Discussing the Logical Argument of a Quiz Given on Unknown Day

[disregardthat]

There is a logical argument I'd like to have some comments on. I read it in a book, and I am not sure about it.

"An instructor teaches a class five days a week, Monday through Friday. She tells her class that she will give one more quiz during the final week of classes, but that the students will not know for sure the quiz will be that day until they come to the classroom. What is the last day of the week she can give toe quiz to satisfy these conditions?"

Consider the following proof:

The teacher cannot give the students their tests on Friday, because then they would know on Thursday (not being tested) when they will get their test. Thus Friday is eliminated, and Thursday is the last possible day of being tested. However, if they did not receive their test on Wednesday, they would know what day they would receive the test - again violating the rules. Thus Thursday is eliminated. The argument continues to eliminate Wednesday and Tuesday, and thus Monday is the only possible day left to receive the test. This also violates the rules, since they would know it beforehand.

I do have some comments on this argument. For example, when eliminating Friday, we assume Friday already is a possible day for receiving the test. However, when eliminating Thursday we use the conclusion while simultaneously considering Friday impossible. Would you consider this a dismantle of the previous proof?

The proof does imply that the students know the test will be given Monday. But if the teacher decide to give it on Wednesday, this knowledge is false. However, they also know that the teacher will not violate the rules: so is giving the test at all a violation of the rules?

I'd like some comments on the possible proof, and the possible dismantling of the proof.

And finally, do you know the correct answer?

 

[HallsofIvy]

No, that proof does NOT "imply that the students know the test will be given Monday". Because the teacher said "the students will not know for sure the quiz will be that day until they come to the classroom", they cannot know that!

This is a very old problem and I doubt that your teach really expects any students to come up with an answer.

 

[Redbelly98]

... is giving the test at all a violation of the rules?

Yes. That is what the given proof tells us.

However, not giving a test is also a violation of one of the conditions: "she will give one more quiz during the final week of classes".

 

[disregardthat]

No, that proof does NOT "imply that the students know the test will be given Monday". Because the teacher said "the students will not know for sure the quiz will be that day until they come to the classroom", they cannot know that!

This is a very old problem and I doubt that your teach really expects any students to come up with an answer.

Indeed, my book is also very old. I don't have a teacher in this subject though.

Yes, of course they do not know they will get the test because they know the teacher will not violate the rules. What I meant was a reference to the last part of the proof, it is not very relevant however to what I want to have comments on.

The obvious conclusion is of course, if the proof is correct, that they cannot get the test at all while the teacher is not violating the rules. (If we exclude the condition that the teacher must give the test at all even if it is logically impossible) This is however strangely unintuitive, and since my book implied it was not true earlier, I find reason to suspect it. So you don't agree with me that the proof, when iterating to another day, assumes a conclusion which is implied by holding a predicate possible - and then goes on negating the same possibility? For example, when excluding Friday as a possibility, one does keep it as a possible option (which is essential to the conclusion) - and the rules has not yet (in the sense of this stage at the proof) been violated until Thursday. At this point the rules have been violated, and we conclude that Friday is an impossibility. But then, as we goes on excluding Thursday we accept our conclusion of the impossibility of Friday. The proof of that however assumes that Friday is possible in the first place. Why is the proof not a committing a logical error with respect to this?

Another point is this, if we accept the conclusion of the proof as a whole, and then observe the steps - they all seem to be invalid. For example, the proof of the impossibility of Friday assumes that Thursday has gone by not violating the rules plus keeping Friday a possibility. But our ultimate conclusion has been that this is not a possible event.

 

[g_edgar]

You should find Martin Gardner's discussion on this, from the "Mathematical Games" column. I don't know which of the collections this one is in, though.

Here is a variant of the same paradox, from that column, although the details may be different...

I draw a card, look at in, don't show it to you, and then I say:

"This card is the king of hearts, and you have no way of knowing what this card is."

What are you to make of that? If I told the truth when I said "This card is the king of hearts," then you DO have some way of knowing, so the second clause is false, and my statement (being an AND statement) is false. On the other hand, if I lied when I said "This card is the king of hearts," then of course the statement is also false. So: since the statement is false, you can get no information from it and you have no way of knowing what the card is.

Now, I turn over the card, and it is, indeed, the king of hearts. Was my statement then, in fact, true when I made it?

 

[disregardthat]

You should find Martin Gardner's discussion on this, from the "Mathematical Games" column. I don't know which of the collections this one is in, though.

