The Epistemological School
Various epistemological formulations have been proposed that show that the prisoner's tacit assumptions about what he will know in the future, together with several plausible assumptions about knowledge, are inconsistent.
Chow (1998) provides a detailed analysis of a version of the paradox in which a surprise examination is to take place on one of two days. Applying Chow's analysis to the case of the unexpected hanging (again with the week shortened to two days for simplicity), we start with the observation that the judge's announcement seems to affirm three things:
- S1: The hanging will occur on Monday or Tuesday.
- S2: If the hanging occurs on Monday, then the prisoner will not know on Sunday evening that it will occur on Monday.
- S3: If the hanging occurs on Tuesday, then the prisoner will not know on Monday evening that it will occur on Tuesday.
As a first step, the prisoner reasons that a scenario in which the hanging occurs on Tuesday is impossible because it leads to a contradiction: on the one hand, by S3, the prisoner would not be able to predict the Tuesday hanging on Monday evening; but on the other hand, by S1 and process of elimination, the prisoner would be able to predict the Tuesday hanging on Monday evening.
Chow's analysis points to a subtle flaw in the prisoner's reasoning. What is impossible is not a Tuesday hanging. Rather, what is impossible is a situation in which the hanging occurs on Tuesday despite the prisoner knowing on Monday evening that the judge's assertions S1, S2, and S3 are all true.
The prisoner's reasoning, which gives rise to the paradox, is able to get off the ground because the prisoner tacitly assumes that on Monday evening, he will (if he is still alive) know S1, S2, and S3 to be true. This assumption seems unwarranted on several different grounds. It may be argued that the judge's pronouncement that something is true can never be sufficient grounds for the prisoner knowing that it is true. Further, even if the prisoner knows something to be true in the present moment, unknown psychological factors may erase this knowledge in the future. Finally, Chow suggests that because the statement which the prisoner is supposed to "know" to be true is a statement about his inability to "know" certain things, there is reason to believe that the unexpected hanging paradox is simply a more intricate version of Moore's paradox. A suitable analogy can be reached by reducing the length of the week to just one day. Then the judge's sentence becomes: You will be hanged tomorrow, but you do not know that.
Read more about this topic: Unexpected Hanging Paradox
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