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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 fac

by jerf 2y ago

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.

bongodongobob 2y ago
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.
astrobe_ 2y ago
It is like the infamous 0.999999... = 1. That one uses sloppy notation (what is "..."?) to make students think and talk about math.
- mcphage 2y ago
  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).
  - dwohnitmok 2y ago
    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.
    - mcphage 2y ago
      There's also another numeral system called "Bijective Numeration" (https://en.wikipedia.org/wiki/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.
  - cwmoore 2y ago
    Thanks for the reminder of the trailing zeroes. 1.000000000 is also a big number.
- jjmarr 2y ago
  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.
  - astrobe_ 2y ago
    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 https://en.wikipedia.org/wiki/Infinity#Real_analysis
ogogmad 2y ago
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]
- mcphage 2y ago
  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.
- mcphage 2y ago
  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.
  - jerf 2y ago
    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.

by jerf 2y ago

bongodongobob 2y ago

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.

astrobe_ 2y ago

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

mcphage 2y ago
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).
- dwohnitmok 2y ago
  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.
  - mcphage 2y ago
    There's also another numeral system called "Bijective Numeration" (https://en.wikipedia.org/wiki/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.
- cwmoore 2y ago
  Thanks for the reminder of the trailing zeroes. 1.000000000 is also a big number.
jjmarr 2y ago
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.
- astrobe_ 2y ago
  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 https://en.wikipedia.org/wiki/Infinity#Real_analysis

ogogmad 2y ago

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]

mcphage 2y ago
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.
mcphage 2y ago
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.
- jerf 2y ago
  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.