Remember that we are reasoning within the larger context, which is:
Can I accept both of the judge’s statements and have both events come true?
That’s 4 different things.
In the proof you constructed, you eliminate Mon-Thu and of the 4 things we are trying to test, you chose the following, with respect to that last day Friday:
Accept hanging statement
Accept surprise statement
Assume hanging event
DO NOT assume anything about actual surprise event
With this starting point, you conclude not Friday. But note, it is conditioned on this starting point.
Here is an alternate setup with a different result:
Accept hanging statement
Accept surprise statement
DO NOT assume anything about actual hanging event
Assume surprise event
As you can imagine, using the same format and same basic approach you did, we would reach a contradiction and then conclude, in this case, that we will not be surprised (as opposed to not hanged).
Which of the 2 proofs is the logical person to assume? The correct answer is both, they are to assume (not Friday or not Surprised) but they do not know which one. If you think you can just choose one of them over the other, why? If you do think that, try replacing hanged and surprised with just X’s and Y’s and tell me if you think you can just choose the X proof over the Y proof (hopefully that will appear arbitrary).
Note: here’s the alternate logic
From 4 (surprise) then hanged
if 1 (accept hanged stmt) and hanged then not surprised
1 (accept hanged stmt) and hanged therefore not surprised
Surprised (assumption) and not surprised
Contradiction, therefore not surprised
Premise A: Batman isn’t shorter than Robin.
Premise B: If Batman isn’t shorter than Robin, then Robin isn’t taller than Batman.
Suppose Y: that Robin is taller than Batman. In such a case, by Premise B, we’d conclude not-X: Batman is shorter than Robin. From that and Premise A, we’d have a contradiction. So, do you jettison Premise A to incorrectly conclude that Batman is shorter than Robin? Or do you just stop supposing Y?
I see it similarly, except I think the logic breaks down at the beginning.
The prisoner’s logic is ignores important information included in the judge’s sentence.
1.) He will be hanged
2.) It will occur on a weekday in the following week.
3.) The executioner will knock at noon on said day.
4.) It will be a “surprise hanging”.
Now, if the executioner were to knock on the Friday...that would be a surprise!
To address this, (my comments below are also directed at Raftpeople here,) you’re further elaborating on your suggestion that the prisoner errs in considering one premise defeasible but not the other. As someone (maybe it was you) said above, we should wonder why the prisoner concludes there will be no hanging, rather than that there will be no surprise.
In response, and I know I’m just harping now, but my response is I want to see what the error is in the reasoning that leads to the conclusion “I will not be hung on Friday.” Your suggestion is that as I reconstructed it, the reasoning relies on a false premise about the prisoner’s own memory and capabilities for rationality. But the premise simply says the prisoner will remember the judge’s statements (as you agreed is reasonable) and will continue to be able to draw simple logical inferences from them. You objected to the latter by arguing that the prisoner demonstrates an inability to draw simple conclusions about those statements. Namely, he fails to draw the simple conclusion that he should not find the judge’s statements trustworthy. But even granting this, that fails to falsify premise C. For the premise says the prisoner will be able to draw inferences from them, not that he’s be able to draw conclusions about them. So I still have not been shown a flaw in the reasoning I reconstructed on the prisoner’s behalf. The premises seem true, the reasoning following them seems valid.
Right now I think the reasoning is valid. That would mean that analogous reasoning about Thursday, Wednesday, and Tuesday is also valid. When it comes to Monday, I think the form of the reasoning probably changes, but I haven’t got there yet. I just want to make sure the Friday-reasoning is valid.
If the Tuesday-through-Friday reasoning is valid, and if there is valid reasoning to eliminate Monday as well, then it turns out there is a contradiction, a straightforward one, between the judge’s statements, at least given the prisoner’s basic memory and rationality capabilities. For if all the days can be eliminated by valid reasoning, then the judge’s statement that the prisoner will be hung–Premise A–is falsified. That means that a contradiction can be derived, not from a supposition, but from the premises themselves. What a reasoner should do in this case is realize about the premises that they are faulty. (Kind of like what you say the prisoner should do in fact!) If the prisoner can eliminate each day of the week, then the prisoner must revise his premises, i.e., must either become convinced he will go insane or be a victim of accident in the following week, or else become convinced the judge’s testimony is (whether accidentally or not) untrustworthy.
All of that follows if the prisoner’s reasoning from those premises is valid. However, it looks like most here want to argue that the prisoner’s reasoning is invalid. That’s why I’m trying to check the reasoning step by step. I want to know if the prisoner should be revising his reasoning from the premises given by the judge, or rather, whether the prisoner should simply be ignoring the judge’s statements in the first place.
