But actually, the inevitability is PART of the paradox itself. If we accept the judge's reasoning, the hanging is, in fact, INEVITABLE. If we accept the prisoner's reasoning, the hanging is, in fact, IMPOSSIBLE. The problem is: Why are we unable to come up with a definite criterion that would allow us to decide between these arguments?
<br>The pop quiz is flawed because it brings in the extra-logical concept of &quot;surprise&quot;; thus, if the professor tells me that there will be a surprise quiz sometime during the semester, I could (at least theoretically) come to class every day &quot;expecting&quot; that quiz, and this undermines the paradox without clearing it up.
<br>The version I gave does away with all such extraneous issues, reducing the paradox to a set of purely logical conditions, and thus elucidating its structure.

I think that the hanging is NOT inevitable, even to comply with the judges instructions. That is, by introducing the conditional statement that (under certain circumstances) the hanging cannot occur (if the hanging must occur on a particular day in order to satisfy the five day rule). Perhaps the logical is now entering the realm of the semantical, but it seems to me that pursuant to the judges order the hanging may occur (under the specified conditions) or it may not occur (if the situation is such that the conditions cannot be met). So, on Friday (or Sunday thinking ahead to Friday) there are still two possible outcomes 1) that there is no hanging, or 2) that there is a hanging. #2 is possible because if the instructions cannot be complied to, the prisoner is not hanged. #1 is possible because, it is not a logical necessity that the prisoner MUST be hanged, therefore he CAN be hanged.
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<br>It seems that the judge is really saying something like; the prisoner must be hanged if (and only if) it is done in the next five days, otherwise he is not to be hanged. The second condition is merely a logically superfluous condition that removes the inevitability.
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<br>The other way of reading the judges order is to give undo weight to the word MUST. If the hanging MUST occur and the hanging MUST occur in a way that satisfies the other two conditions without the possibility of the hanging not occurring, there is no longer a paradox, just an order by a judge that may not be possible to comply with under some circumstances.
<br>The fact that the instructions are subject to violation is not paradoxical. So it would be logical for the prisoner to incorporate the possibility of no hanging in order to accept that he might be hanged on Friday. Once he gets this the rest of the regression folds.
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<br>Of course my analysis begs the question of whether it is permitted to add semantic context to the premises of the proposed paradox in order to strip away the contradictions by examining what is &quot;really&quot; being said. I think we can and should. What do you think?
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<br>Ken

I still think we're getting hung up on an incidental point here, Ken. There is no more inevitability to the Unexpected Hanging than there is to the Pop Quiz. The professor is saying &quot;I will give you a quiz that must satisfy two conditions: one, it must be given on one of the remaining school days, and two, it shall not be given on any day on which it would be expected&quot;. This is a direct parallel to the two conditions given by the judge for the prisoner's hanging. I suppose the professor could decide not to give the quiz at all, but this wouldn't affect the paradox, just as the judge's decision not to hang the prisoner wouldn't affect that version of the paradox.
<br>The paradox lies in the fact that the prisoner's argument FEELS right to us when we consider it, but the judge's decision also feels right. But it would seem that they can't both be right, since they come to opposite conclusions on the same issue. So why can't we devise a criterion that will decide between them?

I don't know that I want to give up on my inevitability angle just yet, but I'll table the arguement. To answer your question about establishing a criteria by which we can choose between prisoner and judge in the unexpected hanging paradox... I would choose the side that contains the more complete knowledge. That is, the judge. The judge knows on which day the hanging will take place and the prisoner does not. Despite his attempt at a logical escape, the prisoner can still wake up on any given day and be executed.
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<br>I read a quote by Michael Scriven in which he is talking about this sort of paradox where reality can trump logic: &quot;I think this flavour of logic refuted by the world makes the paradox rather facinating. The logician goes pathetically through the motions that have always worked the spell before, but somehow the monster, Reality, has missed the point and advances still.&quot;
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<br>Ken

But a logical paradox is just that, a LOGICAL paradox. If the amount of knowledge in someone's head determines the solution, then by definition, it isn't a logical paradox. To side with the judge because he already knows when the prisoner will be hanged is to obviate the paradox altogether.
<br>No, the key is that the paradox exploits the difference between two kinds of disjunction, PRACTICAL (the way we actually PRACTICE disjunction) and LOGICAL (the pure logical FORM of disjunction).
<br>Practically, if we're given 5 statements (The prisoner is hanged on Monday, The prisoner is hanged on Tuesday, and so on...)so that 1 of them is true and 4 of them are false, then, since we're merely trying to &quot;get to&quot; the true statement, as soon as we can determine any 1 of the statements to be false, we ELIMINATE it from the disjunction, and we keep doing this until we &quot;capture&quot; the true statement.
<br>But logically, this is not what is happening. Here we're simply assigning the value &quot;true&quot; to 1 of the statements and &quot;false&quot; to the other 4, and the latter value is in no way inferior to the former value. Thus the assignment of the value &quot;false&quot; to a statement does not eliminate it from the disjunction.
<br>In this paradox, the prisoner is engaged in practical disjunction, while the judge is engaged in logical disjunction. This is done by bringing the empirical concept of time into the timelessness of (classical) logic. When Monday has passed and the hanging has not taken place, the prisoner eliminates the first statement from the disjunction, but the judge regards it just as much a part of the disjunction as before. Thus Friday represents the &quot;last&quot; day to the prisoner, but only 1 of 5 days to the judge.
<br>So much for the structure of the paradox; in the STORY of the paradox, the prisoner is right and the judge is wrong. For the judge is being sadistic, since he knows that the logical conditions which he sets for the hanging put no restrictions on it whatsoever from HIS point of view, and therefore can only serve to torment the prisoner with the hope of a reprieve before his inevitable demise. Besides, a hanging should be a matter of practical, not logical, consideration. Thus, my sympathy is with the prisoner, a commendable attitude which has only the one drawback of leaving him just as dead as he was before.

Thank you. Yours is a very good solution to the paradox. I was actually starting along similar lines, but by including &quot;non-hanging&quot; into the judges disjuction rather that the passed days. Ken