In a harsh totalitarian country an innocent person is arrested on Sunday evening and summarily condemned to execution, which they are told will take place on one of the following five mornings. To make matters worse, they are told that they will not know the day before which morning it will be. After several hours torment the prisoner fails into a peaceful sleep as they realise that such a threat cannot be carried out. They reasoned thus:
The execution cannot take place on Friday morning; for if they are still alive on Thursday night then the execution must take place on Friday. But they were told that they would not know the day before which day it would be. So it cannot be Friday, and so Friday can be counted out as a possibility. But by the same reasoning it cannot be Thursday either. For if they are still alive on Wednesday night then the execution must take place on Thursday. But they were told that they would not know the day before which day it would be. So it cannot be Thursday, and so Thursday can be counted out as a possibility. But by the same reasoning it cannot be Wednesday either… The same reasoning covers Wednesday, Tuesday and Monday and so the prisoner can have a sound nights sleep.
The prisoner is, however, greatly surprised to find themselves facing the executioner on Wednesday (or indeed any other) morning. What went wrong with the prisoner’s reasoning?
This paradox, known as the Hangman, is an instance of the Paradox of Prediction. Another instance is the Surprise Exam: a teacher tells the class that there will be an exam sometime during the term but the students will not know the day before when it will be; it can’t be last day of term – if the second to last day has been finished with no exam then it must be this day and so no surprise – nor can it be the second to last day for the same reason, and so on through the other days of term. Yet we know that surprise examinations do occur.
Both the Hangman and the Surprise Exam have a common form, that is
You will be hung one morning this week (there will be a surprise exam during term) but you cannot predict on the basis of this statement what day that will be.
This statement is clearly self-referential and we know from previous experience (see The Paradox of the Liar and Russell’s Paradox) that such statements are problematic. But is self-reference all that is causing the problems here?
As well as self-reference, the paradox also involves prediction and prediction involves knowledge and knowledge involves truth. Consider the following
No-one knows this proposition.
Is this true or false? If it is false then someone does know the proposition. But knowing something implies what is known is true, and so the proposition is true. Hence we have shown that the proposition is true and therefore we now know it to be true. So if we know it to be true then someone knows it to be true, you and I now know it to be true, for example, and so it is false!
So is the problem more to do with knowledge, specifically the prisoner’s claim to knowledge? Suppose that the prisoner is told on Sunday evening that
You will be hung tomorrow at dawn but you will not know that beforehand. Can such a sentence be carried out? The prisoner may claim that such a sentence is self-contradictory but is this simply the prisoner making a mistaken claim to knowledge – which fact would be shown by the sentence being carried out. To be sure the prisoner has a belief that the sentence cannot be carried out but a belief and certainly as in this case a false belief – is a long way from knowledge. The sentence only claims that the prisoner will remain in ignorance up until the time of the actual hanging:
but there remains the possibility that the sentence is a cruel joke intended to further torment the prisoner. The prisoner is only allowed to argue with regard to the above sentence
If it is true then I will know I will be hung in advance.
And the prisoner does not know whether the judge who issued the sentence spoke truly or not. Only when the hanging occurs does it becomes clear that rather than a mere possibility, it is an actual fact.
This response argues that the prisoner is claiming to know more than they are entitled to claim, and here lies the fallacy in the prisoner’s reasoning. The claim must be wrong because surprise hangings (or examinations) do occur.
One point about this reply to the Paradox of Prediction: It supposes that there are statements that are true or false – but that we cannot know which. Consider a prediction that is made, not about tomorrow or next week or the end of term but about some time in the far future when we will not be here to see the outcome: does it make sense to say that the prediction is true or false if we cannot tell which?