FWIW, even if you do disallow self-reference, we can sneak it back in. That’s the whole point of Gödel’s work.
Oh, one other thing: the paradox referenced in the OP is often cast as a teacher attempting to give a surprise exam to her students, so we might get somewhere searching on that.
Ok, but understand that my thinking changed after I realized that the prisoner was wrong, and that there was a way he could be executed (see above). I fell into the same trap the judge did: Believing what the prisoner said.
At first, I maintained that the judge had made inconsistent statements. Then I realized that the prisoner and the judge only believed the judge had made inconsistent statements. Ok. Now the question is “What did the prisoner and the judge believe, that was inconsistent?”
I’m not sure where our understanding isn’t meshing. The judge said he would do thing X only when thing Y happened. His belief is that X must happen, sooner or later. But actually Y couldn’t happen at all. So maintaining that X and Y must both happen is “inconsistent”.
(Note that the last two paragraphs only relate to the judge’s and prisoner’s understanding of the situation – that they’re really both mistaken. The prisoner can be executed by confusing him as to what day it is. And the judge keeps his word.)
IMHO, jb_farley hit the nail on the head. To review, his explanation was this:
As far as I can tell, no one has commented on this analysis.
Nope. Your post is a group of intelligible sentences which make reference to one another. None of those sentences (except the one under discussion) makes reference to itself.
The sentence “This sentence is false” isn’t just outside the ordinary use of language. I was noting how non-standard it was. Read: “it breaks the rules”. The important thing, though, is that it has no meaning. There’s no information content. Someone saying the sentence is not communicating to someone else (unless it be their communication is to confuse them, but as I pointed out, any number of nonsense statements could achieve exactly that effect).
The language could theoretically be changed, as you suggest, to attempt to handle self-referential sentences. Offhand, I can’t think of any reason to do so. I have a feeling you are hinting that if language could be modified to handle “more universal” truths, that is should be. Broadly speaking, that seems like a good thing.
Howsoever, it is not possible to remove inherent contradictions in language, any more than it’s possible to remove them from mathematics. (Referring to Godel’s Theorem.) “Paradoxes” mostly highlight imprecise speech, inexact semantics and misunderstandings. However some paradoxes must reflect the underlying inconsistency between paradigms. Those “paradoxes”, when we get to them, are just manifestations of variations between different theories. Theory A isn’t complete, Theory B isn’t complete, and some “paradoxes” describe the differences. We haven’t been dealing with those sort, however. We’ve been dealing with misuse of language, misdirection, and reading into the meaning of language things that were never intended to be there, such as self-referential truth/validity.
Hmm…
This argument has gotten just the tiniest bit off track, but that tiny bit makes all the difference. The OP states that the prisoner can not know which day he will be executed and the matter of whether the prisoner is surprised has absolutely nothing to do with it.
Consider this:
You are driving 50mph in a 35mph zone. You notice a cop at the next intersection with a radar gun pointed at you. Although you would not be surprised if you were given a citation, you do not know that the citation is forthcoming.
Now consider jawdirk’s game in a different manner: The prisoner is given 7 cards; 6 have ‘breakfast’ printed on them, and the 7th has ‘blindfold’ printed on it. Each morning at 09:00, the guard, carrying either breakfast or a blindfold, knocks on the door. At this point, the prisoner has to present one of the cards. If the prisoner presents the blindfold card and the guard is carrying the blindfold, then the prisoner has demonstrated that he does, indeed, know which day he has been slated for execution and, thus, cannot be executed. However, should he present the ‘blindfold’ card and the guard has brought breakfast, the prisoner has forfeited his chance for freedom. On what day can the prisoner present the ‘blindfold’ card with 100% certainty that the guard will be carrying the blindfold? By the reasoning in the OP, presenting the card on Monday will save the man’s life.
If i were the guard, i’d bring the blindfold Tuesday.
Good points, octothorpe, and your post made me think of this problem in two new ways!
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What does it mean to know something? Suppose I have a 6-sided die, and I know that when I roll it, it will come up 1-6. Then I roll it and it comes up 9. Well what the heck am I supposed to do then? Pretend like it actually did come up 1-6? More likely I would change what I know based on new information. Same thing with the prisoner. On Monday, he knows that he will be executed on Monday. If he’s not, then with his new information, he knows that he will be executed on Tuesday. And so on.
