# Logical Paradox **URL:** \ **Category:** Factual Questions\ **Created:** [September 6, 2002, 9:55am UTC](https://boards.straightdope.com/t/logical-paradox/127260 "2002-09-06T09:55:07Z")\ **Posts on this page:** 10\ **Page:** 4
**Author:** ![Princhester](https://avatars.discourse-cdn.com/v4/letter/p/3e96dc/32.png) [@Princhester](https://boards.straightdope.com/u/Princhester)\ **Post date:** [September 9, 2002, 2:38am UTC](https://boards.straightdope.com/t/logical-paradox/127260/62 "2002-09-09T02:38:00Z")
> [@](#): > > \*Originally posted by ianzin \* > 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. Disagree. You say that if A2 is true then the event does not happen. Why? A2 is a statement about the prisoner’s state of mind. He may well not know whether A1 is true or not until it happens. Therefore he may be uncertain. Indeed, the fact that the judge has said that he _will be_ uncertain of itself carries the implication that the truth or otherwise of A1 is in doubt. _If the prisoner assumes A1 to be true_ then A2 is false. But the problem as stated by you does not state that he _does_ make that assumption. Therefore he may well be uncertain. A1 may be true, and A2 may be 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. No. It is not that you are “trying not to expect something which you have been told will happen”. It is something much simpler. You simply do not know whether what you have been told is the truth. So you don’t know whether to expect it. > [@](#): > > 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. No, again you are making a contradiction where none exists. _If_ you accept A1 then A2 is false, but nothing about the problem allows you (as the prisoner) to know whether to do so or not. Therefore, whether A1 is or is not false does not have any bearing upon whether or not A2 is true or not. A2 is dependant only upon your subjective state of mind. If you accept A2 that means that you do not know whether to accept A1. That does not mean that A1 is false, it just means that you do not know whether it is false or not. > [@](#): > > 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. Well IMHO you are starting to get somewhere here, but go wrong in the fifth sentence. A2 only becomes a nonsense _if_ A1 is taken to be accurate by the subjective ‘you’ in A2. So in other words your fifth sentence should read “Assertion 2 becomes a nonsense, _if ‘you’ have reason to know A1 true_”. ---
**Author:** ![jb\_farley](https://avatars.discourse-cdn.com/v4/letter/j/9d8465/32.png) [@jb\_farley](https://boards.straightdope.com/u/jb_farley)\ **Post date:** [September 9, 2002, 6:20am UTC](https://boards.straightdope.com/t/logical-paradox/127260/63 "2002-09-09T06:20:22Z")
i was watching a movie the other day. at one point, two actors (one kinda looked like randy quaid, but he wasn’t randy quaid- more like randy quaid mixed with pete posthelwaite (sp?)) were arguing with one another over a tabletop. the first (quaidish) gestured wildly to a box on the table, and stated, “Not only is there not an apple in this box, but you are not listening to me _right at this moment_.” the second (I recognized him, but in that non-recognizing way, as in, "Wasn’t he on… The Hogan Family? That friend who died of AIDS? No, wait, or was he Greg on Head of the Class? Damn!) looked at the box, swallowed once, and shouted “DAMN!” best movie i ever saw. jb ---
**Author:** ![partly\_warmer](https://avatars.discourse-cdn.com/v4/letter/p/bc8723/32.png) [@partly\_warmer](https://boards.straightdope.com/u/partly_warmer)\ **Post date:** [September 9, 2002, 1:47pm UTC](https://boards.straightdope.com/t/logical-paradox/127260/64 "2002-09-09T13:47:17Z")
**lucwarm** , I’ve made several points demonstrating the problem: 1) Isn’t a paradox, 2) Can be solved, 3) Isn’t standard English, etc. Your question “Are you saying that the judge’s claim - that the prisoner would be surprised - is false?” is irrelevant to my position. (I didn’t answer before because I assumed you’d gleened what you needed from the other posts.) I’m saying the scenario is, essentially, nonsense. That’s my claim. So to make statements about what is and is not in a nonsense statement is very, very close to pointless. ---
**Author:** ![muppeteer](https://avatars.discourse-cdn.com/v4/letter/m/57b2e6/32.png) [@muppeteer](https://boards.straightdope.com/u/muppeteer)\ **Post