisoner is not allowed to know.
Returning to his cell the prisoner is a bit disturbed at the prospect of being hung without knowing when (presumably it worse when you don’t know what day you are to die) – and confides his fear to his lawyer. His lawyer tells him not to worry.
“Look,’ he says with a smile. “They can’t hang you at all now. The judge has made it a condition that you must be surprised. But think about it. If you make it to Saturday without being hung then Sunday is the last day they could do it. But then it wouldn’t be a surprise would it? So that makes Saturday the last day they could possibly hang you. But hang on
The prisoner is comforted by this line of reasoning and stops worrying about the prospect of being hung at all. When suddenly on Wednesday, much to his great surprise, he is taken from his cell and hung.
This is a great puzzle, not so much because we have two different kinds of legitimate reasoning in conflict with one another, but because one strand of reasoning is shot down by reality itself! I’ll leave it to the reader to puzzle further on this one.
Hempel’s paradox is a paradox of induction. Induction is method of reasoning we use to make generalisations about the world. Consider all the ravens you have seen in your life time. Hopefully they were all black. Now on the basis of seeing nothing but black ravens, it would be reasonable to generalise and claim that all ravens are black. This is a natural step – and it forms the basis to all our scientific reasoning.
Now consider the following statement: ‘All non-black things are not ravens’. This statement is logically equivalent to our generalisation. For if all ravens are black, then something which is not black can not be a raven. We could then go observe non-black things – and each time we saw that a non-black thing was not a raven, we would confirm that all ravens are black.
So it seems by observing a pink flamingo – it would confirm that all ravens are black.
But hold up! Seeing a pink flamingo would also confirm the statement: ‘All non-white things are not ravens’ and this is logically equivalent to ‘All raven’s are white.’ So it seems that the observation of a pink flamingo seems to confirm both that all ravens are white and that all ravens are black. But this is a contradiction!
Again I will leave it to the interested reader to go seek out the solution.
The Sorites Paradox
This is another ancient paradox bequeathed to us by those awfully intelligent Greek fellows.Imagine you have in front of you a heap of sand. It is unquestionably a heap of sand. It has more grains of sand in it than you would care to count. Now it seems fair to say that given any heap of sand, if you were to take one grain of sand away, then it would still be a heap of sand. And if you were to take another grain of sand away from the remaining heap, it would still remain a heap.
But what if you were remove some many grains of sand you were left with only one grain? Well it would no longer be a heap. But we have just proved that it must be! For the removal of one grain of sand from a heap, still leaves us with a heap. By repeated application, we must always be left with a heap. But at some point along the line it stops being a heap and our reasoning leads us astray. What has gone wrong?
This is a paradox that arises out of the vague use of our language. We know what a heap of sand is – but we do not know how to draw the line between a heap and a non-heap. Our use of language does not prescribe an answer.
Solutions to this paradox become very complicated very quickly – and again I’ll leave it to the interested reader to find his/her own solutions.
If you want to learn more about various logical and philosophical puzzles and paradoxes, I can’t recommend highly enough the book: Labyrinths of Reason: Paradox, Puzzles, and the Frailty of Knowledge by William Poundstone. It’s a fantastic introduction to the world of paradox. He is comprehensive, yet explains everything in a clear and easy to understand language. If you’re up for the challenge, check him out today.
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