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Amanojack comments on Open Thread: April 2010 - Less Wrong
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I've become a connoisseur of hard paradoxes and riddles, because I've found that resolving them always teaches me something new about rationalism. Here's the toughest beast I've yet encountered, not as an exercise for solving but as an illustration of just how much brutal trickiness can be hidden in a simple-looking situation, especially when semantics, human knowledge, and time structure are at play (which happens to be the case with many common LW discussions).
Extensive treatment and relation to other epistemic paradoxes here.
Let's not forget that the clever student will be indeed very surprised by a test on any day, since he thinks he's proven that he won't be surprised by tests on those days. It seems he made an error in formalizing 'surprise'.
(imagine how surprised he'll be if the test is on Friday!)
Since the student believes a surprise test is impossible, it seems this wouldn't surprise him.
Why not give a test on Monday, and then give another test later that day? I bet they would be surprised by a second test on the same day.
True, there's nothing saying there won't be two tests.
Rather than solve this, I was hoping people'd take a look at the linked explanation. When phrased more carefully, it becomes a whole bunch of nested paradoxes, the resolution of which contains valuable lessons on how words can trick people. It covers some LW material along the way, such as Moore's Paradox.
But if there's a solution, it's not really a paradox.
And I don't like word arguments.
Ugh, yes. Why are we speaking of "paradoxes" at all? Anything that actually occurs is not a paradox. If something appears to be a paradox, either you have reasoned incorrectly, you've made untenable assumptions, or you've just been using fuzzy thinking. This is a problem; presumably it has some solution. Describing it as a "paradox" and asking people not to solve it is not helpful. You don't understand it better that way, you understand it by solving it. The only thing gained that way is an understanding of why it appears to be a paradox, which is useful as a demonstration of the dangers of fuzzy thinking, but also kind of obvious.
Maybe I'm being overly strict about the word "paradox" here, but I really just don't see the term as at all helpful. If you're using it in the strict sense, they shouldn't occur except as an indicator that you've done something wrong (in which case you probably wouldn't use the word "paradox" to describe it in the first place). If you're using it in the loose sense, it's misleading and unhelpful (I prefer to explcitly say "apparent paradox".)
We're all saying the exact same thing here: words are not to be treated as infallible vehicles for communicating concepts. That was the point of my original post, the point of Rain's reply, and yours as well. (You're completely right about the word "paradox.")
Also, I'm not saying not to try solving it, just that I've no intention of refuting all proposed solutions. I didn't want my reply to be construed as a debate about the solution, because that would never end.
Words frequently confuse people into believing something they wouldn't otherwise. You may be correct that this confusion can always be addressed indirectly, but in any case it needs to be addressed. Addressing semantic confusion requires identifying it, and I found this riddle (actually the whole article) a great neutral exercise for that purpose.
EDIT: Looking back, I should probably just have posted riddle and kept quiet. Updated for next time.
...and yet...
Probably.
If a 50% chance of having a test that day would leave a student surprised he can be 87.5% confident in being able to fullfil his assertion.
However, if the teacher was a causal decision agent then he would not be able to provide a surprise test without making the randomization process public (or a similar precommitment).
The problem with choosing at day at random is, what if it turns out to be Friday? Friday would not be a surprise, since the test will be either Monday, Wednesday or Friday, and so by Thursday the students would know by process of elimination that it had to be Friday.
How do you get that result while requiring that the test occur next week? It is that assumption that drives the 'paradox'.
The answer to the question 'Can the teacher fulfill his announcement?' is 'Probably'. The answer to the question 'Is there a 100% chance that the teacher fulfills his announcement?' is 'No'.
You misunderstand me - I maintain that an obvious unstated condition in the announcement is that there will be a test next week. Under this condition, the student will be surprised by a Wednesday test but not a Friday test, and therefore
and, if I guess your algorithm correctly,
[edit: algebra corrected]
The condition is that there will be a surprise test. If the teacher were to split 'surprise test' into two and consider max(p(surprise | p(test) == 100)) then yes, he would find he is somewhat less likely to be making a correct claim.
I maintain my previous statement (and math):
Something that irritates me with regards to philosophy as it is often practiced is that there is an emphasis on maintaining awe at how deep and counterintuitive a question is rather than extract possible understanding from it, disolve the confusion and move on.
Yes, this question demonstrates how absolute certainty in one thing can preclude uncertainty in some others. Wow. It also demonstrates that one can make self defeating prophecies. Kinda-interesting. But don't let that stop you from giving the best answer to the question. Given that the teacher has made the prediction and given that he is trying to fulfill his announcement there is a distinct probability that he will be successful. Quit saying 'wow', do the math and choose which odds you'll bet on!
I never intended to dispute that
only the specific figure 87.5%.
It's a minor point. Your logic is good.