Here is a variant of the same paradox, from that column, although the details may be different...

I draw a card, look at in, don't show it to you, and then I say:

"This card is the king of hearts, and you have no way of knowing what this card is."

What are you to make of that? If I told the truth when I said "This card is the king of hearts," then you DO have some way of knowing, so the second clause is false, and my statement (being an AND statement) is false. On the other hand, if I lied when I said "This card is the king of hearts," then of course the statement is also false. So: since the statement is false, you can get no information from it and you have no way of knowing what the card is.

Now, I turn over the card, and it is, indeed, the king of hearts. Was my statement then, in fact, true when I made it?

That would be very interesting to read, do you know any source for these articles which I can browse?

 

[bpet]

"An instructor teaches a class five days a week, Monday through Friday. She tells her class that she will give one more quiz during the final week of classes, but that the students will not know for sure the quiz will be that day until they come to the classroom. What is the last day of the week she can give toe quiz to satisfy these conditions?"

This is a neat example of inconsistent axioms. The axioms (the teacher's statements accepted as fact) lead to a contradiction when we try to answer the question "What is the last day...". Although the question doesn't have an answer, the resulting contradiction forms the backbone of an elegant proof that the axioms are inconsistent.

There are lots of interesting paradoxes arising from inconsistent axioms (this has been a major driver of the development of mathematics). Some interesting related results you might come across are Russell's paradox, Godel's incompleteness theorems and Arrow's theorem of fair elections. Have fun!

 

[Office_Shredder]

Godel's incompleteness theorem and Arrow's dictatorship theorem are not examples of paradoxes caused by inconsistent axioms.

 

[disregardthat]

Godel's incompleteness theorem and Arrow's dictatorship theorem are not examples of paradoxes caused by inconsistent axioms.

He said was that they were related to, not caused by.

 

[disregardthat]

This is a neat example of inconsistent axioms. The axioms (the teacher's statements accepted as fact) lead to a contradiction when we try to answer the question "What is the last day...". Although the question doesn't have an answer, the resulting contradiction forms the backbone of an elegant proof that the axioms are inconsistent.

There are lots of interesting paradoxes arising from inconsistent axioms (this has been a major driver of the development of mathematics). Some interesting related results you might come across are Russell's paradox, Godel's incompleteness theorems and Arrow's theorem of fair elections. Have fun!

Hi, I am reading the book called Gödel, Escher, Bach; have you read it? It touches many aspects of logic, especially self-reference and the causes of paradoxes.

 

[bpet]

Hi, I am reading the book called Gödel, Escher, Bach; have you read it?

No - I did an undergrad course on the Godel bit a while back, and have been meaning to read it ever since. Sounds very interesting!

How is Bach linked to Godel though?

 

[disregardthat]

No - I did an undergrad course on the Godel bit a while back, and have been meaning to read it ever since. Sounds very interesting!

How is Bach linked to Godel though?

He compares logical systems to music, especially classical music which has certain "rules" one tend to follow. Also, there can be drawn the analogy between self-reference in logic and in classical music. Hofstadter explains it well in his book.

 

[mdnazmulh]

Hi Jarle, I thought I would also write something after reading ur thread. Today was a significant day in my life. First time I have been reading about mathematical logic since today morning and surprises after surprises were waiting for me. Before today I thought that math is always perfect ( just a loose term). But after knowing about Hilbert's program and Godel's incompleteness theorem I was stunned. I've never studied logic in my life before and being an engineering student and to engineers these pure math stuffs are not taught usually. Just from personal interest I started self-study of mathematical logic and I had to struggle to grasp the concept of consistency, completeness and other formal terminologies. But the subject is really interesting.

 

[disregardthat]

Hi Jarle, I thought I would also write something after reading ur thread. Today was a significant day in my life. First time I have been reading about mathematical logic since today morning and surprises after surprises were waiting for me. Before today I thought that math is always perfect ( just a loose term). But after knowing about Hilbert's program and Godel's incompleteness theorem I was stunned. I've never studied logic in my life before and being an engineering student and to engineers these pure math stuffs are not taught usually. Just from personal interest I started self-study of mathematical logic and I had to struggle to grasp the concept of consistency, completeness and other formal terminologies. But the subject is really interesting.

I had a similar experience when I first read about the incompleteness theorem and related discoveries. You should read "Gödel, Escher, Bach" if you are interesting in this.