date:** [September 9, 2002, 2:35pm UTC](https://boards.straightdope.com/t/logical-paradox/127260/65 "2002-09-09T14:35:13Z")
Ok so isn’t the flaw: the prisoner knows that when he wakes up on day one, he has a 1 in 7 chance of dying. Day two - 1 in 6, day 3 - 1 in 5 and so on until day 7 which is certain death. Taking the prisoners line of reasoning: Assertion: Killing is invalid on day 7 (true because death is certain) =\> Killing is invalid on day 6 because the set size of maximum days in prison has been reduced and day 6 is now the last day. (and so on). This is flawed because the set size is not reduced any farther; there are still six (original number of days - 1) days for the warden to spring the blindfold. So if he gets to day 5 alive, he knows he may or may not die (1/2 prob). Day 4 - 1/3 prob. Day 3 - 1/4 prob. etc. Assuming the prisoner is a reasonable man, you would expect him to be less surprised of dying with a 1 in 1 chance than 1 in 6. So exactly how surprised does he have to be? ---
**Author:** ![lucwarm](https://avatars.discourse-cdn.com/v4/letter/l/e19adc/32.png) [@lucwarm](https://boards.straightdope.com/u/lucwarm)\ **Post date:** [September 9, 2002, 4:21pm UTC](https://boards.straightdope.com/t/logical-paradox/127260/66 "2002-09-09T16:21:11Z")
> [@](#): > > \*Originally posted by partly\_warmer \* > \*\ ***lucwarm** , I’ve made several points demonstrating the problem: 1) Isn’t a paradox, 2) Can be solved, 3) Isn’t standard English, etc. > > Your question “Are you saying that the judge’s claim - that the prisoner would be surprised - is false?” is irrelevant to my position. (I didn’t answer before because I assumed you’d gleened what you needed from the other posts.) > > \*\* Well, here’s what you said in a previous post: > [@](#): > > 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.** (emphasis supplied). It sounded to me like you were saying the judge’s claim - the the prisoner would be surprised - was false. If you’re not saying that, then I have no idea what you are saying. \*\* > [@](#): > > I’m saying the scenario is, essentially, nonsense. That’s my claim. \*\* Well, I have no idea what you mean when you say that. It’s certainly possible to set up the situation I described earlier and have the judge’s claims turn out to be 100% correct. ---
**Author:** ![PBear](https://avatars.discourse-cdn.com/v4/letter/p/839c29/32.png) [@PBear](https://boards.straightdope.com/u/PBear)\ **Post date:** [September 9, 2002, 7:55pm UTC](https://boards.straightdope.com/t/logical-paradox/127260/67 "2002-09-09T19:55:12Z")
Agreed, muppeteer. IMHO, that’s another very good (and succinct) statement of the solution. ---
**Author:** ![Somnambulist](https://avatars.discourse-cdn.com/v4/letter/s/ed655f/32.png) [@Somnambulist](https://boards.straightdope.com/u/Somnambulist)\ **Post date:** [September 10, 2002, 7:46am UTC](https://boards.straightdope.com/t/logical-paradox/127260/68 "2002-09-10T07:46:24Z")
Consider the problem as drawing balls from a jar. All 7 balls in the jar are the same except one, which is marked with a letter ‘E’. The Judge (J) makes the rules; Condemned man © plays the game. C must draw balls from the jar and if it is the one with the letter ‘E’ he is for the chop. The first chance is an event called ‘Monday’, the second event ‘Tuesday’, etc. J’s Rule 1: C will be surprised when he draws the ‘E’ ball. This means that C will not know which event he will draw the ‘E’ ball and implies that each event must have a probability less than 1 (say 1 in 7). This also means that in 7 events there is a possibility that the ‘E’ ball will not be drawn at all. This would be true if each ball is replaced back in the jar after it has been drawn (call this GAME 1). J’s Rule 2: C will draw the ‘E’ ball next week (in the next 7 events) This implies that after the first event, if the ‘E’ ball is not drawn, the probability that it will happen in the next event is higher. In this instance the last event that uncertainty can occur is ‘Saturday’ where the probability of drawing ‘E’ is 1 in 2, and allows for the possibility of and event ‘Sunday’ with a probability of 1. This is the equivalent of each ball being discarded after it is drawn (GAME 2). These rules contradict. We either play GAME 1, replacing each ball after it is drawn (which maintains uncertainty but allows for the possibility of not drawing ‘E’). Or we play GAME 2, discarding each ball after it is drawn (which ensures an outcome within 7 events but allows for no uncertainty in the 7th event). C exploits this inconsistency and asks to play GAME 1 but with the ‘E’ ball removed. J agrees but cheats and secretly replaces all the balls with ‘E’ balls before the second event on ‘Tuesday’, achieving his desired outcome but in blatant disregard of his original rules. ---
