This is a very good way to expose the difference. If they were truly the same, then making the hanging an absolute fact would change both in the same manner, but it clearly doesn’t.
But that’s still my initial point: our dialogue corrects the same mistake in both the 5-Day and 1-Day scenario, since the problem either way is that he concludes the hanging won’t occur at all.
Which is why I reject RaftPeople’s breakdown of the issue:
Let me rephrase what I’d said earlier; I don’t yet want to move on to discussing some new variant until we’ve first established whether the original situation from the OP (and right there at the start of Hall’s piece) can be expressed as follows: “he mistakenly concluded that the event wouldn’t occur at all.” I don’t care whether he makes that mistake in a 5-Day timeframe or a 1-Day timeframe; I only care whether he makes that mistake at all.
If everyone on both sides – plus maybe even the OP – can agree with that, then we can move on to discussing what a reasonable prisoner should conclude rather than concentrating on what this prisoner did conclude. Once we’ve agreed that he mistakenly assumed there wouldn’t be a hanging or a quiz or whatever, we can move on to talking about how people would reason in the real world or whatever. But I don’t want us to skip over the actual content (and resolution) of the initial problem:
I don’t yet want to argue whether the 1-Day scenario is only the same if you include the same reasoning by the prisoner. I want to argue that the 1-Day scenario is the same if you include the same reasoning by the prisoner. So long as it’s a given that he mistakenly concludes the hanging can’t occur, what matters is that he reaches that same mistaken conclusion – which can be remedied, as Bear says, by pointing out the same fact in either scenario.
We could then get different results if we corrected that same mistake by pointing out the same fact, but IMHO that’s irrelevant in analyzing whether the situations are the same: so long as someone does conclude the hanging won’t happen, the situations are equivalent: they make the same mistake for the same reason.
Hypothetical situations in which the prisoner reasons differently wouldn’t be equivalent, but that’s irrelevant in pinpointing the error he actually made: he mistakenly concluded that no hanging would occur at all, and that’s incorrect whether it’s a 1-Day or a 5-Day or a 365-Day scenario.
You and I have very different definition of the word “the problem” in this case, which is ok, as long as we know what the other means. For me, “the problem” encompasses the entire setup, conclusion and everything in between. Kind of like if a teacher said “do problem number 27”, they are referring to the whole thing.
It seems that you consider “the problem” to be just the portion in which the prisoner concluded he won’t be hanged, that you don’t refer to the entire scenario as “the problem”, is this correct? If so, using your terminology, I agree that “the problem” is the same in both cases in that the prisoner did come to the wrong conclusion, that the statement regarding his future hanging was not true when in reality it was.
Note: I do, at this point, consider this to be a trivial position because we are told both of these facts right in the problem (using my terminology in this case). But I will try to keep an open mind about this.
No, from my perspective I’m using “the problem” to encompass the entire setup, conclusion, and everything in between, and you’re using “the problem” to encompass the entire setup, conclusion, and everything in between – plus some counterfactual stuff that didn’t actually happen but could have.
You lost me at the end there: what’s trivial about that, and which facts are you referring to?
The Other Waldo Pepper, even if we just focus on the prisoner’s invalid logic, there is still a difference:
In the 1 day scenario the prisoner rules out that both the judges statements can be true at the same time the events are true. This is simple straight forward valid logic, he correctly knows that all of that can’t be true at the same time. He then reasons from there, but his first conclusion is correct.
In the 5 day scenario the prisoner rules out Friday (is this first step even valid? unknown), and then proceeds to use that information for subsequent steps.
You don’t consider those 2 sets of logic to be the same, right?
See, you just glossed over the whole equivalence I’m on about: that “first” step, the one you ask after the validity of and declare unknown, is the 1-Day scenario – and when “the prisoner rules out Friday” in that scenario, he’s making the 1-Day mistake right there.
No, it’s not. Both of the judge’s statements can be true – and, in fact, turn out to be, if the prisoner concludes that there won’t be a hanging and then is surprised when the hanging occurs at all.
You can’t have a 1 day scenario in which all of the following are true:
- Judge’s statement about the hanging
- Judge’s statement about the surprise
- The actual hanging event
- The actual surprise event
It is simple logic to show that you can’t have all 4 of those true at the same time in the 1 day scenario. And, I think it’s fair to say that the prisoner in the 1 day scenario will start with the simplest valid logic available to them, I don’t think it’s reasonable to force the prisoner to ignore this simple logic. Note: if you require a symbolic proof of this simple logic then I will provide it, but I assume we can all do that one in our heads.
On the other hand, it is far from simple logic to show that you can’t have all 4 of those true at the same time in the 5 day scenario. Maybe you can ultimately show it, possibly, probably not. But the logic will be far more convoluted than that of the 1 day scenario.