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Causality. What the prisoner knows informs what decision the judge makes, and what decision the judge makes informs what the prisoner knows. The prisoner knows he’s being killed on Monday, so the judge declares that he be killed on Tuesday (for example). And the prisoner can now reasonably expect that Monday is safe, so he knows he will not be killed on Monday. And so it goes, back and forth, an infinite sequence which does not converge. “You and I can change teams freely. You will be happy when you’re on my team, and I will be happy when I’m on the opposite team as you. When we’re both happy with our teams, we’re decided.” No logical inconsistency here, but we can never be decided.
octothorpe, there is no distinction between “not knowing” and “being surprised” in the OP. The phrases were intended to be synonomous.
The judge is not inviting the prisoner to play “Russian roulette” with seven cards, and telling him he can get out if he guesses. Don’t change the nature of the problem! He’s threatening the prisoner – using a statement that the prisoner convinced the judge was inconsistent.
As pointed out, the prisoner has hoodwinked the judge. He can be executed.
Achernar, full speed ahead with the creative thinking, but the example with the die violates the basic principle of game-playing that the players all agree on: One of six sides must come up. If another number appears, or the dice is lost, or explodes, then any player would have a right to claim the throw was invalid.
True about the die-rolling rules, partly_warmer, but I didn’t mean it in the context of gameplay (although, since this problem involves game theory, I should have realized that it would come up). I could just as easily have used another example. I know that my cat can’t start talking to me. If my cat does start talking to me, I’ll have to change what I know.
There’s no paradox. Jb farley’s post is the answer. The prisoner is making the false assumption that because he can logically eliminate some days, he can then use that same logic to eliminate all days.
Sorry to butt in, but it seems to me that ultrafilter’s entire post makes reference to itself.
Anyway, as far as “this statement is false” goes, I don’t think there’s a paradox. It only seems that way, IMHO, because we assume that all statements must be either true or false. IMHO, this assumption is incorrect.
Suppose I present you with the following problem in boolean algebra:
The answer is that there is no value of X for which X <—> ~X.
In algebra, we don’t assume that all equations have solutions, and there’s no logical reason to make the analogous assumption in dealing with ordinary language.
Anyway, partly, it looks like you still haven’t answered my earlier question:
Are you saying that the judge’s claim - that the prisoner would be surprised - is false?
This paradox is very well-known, albeit in many different guises, and has been resolved. Martin Gardner’s books contain several references to the paradox of the Unexpected Hanging, which is the name typically given to just one common version.
The flaw lies in the Judge’s original assertions. Specifically, it lies in the fact that while the Judge’s assertions seem, intuitively, to be straightforward, they are actually inconsistent with each other. It is this subtle, hidden logical inconsistency which sets up the paradox.
Because there are so many themes and variations, let me strip the story downto its basics.
An authority figure makes two assertions:
A1. An event will happen on one of several consecutive days in the future.
A2. You will not know on which one of these days it’s going to happen. That is, the event will be unexpected.
Upon first reading, and if we just go by common sense and intuition, there seems nothing wrong with these statements. However, to see the flaw, try to assess whether these statements are TRUE or not.
Plainly, if either statement is not true, then there is no contradiction, no puzzle, no paradox. So the paradox only arises if both assertions are true. Now, consider the case where only one possible day remains.
If (A1) is true, then the event must happen on that day, and is therefore expected.
If (A2) is true, then the event must not happen on that day.
So it follows that in this context, it is not possible to regard both of the assertions as true. In fact, since they are mutually exclusive, it is only possible for one to be true, at most. And we have already seen that there is no paradox unless both assertions are true.
Elaborating on this, consider a special case where the authority figure only mentions one possible day, like this:
A1. An event will happen tomorrow.
A2. You will not know when it’s going to happen. That is, the event will be unexpected.
If you try to accept both of these assertions as true, you put yourself in the position of trying not to expect something which you have been told will happen. If you accept (A1) then the event is expected. If you accept (A2) then it isn’t. This doesn’t make sense, because if you form an expectation based on true information, you cannot consistently be asked to un-expect it.
Martin Gardner’s analysis (working from memory here) continues with the further example of someone placing a box in front of you and saying:
A1. There is an apple in the box.
A2. You will not expect to find an apple in the box.
This tries to subvert the very notion of what it means to ‘expect’ something. An expectation is based on information. If A1 is taken to be accurate information, then of course one expects to find an apple in the box. Assertion 2 becomes a nonsense, unless there is reason to declare A1 as untrue.
It is the same with the story in its original form. The two separate assertions sound harmless enough, and the range of ‘several possible days’ encourages us to start going through the regressive analysis of eliminating the last day, then the one before that, and so on. But in doing this we’re falling for a three card trick, and missing the real flaw which lies in two two assertions. As we have seen, they cannot both be true and consistent precisely because of what would happen on the last available day.
