Surprises in Logic (2016)

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.

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.

I would encourage anyone who's intrigued by this paradox to read Timothy Chow's comprehensive paper on the subject (https://arxiv.org/abs/math/9903160).

In particular, he discusses what he calls the meta-paradox:

> The meta-paradox consists of two seemingly incompatible facts. The first is that the surprise exam paradox seems easy to resolve. Those seeing it for the first time typically have the instinctive reaction that the flaw in the students’ reasoning is obvious. Furthermore, most readers who have tried to think it through have had little difficulty resolving it to their own satisfaction.

> The second (astonishing) fact is that to date nearly a hundred papers on the paradox have been published, and still no consensus on its correct resolution has been reached. The paradox has even been called a “significant problem” for philosophy [30, chapter 7, section VII]. How can this be? Can such a ridiculous argument really be a major unsolved mystery? If not, why does paper after paper begin by brusquely dismissing all previous work and claiming that it alone presents the long-awaited simple solution that lays the paradox to rest once and for all?

> Some other paradoxes suffer from a similar meta-paradox, but the problem is especially acute in the case of the surprise examination paradox. For most other trivial-sounding paradoxes there is broad consensus on the proper resolution, whereas for the surprise exam paradox there is not even agreement on its proper formulation. Since one’s view of the meta-paradox influences the way one views the paradox itself, I must try to clear up the former before discussing the latter.

> In my view, most of the confusion has been caused by authors who have plunged into the process of “resolving” the paradox without first having a clear idea of what it means to “resolve” a paradox. The goal is poorly understood, so controversy over whether the goal has been attained is inevitable. Let me now suggest a way of thinking about the process of “resolving a paradox” that I believe dispels the meta-paradox.

I agree. The premise itself is spurious. I've never liked this paradox because I don't think it makes sense from the get go.

It is like the infamous 0.999999... = 1. That one uses sloppy notation (what is "..."?) to make students think and talk about math.

That sounds interesting—thanks for sharing, I'll check it out.

I'm not sure the "..." is sloppy notation—it can be made rigid pretty easily. The surprise is that students' expectations that if two decimal expressions are distinct, that the real number they correspond to must be distinct also. (Even there, students have already gotten used to trailing zeros being irrelevant).

But it's stipulated that the test will happen on one of the days - it's not a possibility that it won't happen at all, hence the paradox.

One resolution is that what the teacher stipulates is impossible. It should really be

"You'll have a test within the next x days but won't know which day it'll be on (unless it's the last day)"

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]

> 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.

Your critical thinking is bad. The first paradox happens when the prisoner concludes that the judge lied, using a rational deduction. A second paradox happens when it transpires the judge told the truth.

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.

> 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.

It’s amusing that you stopped here without giving an actual solution. Please do tell us, which day is the test on?

Yeah; students often conflate numerals with numbers and assume that if two numerals are distinct then the numbers they represent must also be distinct. It's not trivially true! You need to prove that distinct numerals lead to distinct numbers and as 0.999... vs 1.000... demonstrates it is not always true for various kinds of numerals and numbers!

I think teasing apart numerals and numbers is a good first step on one's journey in mathematical logic.

Thanks for the reminder of the trailing zeroes.

    1.000000000

is also a big number.

It could be on any day—even the last—and would be a surprise.

The prisoner concludes that the judge’s instructions were impossible, which is true. Their conclusion was that there’s an additional possibility: that they don’t get hung at all. Which is also true. Their mistake is believing that this new possibility will come to pass, instead of realizing that the new possibility means that the judge’s instructions aren’t impossible after all.

So in the end, the judge was telling the truth, and the prisoner was mistaken, and then dead.

That’s not a paradox, though, that’s just impossible instructions. There’s nothing paradoxical about impossible instructions. The teacher should have stipulated “You’ll have a test within the next X days but won’t know which day it’ll be on, or else it won’t happen at all.” The students realize that no test is a possibility, but they wrongly conclude that it’s what will happen, instead of realizing that that possibility merely makes the teacher’s instructions valid.

Really? Kids show up on the last day and know with certainty that the test is coming that day.

Not if they consider “we won’t have the test at all” as a possibility.

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.

Well, yeah, a resolution to something that's called a paradox means it's no longer a paradox. It seems we agree that the original instruction as stated,

"You’ll have a test within the next X days but won’t know which day it’ll be on"

is impossible

What if the teacher writes the test day on a piece of paper at the beginning of the week and hides it in his desk? That way "we won't have the test at all" is no longer a possibility.

I think that gets you back to impossible instructions—if the teacher had written "Friday", then on Friday the students wouldn't be surprised by the test on Friday. I guess, unless the students think it's possible the teacher left it blank or wrote "No Test This Week" or something on it?

There's also another numeral system called "Bijective Numeration" (https://en.wikipedia.org/wiki/Bijective_numeration) which does is a bijection between numeral sequences and numbers (hence the name). It's pretty neat... it doesn't have a 0, so you can't get an infinite string of trailing 0s.

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.

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.

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