Logical Paradox

I have a logical paradox, I know that there is a flaw in the agument but I can’t find it.

A man is to be killed by firing squad. the judge says that his crime was so heinous that, to increase his suffering, he cannot know which day he will die until the morning of the execution.

The execution will happen one day next week, from Monday to Sunday. Each morning at 09:00 there will be a knock on his cell door; the guard will either give him breakfast, or a blindfold. But the judge stipulates that until he opens the door he will not know which one.

The man thinks for a while then tells the judge he can’t possibly do it.

“You cannot kill me on the last day, because if I wake up on Sunday morning I’ll know that a blindfold awaits me that day”

The judge reluctantly agrees that the last day is out.

“So you can’t kill me on the Saturday either, we have already discounted Sunday, so if I wake up on Saturday morning I’ll know that you have run out of options and that’s the day you have chosen”

Again the judge agrees. And so it continues until the man has discounted all days and the judge has to let him go.

What’s the dope ?

By the reasoning set out above the man concludes that he cannot be executed at all. He is therefore surprised when, on Tuesday, the guard knocks at his door and gives him a blindfold. Thus the judge’s prediction that he would not know in advance which day he would be executed is correct.

I know, but all that you have done is prove that the man’s conclusion is incorrect, which I already know. I want to find the flaw in the logical step that brought him there.

This has also perplexed me greatly for years. I think the fallacy is that on Saturday (or any day before), the days that have been discounted subsequent to that day don’t count. But I can’t work out why.

There are no such things as paradoxes. I do realize that people will want to contest this, however, this story is a pretty much ideal example of the problem.

These situations are created by inexact language, and various other logical flaws.

In this case, the judge has made a statement that is logically incoherent, and one which he cannot act upon. There’s no paradox at all. The judge made a claim that was incorrect. Pigs don’t fly, no matter what he might say.

One can get into the details of exactly why the judge said something nonsensical, but there’s really no need. This is like grammar school arguments: “If God can do anything, how can he lift an immovable rock?”

The problem isn’t with God, but with the definition of a word that man created out of whole cloth, in this example, “immovable”. The cosmos is not required to conform to arbitrary, inconsistent labels humankind places on it.

Well now I’m more confused

I thought that the god example was a thought experiment to disprove omnipotence.

As far as the rest of you argument goes, are you saying that you consider the man’s reasoning to be sound and the judge’s statement (that the man will not no which day he is to die) to be logically impossible ?

If you think that the lanuage is in exact please feel free to re-phrase it.

I disagree. When the prisoner is woken up to be executed on Tuesday, the judge’s claim is pretty clearly correct - the prisoner is surprised.

I think that the logical problem in the paradox is that the prisoner needs to carry his reasoning to its conclusion. If it is truly impossible for the sentence to be carried out (which is what the prisoner is claiming), it follows that the prisoner can be executed on any day - even the last day - and the execution will be a surprise. In effect, the prisoner’s reasoning is self-defeating.

By the way, my instinct is that you are correct - that there are no paradoxes.

The judge could resolve the statement to be logically coherent by specifying that the execution would take place by surprise, but at some point within an indefinite period. I can’t think of any other way that it could remain a surprise.

Ta, lucwarm. (I just noticed your name is a hop, skip and a jump from mine. No offense – and no copying intended! Mine’s an allusion to weather forcasting.)

The problem with the OP question is multi-layered. It’s wrong on every layer. I really don’t want to argue each detail, because that tends to give the impression that each detail is important to support the overall position. This isn’t the case. The paradox falls apart, as all paradoxes do, because we inexactly describe or inexactly understand a situation.

What the judge says or thinks about the future is irrelevant. He can say pigs fly now, or that pigs will fly by the end of the week. Both statements are incorrect. What the judge says or intends to do makes no difference at all. The only problem is with his thinking.

partly_warmer, can you tell me why the OP is inexcatly understood ?

I’m all for the agrgument that there is no paradox and it will fall apart if properly scrutinised, but I’d like someone to explain how to do this.

Also I don’t understand your flying pigs statments, these statements are not logically incorrect, they are factually incorrect.

First, just so we are on the same page, please note that I am talking about the problem in its more traditional formulation, which goes something like this:

Ok, now let me ask you this: Are you saying that the judge’s claim - that the prisoner would be surprised - is false?

I think that you’re cheating.

It’s intuitively obvious that the judge can surprise the man, so I beleive that the man’s conclusion is incorrect (this is all that you have show by suprising him on tuesday).

So if the man’s conclusion is incorrect, which we all seem to agree on, there must be an error in his reasoning.

Where is it ?

The elimination of the last day of the week is based entirely on the assumption that the prisoner will survive that long.

[Post eaten unexpectedly by hamsters]

This is known as the “unexpected inspection” paradox and it casts doubt on a major technique of game theory. AFAIK it has not been solved.

Permit me to alter the story a little, Suppose the OP is entirely the prisoner’s internal monologue. He concludes that it is impossible to be surprised on any day, yet on some day he is handed the blindfold.

As a number of people have already said, the use of backward induction to rule out the last day must be invalid in some way. Imagining yourself to be at the last node of the game which you subsequently rule out as a node you could possibly find yourself at must be a problem.

A similar problem exists with the conclusion that coopertaion is irrational in a finitely iterated prisoners’ dilemma game. Since the gains from cooperating in the last round are zero, it is rational to defect. So in the second-last round there are no gains from cooperation and the whole thing collapses.

Reasoning about what would happen if a circumstance occurred which reason tells you won’t happen is evidently faulty in some way. It’s a paradox, not in the sense of two true things yielding a genuine contradiction, but in the sense of two seemingly true things yielding a contradiction: paradox is" truth standing on it’s head to draw attention to itself".

There is no paradox here.

The prisoner has simply discovered that all the comments by the judge are not consistent or possible. That’s all.

The prisoner must be surprised by a finite, predictable variable. Doesn’t compute.


I like this idea, you’re attacking the premis, but I disagree. The premis is not

“entirely on the assumption that the prisoner will survive that long.”

It is based on the idea that IF the prisoner survives that long then the last day is no longer an option. As the last day was only ever an option IF the prisoner survived that long, it seems that we must now discount it all togther.


Thanks for that, I wonder if you could explain the cooperation paradox to which you allude.

I think you are trusting the logic over your common sense. If they knocked on his door on wednesday and gave him a blindfold, it would be a suprise.

lucwarm and murphyboy99, a person can believe or say something that’s impossible. That’s the problem, here. The long, theatrical explanation of the situation as presented by the story obscures the basic fact: the judge has created a set of conditions that don’t allow the prisoner to be executed.

There is no paradox. It’s as if he said “You will be executed on the first day of the week that doesn’t include the letters ‘day’”. I.e., it will never happen.

Of course, the verbal contradiction is more artfully concealed than that. The listener gets wrapped around the unexpected process of having the days eliminated one-by-one.

But with a simple change of wording the judge’s sentence works. “You’ll be executed soon, but you won’t know what morning until they come to get you.” It only takes the stipulation that it must happen by Sunday to be removed.

Agree partly.

Let me restate my position on this.

In order to have a paradox, one must claim that “A” and “not A” will exist at the same time.

“A” here is the judge’s sentence in its entirety. “Not A” is obvious, but involves the idea that the prisoner can not be surprised LOGICALLY.

Of course the prisoner can be surprised emotionally(even if he knew for certain the time of his execution). But emotions are what we’re discussing .

The prisoner has proven “A” simply is impossible. So there’s no paradox.