Aha aha aha. I think I may understand you better than I did a minute ago.
Maybe what you’re saying is this. The prisoner should consider premises A and B to be defeasible. Premise C should read something like “I won’t forget the judge said A and B, and I’ll be able to make simple inferences from them, and I’ll know when it is reasonable to reject one or the other.” And this is falsified in a scenario in which the prisoner reaches Friday–and fails to reject premise B when he should.
Without saying whether I agree or not, let me just make sure first: Do you think that’s roughly what you’re saying?
ETA: Hadn’t read your last post when I wrote this.
That sounds about right. I’d been taking “able to draw simple logical conclusions from them at all times” to include “know when it is reasonable to reject one or the other” – but if adding that on helps to make the case that “C” is falsified, then I’m fine with so expanding “C”.
I’m not sure how much self-awareness you have, but I feel like I should help you out.
You are insulting posters in this thread. That part I figured you already knew, the self-awareness part is: Do you know why you feel the need to insult posters in this thread?
I can’t answer the question (I can guess), but at least I can try to get you to start thinking about it.
You supposed premises A and C, but not B, when getting from Step 1 to Step 4. Step 5 says “5. Line four contradicts premise B” – but so what? We haven’t yet supposed premise B in Steps 1 through 4. That’s why Step 7 can involve incorrectly ruling out A or correctly ruling out B: you used A to build Steps 1-4, and then concluded not-A when you could just as easily have concluded not-B.
Suppose Y
Then (from the second premise) not-X
From that and the first premise, then, both X and not-X. Contradiction.
Therefore: not-Y
The reason I laid that out is this. You said I hadn’t supposed premise B in steps 1 through 4. But I didn’t need to. The contradiction you derive in an indirect proof doesn’t have to be a contradiction of something you’ve supposed for indirect proof. Any contradiction will do. And the contradiction can be derived from premises as well as whatever you’re supposing. So in this abstract example, the contradiction (X and not-X) involved something in a premise, not something in the supposition. But that’s fine. That’s a perfectly valid way for an indirect proof to go.
So there is no problem with the argument along the lines of what you’re saying here.
On the other hand, I’m still thinking about what you said about the prisoner needing to think of the premises as defeasible.
That’s perfect. Now just note that X is that there will be a hanging, and that Y is that it will be a surprise. Taking X as a premise, supposing Y leads to not-X. Therefore: not-Y.
That’s how the prisoner should reason, when thinking about Friday. Sadly, it’s not how the prisoner does reason when thinking about Friday; instead, the prisoner takes X as a premise, notes that supposing Y leads to not-X, and therefore concludes – not-X.
I see your posts but don’t have time to thoroughly read, this is what I thought on my drive:
We are trying to determine if we can accept both statements as true and have both events come true, we assumed the statements and then we need to assume the events, therefore step 1 needs to be:
Friday and Surprise
Which ultimately leads to not (Friday and Surprise) (or (not Friday or not Surprise) )
But we aren’t allowed to just extract one of those values like not Friday
I disagree; he has no basis for eliminating Friday, and also has no basis for eliminating Tuesday.
It’s not merely that he happens to draw the same conclusion for Tuesday as he does for Friday; it’s that (as per the OP) he eventually concludes “the hanging will not occur at all”, which he can only do by concluding the hanging won’t occur on Friday or Tuesday or any other day. An early step in reaching that incorrect conclusion is when (as per the OP) “he concludes it cannot occur on Friday” – which is (a) explicit, and (b) dead wrong.
You wrote that “all the prisoner can reasonably conclude on Friday is that both conditions can’t be satisfied.” The problem is, the prisoner doesn’t reach that eminently reasonable conclusion; he’s a fool, and explicitly reaches a different conclusion – at which point both conditions can be satisfied. Upon incorrectly concluding that the hanging won’t occur on Friday, he can be surprised if the hanging occurs on Friday.
You wouldn’t reach that conclusion. I wouldn’t reach that conclusion. That’s why it’s so easy to assume the prisoner wouldn’t reach that conclusion. It’s why Frylock inserts a Premise C that (unlike Premise A or Premise B) doesn’t actually appear in the story. It’s why Ned Hall quickly sets to “making explicit certain assumptions about the student’s cognitive abilities” because “the argument falters immediately” if we “render the task of diagnosing the student’s reasoning trivial” by just supposing that “the student is woefully bad at reasoning”.
