4

The link for the paper is available here.

This is the case of the headmistress who says that next week, there will be a surprise exam one of the afternoons. The girl students argue that it is impossible for come Friday noon with no prior exam, the exam is no longer surprise later that afternoon. Arguing inductively backwards, they seek to show that no exam next week is possible.

In the above article, Jackson seems to break down his solution to the problem thus:

EC (Existence condition): An exam will be held one afternoon next week.

IC (Ignorance condition): You won't know until after noon on the day which afternoon it is.

D : One may be justified at t1 in believing p while acknowledging at t1 that should certain things happen between t1 and t2, one would not be justified at t2 in believing p


To the above, he adds the following:

BEC : At every noon prior to the exam, it is reasonable to believe EC.

BIC: At every noon prior to the exam, it is reasonable to believe IC.


Now, his solution to the problem states thus:

Once we have BEC and BIC, Friday afternoon exam is really not possible. (I can understand this.) But exam on any other prior afternoon would not violate any of the above. On any noon prior to Friday, the following are consistent (he claims):

(1) No exam has yet made its appearance (2) Girls are entitled to believe that the exam will occur at some future afternoon (3) Girls acknowledge that should Friday noon arrive without the exam having appeared, they would no longer be entitled to believe that the exam would occur within the specified period.

At this stage, he adds "It is important to see that Modus Tollens is beside the point here." He goes on to state that the girls would be arguing fallaciously if they argued at Wednesday noon "We are entitled to EC; if Friday noon should arrive without an exam, we would not; therefore Friday noon won't" He says that the first premise must be "We are entitled to EC at Wednesday noon" and the conditional premise must be "If Friday noon should arrive without an exam, we would not be entitled to EC at Friday noon. He claims that this follows from D, which is not controversial.

He then goes on to even more somewhat complex arguments to understand.

Can it be pointed out why these different assumptions are being used and what in particular the author is trying to convey by taking recourse to Modus Tollens and how he solves the paradox and not the hard version of the paradox? I am struggling to clearly see the connections.


I posted this over at /r/askphilosophy also but that site seems to prohibit newer posters from responding to posts from what I understand, hence the post here.

1
  • 1
    Ironically, if the students conclude that a surprise exam CAN'T occur on a Friday, then getting the exam on a Friday would be a surprise indeed.
    – TKoL
    Dec 5, 2023 at 13:26

1 Answer 1

2

As I expect you are aware, there have been many published papers on this paradox, and there is a useful summary from 1999 by Timothy Chow. https://arxiv.org/abs/math/9903160

Jackson divides the problem into easy and hard cases depending on an important assumption about the nature of the teacher's statement.

(1) The students have reason to believe the teacher, but this reason is potentially defeasible depending on how events pan out later in the week. The easy case.

(2) The students are entitled to be certain that what the teacher says is true, and no future events may change this. The hard case.

Jackson proposes to solve the easy case by saying that on Friday, the students have no reason to believe the teacher. If Friday noon rolls round and there has been no exam then the teacher is not a reliable source of information and she has forfeited the students' trust. But what she said can still turn out to be true: there may be an exam that the students did not expect. But on Wednesday, the students still trust the teacher and they have reason to believe there will be an exam. On Wednesday the students cannot correctly argue, "Were Friday noon to arrive and no exam, we would have reason to doubt there will be an exam. So today we have reason to doubt there will be an exam". The potential unreliability of the teacher's utterance is not established on Wednesday.

The case is similar to a kind of Moorean paradox. One cannot coherently believe, "it is raining but I don't believe it is raining". Or, it may be the case that, "P is true and Fred does not know that P is true", but that is something Fred himself cannot know.

Imagine the teacher saying, "there will be an exam, but I am not a reliable source of information that there will be an exam". This may turn out to be true, but the students cannot deduce from it that there will be an exam. If they accept the first conjunct they are disregarding the second, while if they accept the second they have no reason to believe the first.

In the hard case, there is no room for the teacher's statement to prove unreliable, and this simply leads to a contradiction.

2
  • Thank you for the answer. When you say "students did not expect", does that mean they were certain there would be no exam, but there was an exam, or that while they thought there was a nonzero probability of an exam, it was still a surprise because this probability was not 1. Also, when you use "reason to belive" vs "entitled to believe". Is the former a case where the probability of the event is strictly between 0 and 1 and latter is the case where the probability of the event is exactly 1? Or, are the two words "reason to xxxx" completely equivalent to "entitled to xxxx" Dec 3, 2023 at 23:59
  • 1
    In the easy case, we are not concerned with what is certain but with what the students have some plausible reason to believe. If Friday afternoon arrives, the teacher's statement has proved to be something they cannot rely on, so they are not in a position to deduce that an exam will happen. Hence they do not expect it. You might say they will be surprised whether or not the exam happens at that point.
    – Bumble
    Dec 4, 2023 at 0:35

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .