Capital_Net_6438 t1_j257nt0 wrote on December 29, 2022 at 7:17 PM

The teacher in the surprise quiz paradox announces on day 1 that there will be a surprise quiz this week, which has 5 days. The paradox involves an argument that purports to show the impossibility of… something. Sometimes the argument is explained as trying to show a surprise quiz is impossible. I don’t think that works for reasons I won’t belabor.

The argument could also be taken as trying to show knowledge of a surprise quiz on day 1 is impossible. So suppose for reductio that the student knows on day 1 that there will be a surprise quiz. Suppose that at the end of day 4 there has been no quiz. We assume if there’s been no quiz by a certain point, then the student knows that. So at the end of day 4, the student knows there’s been no quiz. And therefore, it would seem, he knows there’ll be a quiz on day 5. But a quiz that is known to happen on a certain day is not a surprise. Therefore the quiz can’t happen Friday.

Then you go through the same process for the other days, ultimately proving the quiz can’t happen any day. And therefore the student doesn’t know there will be a quiz.

I assume that the argument is supposed to deduce this or that. I.e. it’s not just that certain assumptions make certain consequences likely but that they follow logically.

The argument fails at the step that says the student knows on day 4 there will be quiz on day 5. It’s a rudimentary mistake. Just b/c he knows on day 1 that there’ll be a quiz it doesn’t follow that he still knows on day 4 that there will be a quiz. It’s not in general true that knowing something one day will guarantee that you continue knowing it later. There’s nothing in the argument to make one think the student’s knowledge does survive changing circumstances here.

The exercise is meant to deduce something. No principle has been presented to suppose the student’s knowledge must survive in this case. So the appropriate response is that the argument does not establish what it set out to establish since there is no reason to credit its critical inference.

But…

At the end of day 4 the student thinks back and remembers believing on day 1 that there would be a surprise quiz. We might wonder whether the student knows on day 4 that he knew on day 1 that there would be a surprise quiz.

Suppose knowledge is true belief in internal and external circumstances conducive to knowledge. The student is special. He will know something in this context iff the proposition is available to be known. The student’s internal and external circumstances on day 4 are conducive to knowing whatever he knew on day 1. So it seems the student should know on day 4 that he knew on day 1 that there would be a surprise quiz.

Knowledge that P at t entails that P is true at t. And being special, the student knows the entailments of the things he knows. So he knows that his day one knowledge entails that it was true on day 1 that there would be a surprise quiz.

It would seem that if the student knows p (that he knows on day 1 there will be a surprise quiz), knows p entails q (that his knowledge on day 1 that there will be a surprise quiz entails that it’s true on day one that there will be a surprise quiz), then he knows q (that it’s true on day one that there will be a surprise quiz).

Now we have that on day four the student knows it was true on day one that there will be a surprise quiz this week. That seems to get us back to the student having the impossible knowledge that there will be a surprise quiz on day five.

That’s where I’ve been stuck for a while. Maybe we can say there’s no guarantee that day 1 knowledge will lead to day 4 knowledge of day one knowledge.

[deleted] t1_j26zhc1 wrote on December 30, 2022 at 2:33 AM

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Capital_Net_6438 t1_j28qpoi wrote on December 30, 2022 at 1:41 PM

As a fan of logic, rejecting logic isn’t an option for me.

Suppose we don’t assume anything about anyone’s knowledge on day 1. And suppose, as is usually the case, that we’re considering proving a surprise quiz is impossible. Then it surely does not follow that a surprise quiz can’t happen on day 5.

The argument is supposed to go that at the end of day 4, the student knows there’ll be a quiz on day 5. But he has no idea really. We didn’t build him having knowledge into the setup at day 1 and therefore he won’t magically have knowledge at day 4. The assertion that he does have knowledge at that point is totally unsupported.

So of course if we don’t assume anything about the student’s knowledge in the setup there could be a surprise quiz on day 5. Day 5 comes; a quiz happens; and the ignorant student says - “wow, I didn’t see that coming.”

[deleted] t1_j28uzg2 wrote on December 30, 2022 at 2:16 PM

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Capital_Net_6438 t1_j2994dl wrote on December 30, 2022 at 3:58 PM

Thanks for the clarification.

Seems like the surprise quiz paradox isn’t unique in illustrating the flaws of formal logic from your perspective. So ideally one would bracket those in thinking about the SQP. Perhaps the paradox isn’t so paradoxical for independent reasons.

On the kk situation, I’m thinking maybe the student doesn’t know on day 4 that he knew on day 1. I feel totally fine resisting the inference that his knowledge has to survive the change in circumstances. Why shouldn’t it be similarly unlikely that his knowledge of knowledge survives? As the student thinks about things at the end of day 4, the argument has given little assurance that he’ll know that he knew. He should think on day 4, “huh, maybe I never knew.”

One way to think of this is that the student knows that he knows in general. If he knows p at t then he knows that he knows p at t. That’s probably an assumption the student needs. And his kk knowledge is no more guaranteed to survive the changing circumstances than his knowledge.