It was actually my favorite course out of all of the classes I took in college. I enjoyed it so much that I did every problem in the 2 books we used, not just the ones assigned for homework, also every extra credit problem on the tests, and I took real statements from politicians (e.g. Edwin Meese) and walked through their convoluted statements to find out what they were really saying which was entertaining in some cases.
In the 1 day case he rules out the entire set of days available (which is 1 day) and he can do that with 1 simple step because and only because there is only 1 day available, the contradiction is immediately apparent.
In the 5 day case he rules out Friday, either correctly or incorrectly, but not for the same reason that 1 day case can be ruled out. He didn’t rule out Friday because there is simply no possible day at all that can satisfy all requirements (which is the 1 day case), he ruled out Friday while there is still the possibility that a different day that week could possibly satisfy the conditions.
Well, I disagree that reaching the same conclusion is the single determinant of whether the 1 day and 5 day scenario is the same. If that were true, wouldn’t Ned Hall agree that the 1 day and 5 day are the same? Yet he clearly states that he doesn’t think so and gives the 100 day as his counter example.
I’m saying he (mistakenly) rules out Friday in the 1-Day scenario in one step – which happens to rule out every day. I’m also saying he (mistakenly) rules out Friday in the 5-Day scenario in one step – which doesn’t yet rule out every day, but so that’s the whole reason I keep saying the 1-Day scenario is the first step of the 5-Day scenario: you first eliminate Friday with the exact same (mistaken) reasoning as you would in the 1-Day scenario, and then go on to eliminate Thursday and Wednesday and so on with (mistaken) reasoning.
Never mind that part about what he goes on to do in the 5-Day scenario, or the part where he’s knocked out every possible day in the 1-Day scenario. I’m just saying he uses the same (mistaken) reasoning to (mistakenly) rule out Friday whether it’s the only possible day to (mistakenly) eliminate or the first of many days to (mistakenly) eliminate.
As you’ve pointed out, the 5-Day – or 100-Day – scenario makes it possible to surprise reasoners who are too savvy to get surprised in a 1-Day scenario; such people wouldn’t reach the same conclusion in both scenarios. But that’s irrelevant so long as we’re discussing someone who does reach the same conclusion in either scenario.
He reasons correctly that he can’t accept both of the judge’s statements and have both of them come true (note: this is correct, unlike the 5 day step #1)
He then concludes he won’t be hanged due to #1 (this is incorrect)
In the 5 day:
0) He reasons that he can’t immediately eliminate the possibility that he could accept both of the judge’s statement and have both of them come true, and so he must continue analyzing the problem. I’ve labeled it step 0 so we can ignore it, but I do think it happens prior to the next step.
He reasons that if he is still alive on Friday morning, then from that point on he can’t accept both of the judge’s statement and have both of them come true. (note: there is a lot going on here regarding what he believes prior to Friday, did he change his mind on Friday, etc., etc. that can’t be ignored and is different from the 1 day and it is far from clear whether this particular step is even correct or not)
other stuff
Key differences:
1 day first step is correct and a full accounting of his predicament sits in front of him, he can walk through all scenarios and determine how it will play out
5 day first step has shed very little light on the issue and has possibly muddied the water
I wanted to start that off with post 181. Looking back at it, I see a couple of things I might have best reworded, but I still think it’s a fine place to start.
But look at why the 5-Day first step sheds comparatively little light on the issue: it eliminates Friday exactly as thoroughly as the 1-Day first step eliminates Friday – and then in one scenario there are more days to address, and in the other there aren’t.
In the 1-Day scenario, that step is a “correct and full accounting” of that one day. In the 5-Day scenario, it’s also a “correct and full accounting” of that one day – and we can then move on to analyzing other days. But insofar as it’s eliminating Friday, the reasoning is identical.
No, it’s not “far from clear” whether the step “is even correct or not.” It’s the exact same step employed in the 1-Day scenario, and likewise eliminates Friday through exactly the same reasoning. Whether more days remain to be eliminated at that point is a completely unrelated matter.
The flaw is Premise C, which isn’t given to us in the initial set-up.
Suppose the prisoner is to be hanged on Friday; on Friday morning, he will presumably know that he is to be hanged – and it follows that he will not be surprised by the hanging. Line 4 contradicts premise B – but if it really is Friday morning, then Line 7 will contradict premise A. At this point, a good reasoner will correctly jettison premise B, and be unsurprised; a poor reasoner will incorrectly jettison premise A, and be surprised. Since premise C isn’t actually in effect, the prisoner can be a poor reasoner who incorrectly jettisons premise A.
