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92 points jxmorris12 | 2 comments | | HN request time: 0.419s | source
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mcphage ◴[] 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 ◴[] 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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astrobe_ ◴[] No.43764550[source]
It is like the infamous 0.999999... = 1. That one uses sloppy notation (what is "..."?) to make students think and talk about math.
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1. jjmarr ◴[] No.43764892[source]
It's not sloppy notation. It's an unambiguous infinite series of the form sum_n=1^infinity 9/10^n that converges to 1.

It's the same reason that 0.333... = 1/3. It's an infinite series that converges on 1/3.

Students learn repeating decimals before they understand infinite series.

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2. astrobe_ ◴[] No.43774507[source]
It is ambiguous, that's the trick. One can interpret "..." as '"any number of 9s", and in this interpretation the equation is false. But the teacher will eventually ask, "what about an infinite number of 9s"? Bam! brilliant introduction to series and convergence.

The correct notations to express exactly what one means feature the infinity symbol [1] - but if you use them, well, of course the students will see the trap.

[1] https://en.wikipedia.org/wiki/Infinity#Real_analysis