Sure you can. So long as the prisoner believes the judge is lying or mistaken or whatever, the judge’s statements can all turn out to be true when the hanging occurs and surprises the heck out of the prisoner. We’re told that our hypothetical prisoner concludes the hanging won’t occur at all; that’s foolish of him, but it nevertheless sets up a situation where all 4 are true.
We’re told that he rules out Friday, that he concludes it cannot happen on Friday, that he doesn’t think it will happen at all in general or on Friday in particular; you phrased it as “that is why I eliminated Friday, because on that day the judge can’t be correct.” Phrase it however you like – but just note that by eliminating Friday you’ve made it a time where all 4 can be true.
The ability of the prisoner to arrive at the conclusion that the judge is mistaken or lying is the critical difference between the 1 and 5 day. You are talking about the reasoning after that point. I am talking about the information the prisoner has prior to concluding that he will or won’t be hanged or surprised.
So what? I don’t care why he reaches his conclusion; I only care that he reaches his conclusion.
He does, whereupon all 4 are true at the same time in the 1-Day scenario. The information that prisoner has prior to reaching his conclusion is irrelevant to whether (a) he reaches that conclusion, and whether (b) all 4 are true at the same time in the 1-Day scenario given a prisoner who does, in fact, reach that conclusion.
Then we’re not all agreed that the problem is to locate a flaw in the prisoner’s reasoning?!
The reason why it is important is because we are discussing whether the 1 day scenario is the same as the 5 day scenario.
In the 1 day scenario you can derive a contradiction prior to any conclusion.
In the 5 day scenario you can’t (I don’t think).
You may personally think that the only thing important to you is that the prisoner’s ultimate conclusion is wrong and that’s fine, but it doesn’t in any way counter the differences in the information available to the prisoner prior to that conclusion. Which means that it doesn’t show that the scenarios are the same, only that the prisoner ended up with the same conclusion.
How about if we define some terms and use those:
“the scenario” will include the setup, number of days, what the judge says, etc. but not the prisoner’s reasoning
“the problem” will be considered any and all flaws in the prisoner’s reasoning within that specific scenario
Sound good?
Again, my assertion is that the first step of the 5-Day scenario is the 1-Day scenario: either way, the mistake is eliminating Friday upon concluding that the hanging won’t happen on that day. That said,
They’re differences, to be sure. But how are those differences relevant?
It wouldn’t change anything relevant to make the hanging a firing squad, or to make the prisoner female instead of male – and it doesn’t matter whether she’s a redhead in a black dress, or a blonde in a white dress, or a brunette in no dress at all. What’s relevant is just that she incorrectly concludes the execution (or quiz, or whatever) won’t occur, and then gets surprised when it does. Why does the prisoner’s information have any more relevance than the color of her hair or the absence of her dress?
Sure. The prisoner’s reasoning will include premises, and to check these premises, we will need to look at the scenario. The prisoner’s reasoning will also include inferences, and to check those inferences, we will need to look at their logical validity.
So it should be simple, shouldn’t it? Lay out the prisoner’s reasoning–including premises and inferences–step by step. Point to the step where he goes wrong. (Whether that means he is reasoning from a premise that isn’t actually verified by the scenario, or else that he uses an invalid inference at some point.) Problem solved!
If I open the box and find Penny Marshall inside it would meet both criteria. However, it won’t be now that I gave you the idea.
Can you show me symbolically how they are the same? When I attempt to do it they are drastically different.
You don’t think that the information the prisoner has to draw on to attempt to determine what is going on is relevant?
Yep should be simple 
I think we should walk through the exercise, do you want to start?
Symbolic Logic was the one philosophy class I got a “C” in. Let me just copy-and-paste:
Ignore that part where you said “and then proceeds to use that information for subsequent steps.” How is the first step in the 5-Day scenario any different from the 1-Day scenario? He rules out Friday somehow; how?
Not if he reaches the same conclusion, no.
Ok. How’s this?
The scenario: The judge says “I will give you a quiz next week, on one of the five weekdays. Every day prior to the actual quiz, I will tell you that there is no quiz that day. On the night prior to the quiz, you will not be able to logically deduce that you have a quiz the next day from statements I will have told you by then (that is, from the very scenario I am telling you now and the prior announcements of days on which you lack a quiz).” After this, the judge does, in fact, announce no quiz on Monday and Tuesday, and hands out a quiz on Wednesday, say.
Is that an acceptable semi-formalization of the judge’s announcement? Or would we like to cast it differently? [Perhaps surprise should also separately include the possibility that one is able to logically deduce the negation of what’s about to happen, for example.]