But the prisoner is woefully bad at reasoning. He does reach the wrong conclusion about Friday in particular, just like he reaches the wrong conclusion about whether the hanging will occur at all.
I must have missed the part where someone asked for your opinion of the logical coherence of our comments in this thread. Could you go back and point that post out to me?
Love to. See your own post, #107 in this thread. And there have been other similar invitations.
But anyway, surely discussing the merits or otherwise of the various arguments people have put forward is the point of this thread, and perfectly appropriate for GQ?
Oddly enough, the post you refer to doesn’t fit the description “asking for your judgment about logical coherence,” but I get what you were trying to do, and assign an A for effort.
Everyone else in this thread has been able to discuss the issue without being huffy or condescending.
You did not discuss the merits of anyone’s argument in the comment Raftpeople and I are responding to. You simply characterized the thread, and by extension, its participants, as logically muddled. There’s nothing GQ about that.
Hmm. Actually, it sounds to me like we mostly agree. See especially paragraph 4 of your reply. We differ, perhaps, on what aspect of the problem is central, but I’m okay with that. As I said to ianzin a few pages back, there’s a reason the problem continues to be discussed sixty years after it was first propounded.
So, what I think we agree on is that the prisoner can’t validly eliminate any day. In the one-day scenario or Friday of the five-day one, he can’t conclude he won’t be hanged; only that if he is hanged, it won’t be a surprise. What he can validly conclude on Thursday and Wednesday is complicated and depends on how we interpret the premises. By Monday and Tuesday, though, it’s pretty clear he can be hanged on either day and it be a surprise. If we expand the multi-day period to 20 days or 100, the fallacy of backwards induction becomes even more clear.
Where we still disagree is over whether one-day and multi-day are equivalent. I’d like to try to explain one more time why I think they are not. In addition to the points I’ve already made (and RaftPeople’s points, with which I agree), there is this. Surprise in the context of the problem can have two meanings, subjective and objective. Subjective is the kind of surprise we get in the one-day scenario. The prisoner reasons poorly and is thus surprised. We could say the same thing is true of the multi-day scenario, but there’s a difference. In the latter, it is possible both to hang the prisoner and have it be an objective surprise, in the sense that he had no legitimate basis for concluding it couldn’t be a surprise. IOW, he couldn’t on Friday be certain he wouldn’t be hanged, but he could conclude such a hanging wouldn’t satisfy both conditions. On Tuesday, he can make no such valid inference.
To close the circle, I gather that what interests you is the problem as stated. In that view, the difference between the two kinds of surprise isn’t terribly important. What interests me is what constitutes valid reasoning by the prisoner. In that view, the difference matters quite a bit. Stated a little differently, to me, subjective surprise is an almost trivial resolution of the paradox. That backwards induction doesn’t work over more than a few days is the meat of the problem. And why most people consider the paradox worth discussing.
Frylock, Waldo, Indistinguishable(not sure if you are still reading or not), (not PBear, I don’t think, because we agree already):
Assuming the context is “let’s construct valid reasoning” and you really really want to start out with the original assumption “Hanged on Friday” only (nothing about surprise), that’s actually ok as long as you realize you have chosen to assume 1 value of 1 of the 2 items (hanging, surprise). You can’t stop there, you also have to do the same reasoning assuming each of the other possible values (not hanged, surprised and not surprised). Once that is done then you can gather all of that data and come to some sort of conclusion, which will be “not Friday or not Surprise”.
Waldo, actually you can ignore that previous post, because it seems that you are indeed using the “no surprise” conclusion as part of a conditional (if friday then no surprise) that is getting added to our set of data, as opposed to ruling out friday. I know, you’ve made that last point explicit multiple times, I was getting very focused on the correctness of step 1 in our reasoning.
I’m sorry I don’t have something deeper to say here, but the simple fact is, I don’t understand why you think this. It seems blatantly wrong to me, which makes me think I’m probably misunderstanding you.
To illustrate why it seems blatantly wrong to me, take the following illustration. It looks to me as though you’re saying something tantamount to:
It seems like you’re saying something analogous to that. And my only reply is a kind of puzzled, “No, I don’t have to do that.” There’s no requirement that I investigate the implications of all possibilities in order for me to perform a valid indirect proof. The argument constructed just now is perfectly valid without having to find implications of a supposition that the sky is colored. It validly proves that the sky is colored. And similarly, it still seems to me that the argument I constructed for the prisoner gives a valid proof that he is not to be hanged on Friday.