I think the reasoning is valid. But if the reasoning is valid, I’m puzzled by a further fact. I said if the reasoning is valid, then a contradiction has been found in the judge’s statements themselves. Yet as I’ve argued, the two statements can both be true. This means they are not contradictory. How can this be? Perhaps it has something to do with the third premise I threw in to get the prisoner’s reasoning going–premise C. Perhaps the judge’s statements themselves aren’t contradictory, but rather, the statements together with C are contradictory. But that seems strange. How could two statements be compatible, yet cause contradiction when combined with a simple statement of one’s own basic rational competence? Especially when the two premises themselves say nothing about one’s basic rational competence? (How can they contradict without sharing some terms?)
That puzzle makes me think maybe I’m wrong and the prisoner’s reasoning is invalid after all. The Friday reasoning still looks valid to me, meaning Tuesday through Thursday can also be validly eliminated at least. What about Monday? Maybe the reasoning there is different, and is not valid? I don’t know, I haven’t gotten that far yet. Right now I’m just checking the Friday reasoning.
One final note: In everything I’m saying, I am assuming that the prisoner is doing his reasoning on Sunday evening, about what will or could happen on future days. At no point am I reconstructing reasoning that happens on any of the future days being discussed in the story. All the reasoning takes place on Sunday evening just after the judge utters his statements. I said that just because I’ve seen some comments on the thread about what the prisoner should think on Friday. I don’t have an opinion about that right now. I’m just talking about what the prisoner should think about Friday on Sunday.
Not just why he concludes that, why does he incorrectly stop after only hypothetically assuming the hanging event and not hypothetically assuming the surprise event.
The results are different.
You haven’t offered any reason why you think it’s ok to ignore the other information and how you reconcile that the 2 assumptions result in opposite (regarding hanging) results.
I’m am not saying anything about his memory, I completely accept your memory premise. I know you said you were responding to me, but I don’t see anything in your response that counters my recent post.
If the reasoning is valid, then you will be able to show why we should choose one conclusion over the other in our logic below (which has been replaced by x’s and y’s). I will be very surprised if you can show that we can separate those 2, that one is clear the one to take and the other should be ignored.
We have 4 things:
stmt about event X (sx)
stmt about event Y (sy)
event X (x)
event Y (y)
Things we can immediately infer on 1 specific day (but not with 5 days):
If (sx and sy and x) then not y
If (sx and sy and y) then not x
If we assume the following:
sx true (meaning that the statement was made)
sy true (meaning that the statement was made)
x true (meaning event happened)
y unknown (no assumptions about this actual event)
Then we can reason that there is a contradiction and so we conclude not x
If we assume the following:
sx true (meaning that the statement was made)
sy true (meaning that the statement was made)
x unknown (no assumptions about this actual event)
y true (meaning event happened)
Then we can reason that there is a contradiction and so we conclude not y
How do you justify taking 1 conclusion and not both? Isn’t it arbitrary?
I need to break it down into something a little less abstract than ‘surprised’ and ‘days of week’. Suppose the judge instead laid out 7 boxes numbered 1-7 (one for each day of the week). Every day the judge comes and places a penny in one of the boxes that are laid out. He then calls on the prisoner to come and open the right-most box. Before he opens it he is entitled to guess whether he thinks the penny is in that box. If the coin is in that box, then that is the day he gets hanged. However, if he was able to guess that the penny is in that box prior to opening it, then he is obviously not surprised and goes free (for this exercise lets also assume that if the coin is not there and the prisoner guesses that it is there then he is also hanged). If no coin and the guess correct, that box is removed - along with the coin - and the prisoner goes back to his cell awaiting the next day. Using this formula I no longer have to think of days, it is simply a matter of opening up boxes sequentially while the coin can randomly jump between the remaining closed boxes. Also ‘surprise’ has a better definition. So now if I were that prisoner I would think, ‘well I’m not stupid, I know it’s not going to be in the last box because the judge couldn’t be that stupid’. And by the same token I can eliminate all the other boxes by only assuming that ‘the judge wouldn’t be that stupid’. It then becomes obvious to me that the judge must be lying to me about the penny and I therefore know I will go free by the end of the week. I can keep saying that there is no penny.
So now let’s assume that god was on my side and I made it to the last box, however, by my logic I am already convinced that there is no penny. What do I do? Do I say there is a coin in there and go against all logic or do I trust that my original logic is sound? If I go against then my original assumption about the judge not being that stupid was wrong. He really is stupid. But if I trust my logic and the coin is there who really is the dumb one?
So back to the original example. All I can really say is that I personally think the problem lies in the assumption that Friday can be ruled out. As soon as this assumption is made, then you can be ‘surprised’ on Friday (this can be carried across for all other days). Of course there could be something wrong with my logic.
You were right above that my previous post didn’t actually address your remarks. I had some vague notion that it did, but on closer examination I see that I missed your remarks completely. Sorry about that.
Are you proposing the prisoner should have reasoned thusly?:
Premise A, B and C given.
Suppose I’m to hanged and surprised on Friday.
Then I’m to be hanged on Friday.
…(insert my argument here)
We reached a contradiction
So I am not to be hanged and surprised on Friday.
So either I am not to be hanged on Friday, or I am not to be surprised on Friday.
In other words, if I am hanged on Friday, it won’t be a surprise.