Therefore, going back to the OP, we are entitled to say that although the Judge’s assertions do not, at first sight, strike us as contradictory, in fact they are logically inconsistent (in that they can give rise to the proposition that one should simultaneously both expect and not expect something). Hence they cannot both be true.
I’d just like to point out that I actually wrote a paper on this in my college days, and that I’ve used my 1000th post here.
I agree with ianzin’s post, however I believe there may be a flaw in the prisoner’s reasoning as well.
Considering his argument as an exercise in probability, the initial statement “I cannot be executed on Sunday” is incomplete as it does not allow for the possibility that he will be executed earlier.
The statement (prediction) should be “I cannot be executed on Sunday provided that I have not been executed earlier”.
This makes the entire logical argument dependant on the condition that he has not already been executed, and when the blindfold appears on Tuesday that condition fails and his entire argument collapses.
IMHO, ianzin, you’ve resolved the paradox the wrong direction. On your reasoning, the prisoner is right and the judge’s orders can’t be executed (so to speak). Whereas, intuitively, we know they can be, and the question is how in face of the prisoner’s reasoning.
Also, FWIW, I find your reasoning unpersuasive. You assert, correctly of course, that the judge cannot sentence the prisoner to be executed on a date certain and also that the prisoner shall be surprised at such execution. Well, yeah. But how do you make the leap from there to the problem stated, where the execution is any one of a number of stated days (that they’re consecutive is irrelevant)? As far as I can tell, all you say is that the day-by-day regressive analysis makes the two cases equivalent. Sorry, but whether the days can be eliminated seriatum is precisely the question in issue.
So this sentence is disallowed, eh?
What rules? Ordinary language doesn’t have any rules; that’s what makes it so hard to deal with. Colorless green ideas sleep furiously and all that.
Being a semi-amateur logician (looking to get a PhD in that area sometime in the next ten years), I’m interested in extending our language to model whatever we can that exists. If someone else can say it, we should be able to too.
Disagree. What about the possibility of believing in a correct fact for the wrong reason?
Let me give an example:
Suppose, in the original puzzle, the prisoner is absolutely sure that he will be executed on Monday (since the other days have been ruled out). Monday roles around, and he’s not executed. So now it’s Tuesday, and the prisoner is again absolutely sure he will be executed today. And he is.
Was the execution a surprise, or not?
Well, it depends on how we define “knowledge,” but I would say that the answer is YES. If so, then it’s possible that (A2) is true AND the event happens on that day.
The sentence “This sentence is disallowed, eh?”, because it doesn’t conform to normal formulations of logic, can’t be taken to necessarily have content. It may, it may not. I used to work as a professional linguist, and off-hand, I’m not aware of any body of “rules” specifically relating to self-referential statements. (Nothing in high school or college texts, for example.) The very fact that there is dispute about the sentence, and not two, but several opinions about it suggests its meaning and/or validity are in question.
While Chomsky is hardly my favorite linguist, I begrudgingly admit that some parts of understanding human language must be encoded at a neurological level. Those cells must follow rules. More broadly, if there were really no rules, you and I couldn’t agree that
*Colorless green ideas sleep furiously
is meaningless collection of words. Which we do, I take it. The rules I’m talking about are rules about the actual use of the language, and not the “rules” that a sixth grade teacher who majored in biology believes are absolutely true. Without some form of rules, no matter how complex, contradictory, vague and misunderstood, communication would be impossible.
I’ve been interested for some time in expanding the language modifying words with emotional content, so that people can express in a quantifiable way, if they wish, how strongly they feel about something. Some modification to avoid misunderstandings like this:
She: “What movie do you want to see tonight?”
He (not really caring): “Star Wars, Part 13.”
She (throwing out another possibility) "How about “Romeo and Hamlet?”
He (really hating the idea): “I’d prefer Star Wars.”
She: “Well, I want to see Romeo and Hamlet.”
Potential impasse. And the couple may be on the way to having an argument.
It would be easier if the “wants” were quantified. I.e., he wants to see Star Wars, but only “strength 1”, she wants to see “Romeo and Hamlet” but only “strength 2”, he hates the idea of seeing “Romeo and Hamlet” – “strength 10”. This would enable her to see that “Romeo and Hamlet”, as a mutual decision, is a poor choice.
Agreed that statements may be other than true or false. Specifically, they may be invalid/meaningless.
ultrafilter’s paragraph as a whole talks about itself, but the single sentences within, taken in isolation, do not.
Answering your question (I’m tussling slightly with your choice of words, which is why I didn’t answer earlier):
- The judge intends for the prisoner to be surprised. He obviously presents the situation to the prisoner in that light.