But it seems like there’s no interesting problem if you don’t throw in premise C. (Ned Hall, I think, touches on this in his article as well.) Without premise C, the prisoner will have to take into account the possibility that the judge could do things that might make him somehow forget what the judge said. Maybe hit the prisoner on the head. I don’t know! But all such possibilities render the problem completely uninteresting, it seems to me. To keep the problem difficult in the way it is supposed to be difficult, it seems like we have to assume C about the prisoner–and assume the prisoner assumes C about himself as well.
Indeed, I don’t think the reasoning given by the prisoner in most plain-english formulations of the paradox can get off the ground unless something like C is thrown in there as an unstated premise.
And C is not an unreasonable premise. They are only two statements, after all, ones it would be pretty hard to forget given the circumstances. It seems reasonable to think–and for the prisoner to think–that the prisoner won’t forget them, and will be able to reason about them at least simply.
It’s worth noting, at the very least, that Premise C implicitly throws in “I am (i.e., the prisoner is) aware of Premises A and B”, which isn’t already automatically entailed by the mere truth of Premises A and B and is certainly necessary to get the “But then I wouldn’t be surprised…” argument going.
But it’s not merely that premise C isn’t given in the set-up; it’s demonstrably not in effect, since the prisoner really does jettison the wrong premise like I was just saying.
Sure, it seems like we have to assume C about the prisoner; that’s why it seems like (as per the OP) there’s a paradox that’s confounding and cannot be resolved. Likewise, that’s why rejecting C renders the problem uninteresting; it’s maybe not all that difficult a problem if he’s just some guy who made a mistake. But that’s what happened, regardless of how interesting or difficult it would be if things were different.
I’m with you so far.
Not the way I read “at least simply.” After all, he could correctly jettison premise B, or incorrectly jettison premise A – and winds up getting surprised upon incorrectly jettisoning premise A, because premise C isn’t actually in effect. (No, not the part of C that says he’ll remember the premises; the part where he can draw logical conclusions from them at all times.)
The prisoner assumes he’ll continue to be able to make basic rational inferences throughout the week. But that assumption is incorrect.
Are you comfortable with that as a resolution of the problem? If so, can you show me again where it is the prisoner fails to be able to make some basic rational inference or other? (“Basic rational inference” is maybe a little vague. But it seems like you need to be able to show some point in the process during the prisoner’s week at which there is some basic, one step inference that most of us can usually make when we’re awake and alert–but which the prisoner, at some point, can not make. Does that seem fair to you?)
The Other Waldo Pepper, remember that at this point we are discussing whether the 1 day scenario and the 5 day scenario are the same.
In both cases the first question to ask is: “Is it possible for me to accept both of the judge’s statements and have both events happen”
1 day answer: No, those 4 items include a contradiction.
5 day answer: Uhhh, not sure. There is no obvious contradiction that I can draw upon like in the 1 day scenario. Real world results show it’s possible, but let’s try some logic to see if we can get anywhere.
The 1 day scenario reasoning uses the contradiction surrounding the entire scenario. It’s immediate and obvious and available to be used.
The 5 day scenario reasoning can’t use a contradiction because it doesn’t exist. At least not yet, you need to work at it and maybe or maybe not you can produce one. Odds are against it though given real world results.
I understand that you want me to see that when analyzing Friday only, a contradiction can be inferred just like in the 1 day case. That doesn’t make the scenario the same and it doesn’t make their reasoning the same in that at that point in their reasoning, the 1 day prisoner can draw upon the contradiction surrounding the entire scenario, but the 5 day prisoner can not draw upon a contradiction surrounding the entire scenario and thus their reasoning must be different. If it were the same the 2 prisoners would have the same number of steps and have the same information available to them at each step, but they don’t.
But maybe you think that is too hasty and if we give the 5 day prisoner a little time and a few more details, then we will see they are the same substantially (even though the 5 day prisoner had to take a detour to produce a contradiction surrounding the entire scenario). But there are 2 problems with that:
It hasn’t happened yet to a degree that in academic circles the problem is considered resolved and that resolution includes a contradiction substantially similar to the 1 day case
Real world results show that it is possible to accept both statements and have both events occur
As an additional point regarding the difference between the 1 day and 5 day scenario: I don’t think the 1 day scenario is controversial, is it? If it is not, how can the 1 day and the 5 day be the same?