**Author:** ![Princhester](https://avatars.discourse-cdn.com/v4/letter/p/3e96dc/32.png) [@Princhester](https://boards.straightdope.com/u/Princhester)\ **Post date:** [September 10, 2002, 8:06am UTC](https://boards.straightdope.com/t/logical-paradox/127260/69 "2002-09-10T08:06:15Z")
Why is J cheating, Somnabulist? You only stated two original rules for J, one being that C will be surprised, (which C will be, come Tuesday) and the other being that C will draw an “E” ball, which he does. The rules about drawing balls blah blah are _yours_ not J’s. Going back to the original OP, there was _never_ any rule that on a particular day execution would not be imposed (ie that drawing of the E ball would be made a certainty in terms of your re-writing of the problem). ---
**Author:** ![Somnambulist](https://avatars.discourse-cdn.com/v4/letter/s/ed655f/32.png) [@Somnambulist](https://boards.straightdope.com/u/Somnambulist)\ **Post date:** [September 10, 2002, 9:16am UTC](https://boards.straightdope.com/t/logical-paradox/127260/70 "2002-09-10T09:16:36Z")
Princhester, My mistake, sorry, in the OP J does not cheat, he simply lets C go when C exposes the inconsistency in his rules. Cheating In the extension of the problem (which allows J to complete his originally stated intentions) he concedes the inconsistency and agrees not to execute C (but then does so on Tuesday). By agreeing not to execute C, they are no longer playing the execution game and the rules do not apply. Let’s say they are playing the Jailing game (C may not be detained but not executed - there is no E ball in the jar). J then breaks the rules of the new game in order to satisfy the rules of the old game. Sounds like cheating to me. Drawing balls blah blah Yes they are my rules to the game in my analogy of the problem (apologies if you find it tedious). If you think they do not illustrate the inconsistency in the Judges’ ‘rules’ please elaborate. I contend that in the OP there is a rule that on a particular day execution would not be imposed, namely any day when C is not surprised. That would arise by default on the Sunday had C survived that far and it is that argument that C uses to win his reprieve. If you do not think that is the case then one of us is misinterpreting the problem. ---
**Author:** ![Princhester](https://avatars.discourse-cdn.com/v4/letter/p/3e96dc/32.png) [@Princhester](https://boards.straightdope.com/u/Princhester)\ **Post date:** [September 10, 2002, 11:51am UTC](https://boards.straightdope.com/t/logical-paradox/127260/71 "2002-09-10T11:51:04Z")
Firstly Somnambulist an apology. I used the phrase “blah blah” just meaning “etc” I wasn’t trying to be dismissive but I accept I came across that way. I apologise for that. The scenario has been formulated by so many people in so many ways that perhaps what I am about to say is wrong, I confess I haven’t read over the complete thread again to check. But nonetheless, as I understand it, no one is suggesting a formulation of the scenario in which the judge indicates that he will not execute the prisoner but then does so. I suspect that is a misreading. Two scenarios have been discussed. In the original OP, the judge lets the prisoner go because he is convinced by the prisoner’s argument that by his (the judge’s) own rules, the prisoner can never be executed. But it is obvious that the judge is being hoodwinked by the prisoner in such a scenario, as is easily demonstrated by the second scenario: namely where the judge quite correctly dismisses the prisoner’s argument as being nonsense and maintains his order that his sentence is to be carried out in accordance with his rules. On the Tuesday the prisoner opens the door not knowing whether he will receive breakfast or the blindfold. He receives the blindfold. Thus, both the judge’s rules are fulfilled despite the prisoner’s seemingly coherent argument that this could not occur. [Previous page](https://boards.straightdope.com/t/logical-paradox/127260.md?page=3)