The Unexpected Hanging Paradox
Introduction
Description of The Unexpected Hanging Paradox(TUHP) from Wikipedia:
A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.
Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the "surprise hanging" can't be on Friday, as if he hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on Friday. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.
He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.
The next week, the executioner knocks on the prisoner's door at noon on Wednesday — which, despite all the above, was an utter surprise to him. Everything the judge said came true.
The explanation of the problem above seems fairly simple. And the paradox is fairly clear - the logic the prisoner uses to conclude that there is no day on which he can be hanged looks pretty sound - and yet he is hanged. This paradox, like all other paradoxes, started with catching my attention, and disturbing me a little (like all paradoxes should). But unlike other paradoxes like the Potato paradox, which eventually get resolved in our brains, leaving a bitter-sweet taste of being introduced to one's failings of Intuition, TUHP kept troubling me, ever more than before. One reason why this paradox is troublesome might be, as one might expect, that this simple-looking paradox still has no satisfactory, generally accepted resolution despite being discussed for a fairly long amount of time by philosophers. The second reason which I think makes this paradox a potent one is how it apparently un-does rationality. If one subscribes to the view that Rationality is the process of using all the available evidence, and logically deducing the best possible path to pursue or actions to perform to achieve one's goal, it seems like this paradox basically debunks the idea that rationality is always the best possible option to achieve maximum efficiency. For someone like me who has long believed that rationality is always the best option and that it trumps irrationality irrespective of the situation, the above inference can be shocking.
But before I conclude that there is an unresolved paradox in TUHP, or that it proves that rationality can be inefficient, there are a lot hidden intricacies in the above explanation of the paradox which need to be looked at, understood or defined thoroughly to be able to meaningfully discuss and move towards a conclusion.
Irrationality > Rationality ?
First let us clarify how TUHP seems to prove that being irrational can be better than being rational. We have already seen in the above explanation how the rational prisoner (RP-1) deduces that he will not be killed, and by virtue of that deduction, gets killed. Now let us replace RP-1 with another prisoner who is a perpetual pessimist. He always believes deeply that only the worst things will happen to him. This makes this prisoner the irrational prisoner (IP-1). So in this situation, when the judge lets him know of the conditions regarding his death, he immediately concludes (albeit irrationally) that he will definitely be killed on very next day which is Monday since that is the worst that could happen to him. Due to his lack of logical abilities, he is not able to further deduce the implications of this belief of his. If the executioner doesn't come on Monday noon, IP-1 immediately concludes and believes that he will be executed the next day and so on. This means that whenever the executioner knocks on the door, may it be Monday, Tuesday, Thursday or any other day, he will always find IP-1 waiting for the knock on the door and unsurprised. But because the judge told him that the hanging will be a surprise, which can never be the case thanks to his irrational pessimism, he is not hanged. His irrationality saved him! It should be noted however that not all kinds of irrationality will result in a happy ending for the prisoner. For instance, let us change one attribute of IP-1 - let us make him a perpetual optimist named IP-2, who always believes that only the best things will happen to him. What this means is that as soon as the judge tells him the conditions, IP-2 believes that he will never be hanged because of his irrational optimism. This means that whenever the executioner knocks on the door, IP-2 will be surprised, thus satisfying the condition, and being hanged. IP-1's irrationality saved him. IP-2's didn't.
Nevertheless, it still seems depressing that IP-1 was saved but not RP-1... until you start thinking more critically of the situation. My reprieve came from a moment from this episode of the Rationally Speaking podcast where the host Julia Galef talks about how a situation can be rigged to favour irrationality. The example she gives is "What if a billionaire comes to you and says - I will give you a million dollars if you believe that the sky is green". Of course, we have to assume that the billionaire somehow has the technology to measure or detect true belief. A rational person will not believe that the sky is green because he knows the evidence, and his rationality prevents him from believing in a falsity. On the other hand an irrational person maybe can easily fool himself using irrational arguments that the sky is green and win the million dollars. The example can be made more relatable by replacing the "sky is green" phrase with something like "the earth is a flat disk". This kind of example somehow made me feel much better. I am now able to see the paradox in a different light by being provided the possibility that maybe the paradox does not prove that rationality is more inefficient that irrationality, but maybe it is just one such situation which has been rigged against rationality. Maybe it is just a more complex way of saying - all perfectly rational people will be hanged. If that is the case, then it doesn't mean rationality has been undone. But to be able to conclude that this indeed is the case - that the situation in the paradox has been rigged against perfectly rational agents, I will have to prove it, and if I am able to prove it, it will be one way of resolving the paradox. The other more obvious way is to show how RP-1's rationality is not perfect -in other words, that there is a logical fallacy in the way he deduced that he won't be killed.
