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Surprises in Logic (2016)

(math.ucr.edu)
92 points jxmorris12 | 4 comments | 22 Apr 25 15:24 UTC | HN request time: 0.717s | source
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mcphage ◴[22 Apr 25 16:30 UTC] No.43763949[source]▶
I don't think you need anything fancy to tackle the "surprise examination" or "unexpected hanging" paradox. This is my take on it, at least:

> The teacher says one day he'll give a quiz and it will be a surprise. So the kids think "well, it can't be on the last day then—we'd know it was coming." And then they think "well, so it can't be on the day before the last day, either!—we'd know it was coming." And so on... and they convince themselves it can't happen at all.

> But then the teacher gives it the very next day, and they're completely surprised.

The students convince themselves that it can't happen at all... and that's well and good, but once they admit that as an option, they have to include that in their argument—and if they do so, their entire argument falls apart immediate.

Consider the first time through: "It can't be on the last day, because we'd know it was coming, and so couldn't be a surprise." Fine.

Now compare the second time through: "If we get to the last day, then either it will be on that day, or it won't happen at all. We don't know which, so if it did happen on that day, it would count as a surprise." Now you can't exclude any day, the whole structure of the argument fell apart.

Basically, they start with a bunch of premises, arrive at a contradiction, and conclude some new possibility. But if you stop there, you just end up with a contradiction and can't conclude anything.

So you need to restart your argument, with your new possibility as one of the premises. And now you don't get to a contradiction at all.

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jerf ◴[22 Apr 25 16:48 UTC] No.43764093[source]▶
I can't help but think the "surprise examination paradox" rests too much in English equivocation for it to be a properly logical paradox. In particular, the fact that "surprise" changes over time, and the fact that if I've logically deduced that it is "impossible" for the test to occur on the last day then it is ipso facto a surprise if it happens then.

Sit down and make the argument really rigorous as to the definition of "surprise" and the fuzz disappears. You can get several different results from doing so, and that's really another way of saying the original problem is inadequately specified and not really a logical conundrum. As "logical conundrums" go, equivocation is endlessly fascinating to humans, it seems, but any conundrum that can be solved merely by being more careful, up to merely a normal level of mathematical rigor, isn't logically interesting.

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1. ogogmad ◴[22 Apr 25 18:04 UTC] No.43764743[source]▶
You did not understand the paradox.

The word "surprise" here means that the prisoner won't know his date of execution until he is told.

[Edited]

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2. mcphage ◴[23 Apr 25 00:11 UTC] No.43767504[source]▶
I did understand the paradox.

Those are the instructions he was given: he won’t know the date of his execution until he is told. He performs some reasoning, and concludes that he can’t get executed any day that week: therefore he will go free.

But if “he will go free” is a possibility, then his chain of reasoning falls apart. Previously he had argued “if I survive to the last day, I will be executed today. That won’t be a surprise. Therefore I can’t be executed on the last day.”

But once he has “…or I won’t get executed at all” as an option, then his reasoning would begin “if I survive to the last day, then either I’ll get executed today, or I won’t get executed at all” … and that’s as far as he can go. He can’t use that to conclude he won’t get executed on the last day, and he can’t then use that to conclude he won’t be executed on the second last day, and so on. The entire argument breaks apart immediately.

3. mcphage ◴[23 Apr 25 13:21 UTC] No.43771858[source]▶
Ah, sorry—I saw your "You did not understand the paradox", and thought you were replying to me, instead of replying to a reply. In my defense, the way HN uses indentations to indicate depth is hard to follow on a phone screen where the comment and its replies might be screens apart.
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4. jerf ◴[23 Apr 25 16:04 UTC] No.43773646[source]▶
Eh, it works for me. I endorse your answer, though I also endorse more things than that.

I referenced that there were many ways to "resolve the paradox" which isn't really "resolving" anything, based on how you carefully define the terms. It is certainly valid to define the terms in such a way that the prisoner is logically correct. In that case, there is no paradox, just perhaps lies. You can define it such that the prisoner is simply in error. You can also define it such that the answer is "indeterminate"... but that's not a paradox either. "Indeterminate" comes up in logic all the time and if you run around yelling "paradox! paradox!" every time that happens you're going to get hoarse pretty quickly.

The only "paradox" is that people insist on not being careful with their definitions, and any time anyone tries, someone else flips to a different definition (without being clear about it) and then starts arguing from that new point of view. That's not a paradox either. That's just lifting unclear thinking to the level of moral imperative. I have no patience or sympathy for that.