2 Notes:
Ned Hall thinks you can’t rule out Friday (not sure if you read that or not)
Earlier in this thread I was convinced you could rule out Friday. After reading reading through Ned Hall’s article, now I wonder.
Wouldn’t a good reasoner really conclude the following and not jettison anything yet, because we are talking about the entire scenario, we just happened to assume the hanging was on Friday, but that doesn’t mean it can be removed from the entire scenario:
(not Friday or not Surprised)
Note: I think we need C
** revised step 7 **
7. Either I am not to be hanged on Friday, or the hanging will not be a surprise
The prisoner’s next step (which does not mean I think it’s correct, just that it’s the next step the prisoner took):
Assume instead of Friday that I am to be hanged on Thursday while retaining the information in step 7 (it’s unclear to me how we retain the information in step 7 but change our assumption about the day of the hanging, but it’s what the prisoner did so I’m listing it)
On Thursday morning I will not have been hanged on Monday, Tuesday or Wednesday.
From 9 and premises A and C, it follows that Thursday morning it is possible I will be hanged on Thursday or Friday
From 7, 10, it follows that either (I will be hanged on Thursday and surprised), or (I will be hanged on Friday and it will not be a surprise)
From B, and 11, it follows that I will be hanged on Thursday and it will be a surprise (?? just following the rules of logic here, I know this result is different than the prisoner’s)
Sorry, Waldo, no. In the five-day scenario, all the prisoner reasonably can conclude on Friday is that both conditions can’t be satisfied. This, however, provides no basis for drawing the same conclusion on Tuesday. Again, why his reasoning is faulty is the interesting part of the problem.
I’m entirely comfortable, and it seems eminently fair – because, as for needing to show some point in the process where the prisoner fails to make some basic, one-step inference that most of us can usually manage, let me try to specify a bit better:
He shouldn’t have concluded that the hanging “cannot occur on Friday.” He should have concluded that it wouldn’t be a surprise on Friday, rejecting premise B instead of premise A. He then works back with bad reasoning, one day at a time, to become “confident that the hanging will not occur at all.” He’s obviously mistaken; if, say, it happens on Friday, then anyone stupid enough to conclude that “the hanging will not occur at all” will be surprised.
The basic, one-step inference that you could make, but the prisoner didn’t, is that premise B is false; the prisoner instead jettisons premise A. We don’t need to debate whether the prisoner makes that mistake; we’re explicitly told that he reaches that conclusion, sure as he then gets surprised when the judge’s statements all turn out to be true.
RaftPeople helpfully revises Line 7:
…which is the point I’ve been making, and the answer to your question: you asked “can you show me again where it is the prisoner fails to be able to make some basic rational inference or other?” It’s when he supplies the Line 7 you typed out instead of the Line 7 RaftPeople typed out.
(Or, as I’d put it about your Line 7: “if it really is Friday morning, then Line 7 will contradict premise A. At this point, a good reasoner will correctly jettison premise B, and be unsurprised; a poor reasoner will incorrectly jettison premise A, and be surprised. Since premise C isn’t actually in effect, the prisoner can be a poor reasoner who incorrectly jettisons premise A.”)
RaftPeople then goes on to an all-new all-different Line 12: “just following the rules of logic here, I know this result is different than the prisoner’s”. I couldn’t have said it much better myself.
Where RaftPeople and I disagree comes next: he works it out as follows:
Of course you could – provided you phrase it right.
What you can “rule out” on Friday is surprise. What you can’t rule out on Friday is the hanging, because as soon as you incorrectly do so you can get surprised by a hanging that occurs. It’s a crucial bit of equivocation: you can rule out the surprise hanging, which the OP then refers to as ruling out “the hanging” – which then takes over the whole thing when it’s used to reject A instead of B for Friday.
Again, look at that 5-Day quote from up top: “Since the judge’s sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday. He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn’t been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.”
That’s not how good reasoning works.
The prisoner, see, is trying to prove that, if A is true, then A would be false. But all he’s actually proven is that if A is true, then A or B could be false. The prisoner thus incorrectly jettisons A at the first step, when reasoning about Friday; he concludes that “it cannot occur on Friday … cannot be on Thursday either … he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.” His reasoning involves a contradiction surrounding the entire 5-Day scenario, no different than the 1-Day scenario, because his conclusion incorrectly jettisons A while using it as a premise in reasoning that doesn’t rule out A.