Is the paradox rigged against rational agents?
Now we'll see if TUHP is the same as saying "All perfectly rational agents will be hanged" or in other words, is it rigged against rational agents. Honestly I don't know how to go about this. My initial thoughts are pointing me to an intricacy which I might have missed. Let us first take the "sky is green" scenario. The situation is rigged against rational agents because their rationality prevents them from believing something which contradicts evidence. This is, if you think of it, not a shortcoming of rationality itself, but of humans. Humans cannot force belief onto themselves. They cannot unlearn the fact that the statement sky is green is incompatible with evidence. But imagine if humans did have the power to forcefully believe things in the true literal sense of the word believe. Then their rationality (whose aim is to win the million dollars) would dictate to believe that the sky is green, and thus win the reward. An irrational person is just lucky that his lack of rationality is preventing him from going into a situation where his shortcomings as a human start affecting him. The rational person deduces the correct result - that the sky is blue and not green - this conclusion just happens to be at odds with the requirements of winning the reward. So here rationality gives the 'factually correct' answer, but not the efficient answer (even if it does - it is impossible to implement it as explained earlier). We can reasonably define 'rigged' as a case when the factually correct information is at odds with what is most effective. But in case of TUHP, RP-1 not only comes up with an inefficient answer (which leads him to get killed), but he also comes up with the factually incorrect answer (predicting that he can't be hanged). This leads me to think that TUHP is not just a rigged problem at least in the sense of the word rigged as mentioned earlier. It is a paradox not only because RP-1 is inefficient, but also incorrect. So unfortunately, the paradox being rigged does not look like a valid resolution to the paradox. It keeps getting more intriguing - and the question is still open.
Here I have to stop my train of thought, as it has hit a dead end. The other way to reach a resolution to TUHP would be to show that RP-1 is actually not a perfect rational agent - by showing a logical flaw in his argument. But to do that, we will have to first understand the paradox more deeply.
Unravelling the paradox
In the description of the paradox and throughout this post up until now, I have casually used many words whose meaning is not completely obvious if you think deeper, and whose meaning makes a whole lot of difference in the way we perceive the paradox. One such important word is surprise. The description of the paradox says the following:
So surprise here means not knowing. And not surprised means knowing. But what does knowing mean? Is it the same as believing or is it different? This is very important to understand the paradox. In the case of RP-1, since he has deduced to his satisfaction that he cannot be hanged, he both knows and believes that he will live. In the case of IP-1, he does believe that he will be killed the next day. But does he know it? It is very common to use the word 'know' like this instead: 'Does he know it for a fact?'. The answer to this question seems to depend on the person you ask. If you ask IP-1 if he knows it, because he is an eternal pessimist (or he could have seen it in a crystal ball, or a parrot took out a card, or whatever), he will say 'Yes, I know it'. He believes that he knows. (..but does he know that he knows?... :P). But if you ask RP-1, he might say IP-1 believes he will be hanged, but he cannot know, because according to RP-1, the only way to know is by logical deduction. Things become interesting when you ask the Judge. Whenever the executioner knocks on the cell of IP-1 he finds him with the 'knowledge' that he is going to be hanged that day. The executioner takes out his Kopzometer and measures that IP-1 has almost exactly 0 Kopetiums of surprise (± 0.0001 Kopetiums due to device precision issues). Due to the lack of surprise, the executioner does not hang IP-1. The executioner goes to the judge (J) (who set the conditions for the hanging) and reports the same. The judge goes to IP-1.the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.
J: Were you not surprised when the executioner knocked on your cell today?
IP-1: No.
J: Did you know that you were going to be hanged today?
IP-1: Yes
J: How?
