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I came across the below problem in book: 101 Philosophical problems. And wondering for a while what the possible answer could be? Any thoughts?

Problem

Now Judge Dread had had many disagreeable people before him, but this one, who styled himself ‘the Philosopher’, despite never having studied the subject, had really annoyed him. Dread says:

‘I intend to teach you the value of honesty, prisoner. You have been found guilty of being a crook and a swindler and of repeatedly and systematically lying to the court to try to save your wretched skin. Well, justice has caught up with you now, my friend. The sentence of this court is . . . ’ (here the Judge pauses for effect and dons a pair of black gloves and a little black hat) ‘ . . . that you be taken from here to a place of execution and hanged by the neck until you are dead.

‘ . . . BUT, as I am a magnanimous judge, I shall give you one more opportunity to learn the value of truth. If, on the day of your execution, you sign a statement making one true declaration, the sentence will be commuted to ten years imprisonment. If, on the other hand, your statement is, in the view of the Chief Executioner, false, the sentence will be carried out immediately. And I warn you,’ Dread adds, seeing his words having no effect on the crook, ‘the Chief is a member of the Logical Positivist Executioners’ Club and will dismiss any metaphysical nonsense as false, so don’t try any of your tricks on her! There, now you have one day in which to make your choice!’

At this the jury applaud at the severity of the sentence and everyone in the courtroom looks at the defendant, pleased to see such a villain get a heavy sentence, coupled with the humiliating public truth declaration. But, strangely, the Philosopher just smirks back as he is led away to Death Row. The day of the execution arrives and the crook, beaming, signs a declaration which is handed to the Chief Executioner who reads it with growing bewilderment. Then, snarling, she crumples it up and orders the Philosopher be released, with no penalty whatsoever to be imposed.

What could the prisoner have said in the statement to have saved himself?

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Logical positivism can mean many things, but, essentially, it means that the only meaningful truth is that which can be empirically observed and verified.

Thus the accused needs to sign a declaration which the executioner can verify to be true for herself.

The prisoner could then e.g. have signed a declaration saying

I will be executed.

The only way in which this statement can be verified is if the executioner actually proceeds with the execution (assuming, of course, that there aren't other executioners or assassins out after the prisoner). But in that case, the statement will be verified as true, and thus the executioner should not have executed the prisoner.

So the executioner must find another way to declare the sentence false: but this is practically impossible. The only way the sentence can be declared false is if the executioner does not execute. Hence, the executioner is stuck in a limbo of some sorts: she can neither declare the sentence as truth (since that would lead her to wrongfully executing the prisoner) nor can she declare it as false (since she has not yet observed the execution of the prisoner).

Thus, since she cannot declare it as false, she cannot execute, and since she cannot declare it as true, she cannot imprison him for 10 years (those were the conditions). In other words, she can do nothing, and the prisoner simply walks away.

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