Editing Unexpected hanging paradox (section)

==Logical school==
Formulation of the judge's announcement into [[formal logic]] is made difficult by the vague meaning of the word "surprise".<ref name="Chow"/> An attempt at formulation might be:

*''The prisoner will be hanged next week and the date (of the hanging) will not be deducible the night before from the assumption that the hanging will occur during the week'' (A).<ref name="Chow" />

Given this announcement the prisoner can deduce that the hanging will not occur on the last day of the week. However, in order to reproduce the next stage of the argument, which eliminates the penultimate day of the week, the prisoner must argue that his ability to deduce, from statement (A), that the hanging will not occur on the last day, implies that a second-to-last-day hanging ''would not be surprising''.<ref name="Chow" /> But since the meaning of "surprising" has been restricted to ''not deducible from the assumption that the hanging will occur during the week'' '''instead of''' ''not deducible from statement (A)'', the argument is blocked.<ref name="Chow" />

This suggests that a better formulation would in fact be:

*''The prisoner will be hanged next week and its date will not be deducible the night before using this statement as an axiom'' (B).<ref name="Chow" />

[[Frederic Fitch|Fitch]] has shown that this statement can still be expressed in formal logic.<ref>{{cite journal |last=Fitch |first=F. |title=A Goedelized formulation of the prediction paradox |journal=Am. Phil. Q. |volume=1 |year=1964 |issue=2 |pages=161–164 |jstor=20009132 }}</ref> Using an equivalent form of the paradox which reduces the length of the week to just two days, he proved that although self-reference is not illegitimate in all circumstances, it is in this case because the statement is self-contradictory.