IP-1: Because only the worst things always happen to me (OR I saw it in a crystal ball yesterday OR the parrot told me etc etc)
Now the Judge has to make a choice. Does he allow the explanation given by IP-1 as a valid reason for him to say that he knew? If he is as irrational as the prisoner, he might. But if the judge doesn't have the same beliefs as IP-1, he might say 'That is not sufficient information for you to know that you will be hanged today. You think you know, but you do not know. So you shall be hanged. Since the judgement says that you will not know the day of the hanging, and since you satisfy the condition, you shall be hanged'. And thus IP-1 is hanged (YESSSSS!!! Oh.. what a monster I am..)
The conversation above actually doesn't need to end so soon.
IP-1: Wait, but the judgement said 'it will be a surprise hanging', and it was. The Kopzometer proved it. So I should not be hanged.
This is where the statement 'the execution will be a surprise to the prisoner. You will not know the day of the hanging' comes into scrutiny. What does the above statement mean? It can mean one of the following
- The execution will be a surprise OR you will not know
- This is the least likely interpretation. In this case, the decision depends on the guy who gets to decide if IP-1 actually knew. So it depends on the Judge.
- The execution will be a surprise AND you will not know
- This interpretation is more likely than the previous one. In this case, the Kopzometer saves IP-1
- The execution will be a surprise IMPLIES and IS IMPLIED BY you will not know the day of the hanging
- This is the most obvious, and I think common, way to look at it. And on first glance looks completely fine. But as shown above, this statement is incorrect. 'The execution will be a surprise' is not implied by the fact that you will not know the day of the hanging. You can be un-surprised even though you don't know the date of the hanging (depending on whom you ask and what you mean by know, of course)
The resolution
After all this, the paradox is clearer, more precise, but is still a paradox. Now it's time to attempt a resolution to the questions - Where does the paradox arise from? Why is logical deduction resulting in an incorrect answer?
I think the answer lies in realizing that TUHP, as simple as it sounds, is actually much more complex than it needs to be to make the point it makes. The additional complexity creates an illusion of a paradox because it sounds innocent, but leads to something which you don't expect. The question I asked myself is this: Why does it need to be a week? What is the minimum length in number of days that the range of possible days of hanging need to be, for this paradox to exist. It initially seems that there have to be at least two days for there to be a paradox. After all, the judge says that RP-1 will be hanged in ONE OF THE FOLLOWING noons, and that it is going to be a SURPRISE. These two highlighted phrases go well with each other. They tell us that the judge has a choice. Due to this choice that he has, the date is going to be a surprise to anyone, because this choice is going to be executed by a agent of free will. It seems that the prisoner is doomed. But our prisoner is intelligent and he figures out a way to deduce a date of hanging by elimination. And here he seems to escape the clutches of death. But in a bizarre twist he dies.
But it is both important and astonishing to realize that there needn't even be two noons for TUHP to exist. Let us change the sentence to this - 'You will be killed tomorrow noon. And you won't know about it.' This sounds stupid. But still, let us probe this situation for a minute. RP-1 looks at the statement and thinks - 'wait, I know I will be killed tomorrow, so when they knock on my door i will already know it, so I won't be killed. So, i am not going to die'. The next day, the executioner still comes, RP-1 is surprised and is killed. Sound familiar? But suddenly there are so many questions which come-up
- Wait! The judge's statement is self-contradictory! Is he even allowed to say that?
- RP-1 is stupid! Why did he not stop at the point where he knew is going to be killed? Why did he have to go further?
- No!! RP-1 cannot be killed, because he KNEW he was going to be killed.
- The judge's statement just sounds self contradictory. But is not. Think of it this way - he is basically saying, 'I have only one point in time when I can kill you and that is tomorrow noon.' But just to be playful he adds 'But I will not kill you if you know that you will be killed tomorrow'. This doesn't sound that self-contradictory.
- RP-1 pursued the thread of logic as far as he could. That's what everyone does. He stopped after concluding that he is safe - that's how far he could go.
- RP-1 knew? He concluded that he will be safe. He knew the fact before he concluded otherwise. Once he did conclude otherwise, he no longer knew. So you can't say he knew. If he did, he wouldn't be surprised.
