More About the Unexpected Hanging

See the problem "The Inevitable Surprise" below for the initial discussion.

Some authors consider the problem of the unexpected hanging to be an example of an actual paradox, meaning that the sentence of the judge is self contradictory. Assuming that the judge has seven days to choose from, we have shown using game theory that the the sentence is not self contradictory. But consider the reduced problem where the judge pronounces the sentence on Thursday saying thatt the hangman will appear at the jail cell on Friday or Saturday and the hangman will be unexpected.

The assumptions are:

1. The judge's statement is true;
2. Predicate logic holds.

Under these assumptions we can see that

• If the judge chooses Saturday, then the hangman will be expected;
• The judge must choose Friday, so by the previous statement the prisoner ought to expect the hangmen on Friday.
• If the hangman can be expected on Friday, then the judges statement is false and so the statement is self-contradictory.

Some authors claim that the reason for this paradox is that the judge's sentence is self-referential.

What is a self-referential statement?

A self-referential statement is a statement that can contains in it a reference about the statement. For instance:

This statement is false.

If we analyze this sentence we see the following:

• If the statement on its face is true, then we must conclude that the statement is in fact false;
• If the statement is on its face a false statement, then we can conclude that the statement represents a true fact.

We must conclude that the statement is self-contradictory. The fact that a statement is self-referential does not mean that it is contradictory, but it does hold the potential to be self-contradictory since it may contain information which implies two equally valid logical arguments which are not consistent. See comment for more.