For one thing, it’s not too important IMO what argument the prisoner should have made. What’s important is whether the argument he does make is sound or not. Is it valid, meaning does its conclusion follow logically from its premises? Are its premises correct? If the answer to both questions is “yes” then the argument is sound and the prisoner is on firm ground in concluding he can’t be hanged on Friday. If there’s another line of reasoning someone else thinks he should have followed instead, that’s not much of a concern of mine–I’m concerned to know what’s wrong with the reasoning he did engage in. (If someone can find a line of reasoning for the prisoner which starts with true premises and validly concludes that he can be hanged on Friday, then that is important. For it seems to show that two sound arguments lead to conclusions that can’t both be true. That’s alarming! And paradoxical!)
Second line of reply:
From the argument quoted above, you still get a simple deduction that the prisoner can’t be hung on Friday. From the conclusion “If I’m hung on friday, it won’t be a surprise” together with the premise “When I’m hung it will be a surprise” you can immediately conclude “I won’t be hung on Friday.” So we still have an apparently valid argument that the prisoner won’t be hung on Friday.
That is the new data point (“not Friday or not Surprise”) that we can add to our collection regarding the 5 day scenario , and then we can continue reasoning about the rest of the week knowing that.
“When I am hung it will be a surprise” implies, by the use of the word “when” that it is guaranteed.
The correct wording should be “If I am hung, it will be a surprise”.
Remember, all we have are 2 statements (by judge) that do not unequivocally determine future events. They are both suspect. We accept that they have been uttered and so we now have some information to play with. But we truly do not know anything about whether either of the events will actually happen.
Our goal is to determine “can both of those events happen if we accept both of those statements as true”.
Therefore, both the hanging and the surprise are unknowns from the beginning and we are using logic and reason to determine if they are all compatible.
And since Y is standing in for “there will be a surprise” while X is standing in for “there will be a hanging” or “there will be a test” or whatever, your argument doesn’t actually knock out the possibility of being hung on Friday – since you could reject Y instead of rejecting X.
It’s rejecting X when one could reject Y.
It’s not, since the prisoner rejects X when he could reject Y.
No, you don’t. As you just said, you’d just stop supposing Y.
No, it doesn’t. Your explicit advice is to “just stop supposing Y” instead.
You can only conclude that if Premise C is in effect. It’s not; he rejects X when he could just stop supposing Y.
Non-logician here. I just want to point out that a rational person would, in the real life version of the scenario given, never assume that he wouldn’t be hanged. We know that the punishment itself is the stronger of the two premises, as the law requires such a punishment for the crime. There is no legal reason the judge has to keep the surprise.I think this is what Frylock is doing.
I think there is a flaw in line 3. Supposedly, from 2, A and C, it follows that, on Friday morning, I will know that I will be hanged that day. But suppose we distinguish these two statements:
A: I will be hanged on either Monday, Tuesday, Wednesday, Thursday or Friday.
A*: I know that I will be hanged on either Monday, Tuesday, Wednesday, Thursday or Friday.
The difference is that A merely states that I will be hanged some time during the week, whereas A* states further that I know this to be true.
To validly infer that, on Friday morning, I will know that I will be hanged that day, it strikes me that I need my premises to be 2, A* and C, rather than simply 2, A and C.
I really don’t see a paradox at all. Just a bit of faulty assumptions on the prisoner’s behalf.
1 - the surprise comes in the day of the hanging, not the hanging itself.
2 - the “no friday” clause comes with a qualifier that the prisoner ignores - “if i make it to thursday”. this is vital. on wednesday night when he’s faultily eliminating friday, then thursday as possibilities he’s forgotten about this clause. he can’t eliminate friday because it’s only wednesday and not thursday. he could easily be hung tomorrow.
3 - the penny analogy from page 1 isn’t fitting because the surprise isn’t in whether a penny is in there or not. it’s when he finds out about the penny. a time qualifier has to be included. something like “some time in the next 2 minutes i’ll open this box and show you a penny. you will be surprised by when i choose to open this box.”
i skipped the rest of the pages and headed for page 6 to see if had been resolved. i didn’t see anyone arguing what i brought up so i decided to post. sorry if this has been posted already.
1- what exactly is the paradox
2- why does the scenario seem wrong?
The paradox is “how can this prisoner can logically eliminate all the days he’ll be surprised and also know factually he’ll die being surprised?”
This paradox hits a snag because the logic he uses to eliminate the days he’ll be surprised is faulty. I explained why in the last post. If there in fact was a sequence of logic that allowed him to eliminate surprise (like if he made it down to thursday) then the paradox would be achieved.
Even if he does make it to Thursday, it’s still wrong to eliminate Friday. Whether it’s Monday night or even Friday morning when he reaches his conclusion, he can be surprised by a Friday hanging just as soon as he concludes there won’t be a Friday hanging. So it’s not that ‘he can’t eliminate Friday because it’s only Wednesday’; it’s that he can’t eliminate Friday at any point, because doing so is the reason why he can then be surprised.