Logic is not an independent entity. It needs premises. To be able to build mathematics, you need axioms. To be able to argue you will have assumptions. Axioms and premises are inevitable for logic to function or even exist.
But what if a logical deduction undoes an axiom?
What if A implies B and B implies not A?
What if the next sentence is true? What if the previous sentence is false? What if this sentence is false?
What if you can be relieved from the airforce only if you are proven to be insane, the only way to be proven insane is to request for an insanity check yourself, and the definition of insanity states that any person requesting for his own insanity evaluation cannot be called insane?
What if the crocodile tells the mother I will let your child go if you are able to accurately predict what I will do with your child?
All these are examples of logical deductions undoing their own axioms. In these situations, rationality and logical deduction cannot exist. Everything crumbles. You cannot reach the truth. The sentence "This statement is false" does not have a truth value in the traditional sense of the word truth. Such situations can be seen as analogous to the video above. Turning on the switch leads to (instead of implies) the arm coming out. But the arm coming out leads to the switch turning off. The switch turning off leads to the arm going back in. The judge has told you: 'You will live if the arm stays out' or 'You have to know that you will die to know you will be alive which will make you not know that you are going to die which will make you die'. Maybe you are doomed. Maybe there is only one state the arm can be in - inside. Maybe there is only one state you can be in - dead. Maybe that is what this story is telling us.
But what if a logical deduction undoes an axiom?
What if A implies B and B implies not A?
What if the next sentence is true? What if the previous sentence is false? What if this sentence is false?
What if you can be relieved from the airforce only if you are proven to be insane, the only way to be proven insane is to request for an insanity check yourself, and the definition of insanity states that any person requesting for his own insanity evaluation cannot be called insane?
What if the crocodile tells the mother I will let your child go if you are able to accurately predict what I will do with your child?
All these are examples of logical deductions undoing their own axioms. In these situations, rationality and logical deduction cannot exist. Everything crumbles. You cannot reach the truth. The sentence "This statement is false" does not have a truth value in the traditional sense of the word truth. Such situations can be seen as analogous to the video above. Turning on the switch leads to (instead of implies) the arm coming out. But the arm coming out leads to the switch turning off. The switch turning off leads to the arm going back in. The judge has told you: 'You will live if the arm stays out' or 'You have to know that you will die to know you will be alive which will make you not know that you are going to die which will make you die'. Maybe you are doomed. Maybe there is only one state the arm can be in - inside. Maybe there is only one state you can be in - dead. Maybe that is what this story is telling us.
The machine is of the form
A => B => ~A => ~B => Nothing (A = switch on, B = arm out, ~A = opposite of A)So the machine ends up in a stable state. Catch-22 is just like the machine. The system is designed to keep you in a single state - that of flying - inside the box.
The language we use in our daily life is not designed to fit logic. It is designed to fit grammar. We can use this language to cleverly hide illogical/meaningless/rigged situations which sound completely normal and logical. We can build statements which which hide the fact that the conclusions undo the premises. There is nothing in the language to make us feel uncomfortable of the underlying illogical system being expressed - which leads to such paradoxes. You cannot try and succeed in finding the truth when the system doesn't allow for something like truth to exist.
This I think is a satisfying conclusion to the paradox.
THUP is actually slightly different from the machine shown above. It is of the form
A => B => ~A => ~B => A => B and so on... (A = will die tomorrow, B = know that i will die)There is an infinite loop. If you use perfect logical deduction on the above rule-set, you should never end up with a conclusion. RP-1 stopped his logical deduction at ~A (step 3). Which caused him to die. If he would have stopped at A (step 5) he would have lived. At step 7 death, at 9 life. RP-1 did not live. He was two steps of rationality short of saving himself. THUP has made me realize that rationality and logical deduction is a mine you dig to get to the truth, to perfection. You keep digging till you can, till your logical, mental abilities assist you. But if you miss a link or if you stop digging at the wrong moment, you might end up worse than how you had started. But we don't let that realization deter us. We will dig till we can. We might fail. We will fail. But we will not stop digging. The mother tells the crocodile he is going to eat her child. The mother couldn't get back her child.
At least her child lives.
Comments