Thanks for the kind words. For the record, lest silence be construed as agreement, I politely disagree. I think I am, on my best day, slightly less than averagely intelligent, and far from being ‘well-regarded’ I don’t think I’m ‘regarded’ at all, well or otherwise. I am and expect always to remain ‘Ianz… who?’.
I didn’t really want to return to this Tar Baby, given the plain futility of doing so, and I know this thread isn’t about me. But as PBear42’s kind words were addressed to me, I felt it would be okay to post a small comment.
And now back to your regular scheduled convoluted mish-mash of a discussion that could well earn a place in Dope history, like ‘drier lint’ but with less logical rigour.
That’s a valid inference. It skips a step, but that’s not a serious problem. As you probably recall, once you’ve established “X”, you can always immediately infer “X or Y” for arbitrary Y.
If the argument establishes that I am not to be hanged on Friday, then I can immediately infer that either I am not to be hanged or it won’t be a surprise. (I can also just as validly infer that either I am not to be hanged or the sky is green).
But recall that I’m trying to reconstruct the prisoner’s reasoning for the conclusion that he won’t be hanged on Friday. At no point in any explication of the Unexpected Hanging that I know of is the prisoner said to say something like your revised line 7. Rather, the prisoner concludes at one point that he’s not going to be hung on Friday–as per my own line 7. I just wanted to look at the reasoning that leads him to that conclusion, and see if there’s some flaw in it.
Is there a flaw in that reasoning (re-quoted above)? Is there a bad premise in there? A bad inference?
I’m sorry not to respond to most of your post. I had a hard time figuring out what to say about it because it all seemed to rest on a misapprehension contained in the sentence I just quoted. The thing is, in the reasoning I layed out on the prisoner’s behalf, the prisoner doesn’t reject premise A.
Later on, after his Friday-reasoning, (i.e., when he’s reasoning about the rest of the days of the week), he could be said to have rejected A. But in the line of reasoning I laid out, things haven’t gone that far. I’m explicitly stopping just at the point where the prisoner concludes he won’t be hanged on Friday. I want to know whether that specific part of the prisoner’s reasoning has some flaw in it or not. I’m ignoring the later reasoning that leads him to “I won’t be hanged”. I want to know whether he’s even right to conclude simply that he won’t be hanged on Friday.
Is he? If not, where in the reasoning I laid out is the flaw? You were saying premise C is false? Your evidence that it’s false involved and idea that the prisoner rejects premise A. That seemed to you to illustrate some kind of blindness to simple rational inference. But in the argument I laid out, the prisoner never rejects A, so I’m not sure that can be adduced as evidence that C is false.
You said the prisoner shouldn’t conclude that he won’t be hanged on Friday, but rather, that he won’t be surprised on Friday. But I can’t see anything in the line of reasoning up through item 6 that licenses that inference. It starts, in item 1, with a supposition about hanging, not about surprise. If something can be proved from that supposition, it has to involve hanging specifically.
Sorry I’m not following your reasoning as well as I ought. I’m honestly trying, I promise!
Like I’d said, if it really is Friday when he concludes he won’t be hanged, then he’s rejecting premise A instead of premise B. As per what you lay out in #2 and #3, we’re supposed to reason as if it were Friday – and the fact that he eventually concludes that “the hanging will not occur at all” is the direct result of that step where he rules out A instead of B on a hypothetical Friday morning.
He can’t rule out every day unless he rules out Friday. He can’t assure himself that it won’t occur at all unless he assures himself that it won’t occur on Friday.
And that’s his mistake, because he should have concluded that he wouldn’t be surprised on Friday.
No, he’s not. If he concludes that, then he can be surprised if he gets hanged on Friday.
No, he does reject A. You just said that “the prisoner concludes he won’t be hanged on Friday.” You just asked whether “he’s even right to conclude simply that he won’t be hanged on Friday”. If he concludes that on Friday, then he’s rejecting A. If he concludes that while reasoning as if it’s Friday, then he’s rejecting A.
Now, as you point out, he winds up explicitly concluding it regardless: “Later on, after his Friday-reasoning, (i.e., when he’s reasoning about the rest of the days of the week), he could be said to have rejected A.” We therefore know C isn’t in effect. But he likewise rejects A right there in the first step, because C isn’t in effect then either.