- The prisoner, rightly or wrongly, claims he cannot be surprised.
- The prisoner convinces the judge.
- At the end of the story, both the judge and the prisoner believe the prisoner cannot be surprised.
I hope this answers it.
The problem I have with this story is that it’s obviously contrived, and makes several strained assumptions.
- It assumes the judge can’t change his mind.
- It assumes the judge is allowed to put an extra-legal qualification on the prisoner’s sentence. The prisoner was sentenced to death, not to play word games with the judge.
- It assumes the judge buys the prisoner’s argument (It’s apparent that some people here do not.) What’s stopping the judge from saying, rightly or wrongly, “Very interesting, but I don’t agree with your logic. Bye.”
- And most important of all, it assumes that the prisoner really cannot be surprised, which as I have pointed out is not true (just drug him and don’t let him know what day it is).
The person who framed the story is playing semantic games. Notice I can make the problem even more “difficult” by saying:
- The judge can’t change his mind.
- The judge has legal authority to do whatever he wants.
- The judge has to buy the prisoner’s argument if he can’t think of a response that satisfies the prisoner.
- The prisoner cannot be drugged or deceived as to what day it is.
Does the artificial nature of the problem begin to show through? As soon as anybody comes up with an answer, the author of the story can say: “Ok, now suppose you can’t do that.”
No, that’s not an answer at all. For one thing, when I posed the question, I was really careful to describe a version of the paradox where there is no dialogue between prisoner and judge. This was to avoid the very confusion that you are now injecting into the discussion.
Yet you responded with a discussion of the more complicated scenario that in any event is unresponsive to my question.
At this point I’ll assume that you are unwilling or unable to answer what was really a very simple question that sought to clarify your position.
In any event, this seems to be turning into a debate.
Ok, and here’s what he had said (and what you responded “nope” to):
Again, I’m sorry to butt in, and, sorry if I’m helping to turn this into a debate.
Okay, now I’ve done my homework. Reviewing the surprisingly large literature on this subject has confirmed a long-held opinion. Though formal logic may be rigorous, even difficult, it’s not a particularly sharp instrument. I mean, how can it be possible that a problem this simple has resisted strenuous assault for over fifty years?
And that’s the joke. There have been many proposed solutions, both by professors and students, but not one has a consensus behind it. Moreover, a couple hundred pages later, I still haven’t seen anything as clear or incisive as jb farley’s two paragraphs. Two virtues that I’ll take over formal rigor any day.
For those inclined the other direction, I would recommend Ned Hall (philosophy prof at MIT), How to Give a Surprise Exam, 108 Mind 647-703 (1999) (pdf of draft MS available here). He reaches essentially the same conclusion as farley, but with much more rigor. He also takes 66 pages to do it (in MS, the article is a bit shorter). To similar effect, see Elliot Sober (philosophy prof at U. of Wisconsin, Madison), To Give A Surprise Exam, Use Game Theory, 115 Synthese 355-373 (1998) (pdf of MS available here, plus an addendum here).
As mentioned, there is by no means a consensus for the Hall or Sober analyses, though they appear to be the most widely-cited current treatments. Folks looking for support for other theories, or just more information, might find these interesting:
Timothy Y. Chow (mathematician at Tellabs Research Center, Cambridge), The Surprise Examination or Unexpected Hanging Paradox, American Mathematical Monthly (Jan 1998); scan of standalone version of article available here et seq.; postscript version here (Google html conversion here). See also Chow’s “short version” of this article for the rec.puzzles archive.
Mark D’Cruz (philosophy prof at National University of Singapore, The Surprise Exam (p.1 of 6, with navigation links) (note: D’Cruz’s name doesn’t appear on the article, but I was able to identify him by other links on the site).
Roy Sorenson (philosophy prof at NYU), Origin Of The Surprise Test Paradox and A Solution To The Surprise Test Paradox (both undated).
Edward G. Rozycki (Ed.D.), The Pseudo-Paradox of the Surprise Test (last edited 2/25/01; “Rewrite of a paper first presented somewhere, sometime in 1969.”) (another paper supporting the farley approach, albeit achieved a bit differently.
Lucian Wischik, The paradox of the surprise examination (student paper, July 1996) (pdf available here).
Paul Franceschi (philosophy prof at University of Corsica), A Dichotomic Analysis of the Surprise Examination Paradox (April 2002) (I’ve only included this article because it’s so recent; be warned that it’s very long and extremely technical).
Not an exhaustive list. For a thorough bibliography, see the Chow article. And/or search for unexpected examination paradox, etc. on Google.
Incidentally, my apologies if any links don’t work. I tried.