ajuc.

There's not much room to interpret here.

We could program this as a betting game. on every day from monday to friday, the student may place a bet of $1000 at even odds, that the test is on that day.

"Sure" means the bet has a 100% chance of being paid off. At even odds, the $1000 has an expected value of $2000.

It doesn't really matter 'why' they placed that bet. If the expected value of the $1000 bet is $2000 then the details don't really matter. The student wins, and there is a paradox.

The definition of the paradox thus shifts from something vague about 'sure' to something concrete: a bet that has 100% chance of being paid off, made with this intention.

Now, you are saying that student argues thusly. If the teacher is making a guarantee that I will be given no sure-bet (bet at 100% chance to win that the test is on that day), then that is a guarantee that I will not be in a position to place that bet on Friday.

This is true.

What is false is the step: "THEREFORE, if on Friday..." (i.e. if I get to Friday)

That is not a warranted beginning of a sentence.

If a test day of Monday is hard-coded into the algorithm, then saying "THEREFORE, if on Friday.." (if I get to Friday) is as spurious as saying "THEREFORE, if Friday afternoon I still haven't been given a test..."

This is a subtle point. One sounds OK, one sounds like it contradicts what we already know:
1) If I haven't been given a test by Friday (and it's friday morning now, the test can still be given)

2) If I haven't been given a test by Friday afternoon (and it's friday afternoon, the test can't be given today anymore...)

It sounds to you like #2 contradicts your guarantee to be given a test on a weekday next week. But in fact, it doesn't CONTRADICT anything, because it's an "if" about something that has 0.0 chance of happening.

You can legitimately say : "If I haven't been given a test by Friday afternoon" (i.e. no test at all week), but you can't DEDUCE anything from that, since it has 0.0 chance of happening. It doesn't matter what the second half of the sentence is.

Now.

Perhaps the first statement (1) is similar. Perhaps it also has 0.0 chance of happening.

Therefore, perhaps it is no better to say:
If I haven't been given a test by Friday morning...
than
If I haven't been given a test by Friday afternoon...

they both have 0.0 chances of happening.

A die that will always fall on 5. Reasoning "If it falls on six" is no better than reasoning "If it falls on 7" UNLESS you have been given some guarantee that there is a nonzero chance it falls on a six!!!

The whole "paradox" hinges on people thinking that by saying "the test will be given on a monday, tuesday, wednesday, thursday, or friday" means that none of these days has the same probability as "the test is not given at all." But in fact, one or more of these days could have that exact same probability: 0.0.

When it's 0.0 due to the stochastic process involved, "reasoning" with it is no better than "reasoning with" the case that the test isn't given at all.

You can see all this very easily with the pseudocode.

If the students are not given any guarantee that there is a non-zero chance the test is given on Friday or on Thursday, it is a mistake to reason anything at all about Friday or Thursday.

Here is the logical fallacy.

You assume that if there is probability 1 that the day is 1,2,3,4, or 5, then it follows that for each of 1,2,3,4,5 there is probability >0.0. That's a mistake. It's a fallacy. You haven't really been told that.

We can invalidate all assertions of a paradox very easily with two parallel universes.

Suppose the teacher is a mad physicist. He splits (forks) his universe in two, preceding as follows.

In one parallel universe the teacher always gives the test on Monday.

In another parallel universe the teacher always gives the test on Tuesday.

In both cases, the students are told: I guarantee you will be surprised by the test. I guarantee that it will be given next week, and on a Monday, Tuesday, Wednesday, Thursday, or Friday.

If the students DON'T KNOW about his mad experiment... how can they possibly be SURE of being tested (and being correct in this deduction) on either Monday or on Tuesday? None of these students could possibly be sure on any of these days.

They are simply mistaken when they begin to argue: "If we make it to Friday, the test must be Friday". This is no better than arguing "If we are not tested at all, then the test must be Friday". Both of those are equally valid arguments. They're both wrong in this scenario.

The scenario where they're NOT both wrong, where "if we make it to Friday, the test must be Friday" is OK (not violating the guarantee to have a test during the week) whereas "if we are not tested at all, then the test must be Friday" is not okay, is if the students know for a fact htat there isn't a 0.0 chance of having a test on a friday.

you simply can't deduce from something that is false. THe student is (in the case of the scenario just outlined, where the test day is monday or tuesday) arguing from false premises when they say "If we haven't had a test by Friday..."

That's an argument from a false premise (in this scenario) and is invalid for that reason.

The question is: are the students given enough information to know that the premise is NOT false ("if we make it to Friday")? If the teacher guarantees them a stochastic process with a nonzero chance of a test on Friday, then they have enough information to know that the premise is NOT false.

since htis is not stipulated, however, the teacher can meet his commitments.