HL 6. The Surprise Quiz

6. The Surprise Quiz Michael Huemer1
Note:6@@@@@@@@@@POSSIBILE APPLICAZIONE A EAAN E ALL EVOLUZIONIAMOIL TEISMO E' UNA CREDENZA STABILE (UNA VOLTA ADOTTATA NON VA RETTIFICATA)L'EVOLUZIONISMO ATEO E' UNA CREDENZA AUTORIMUOVENTE (UNA VOLTA ADOTTATA DEVE ESSERE RETTIFICATA DA MAGGIORI DUBBI)COME AVVIENE QUESTA RETTIFICA?: QUESTO CAPITOLO LO SPIEGA POICHé ANCHE LA CREDENZA DEL QUIZ SORPRENDENTE E' INSTABILE

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6.1 The Paradox
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A professor announces to his class that there will be a surprise quiz next week.
Note:DESCRIZIONE

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The class meets five days a week, Monday through Friday.
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The students go home and try to deduce when the quiz will occur. “Suppose it comes on Friday,” one student reasons. “Then we’d know by the end of Thursday’s class that it was coming on Friday, since that would be the only day of the week left. So then it wouldn’t be a surprise. So the quiz can’t be on Friday.” “What about Thursday?” says another. “Since we’ve just eliminated Friday, that means that if it hasn’t come by Wednesday evening, we’d know it was coming on Thursday, and it wouldn’t be a surprise. So it can’t be on Thursday either.” By similar reasoning, they eliminate Wednesday, then Tuesday, then Monday.
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E COSÌ VIA

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The students breathe a sigh of relief, knowing that the surprise quiz cannot be given at all. They spend the weekend partying and getting drunk instead of studying. The next week, the professor gives a quiz on Wednesday. Everyone is surprised.
Note:CONCLUSIONE

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6.2 Rejecting the Assumptions
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The students believe without question everything the professor asserts
Note:PRIMO ASSUNTI

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The students believe all the logical consequences of their beliefs.
Note:SECONDO ASSUNTO...RAZIONALISMO DEI RAGAZZI

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The students are fully aware of their own mental states and abilities.
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these are not realistic assumptions
Note:GIUDIZIO

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Perhaps it would be impossible to surprise ideal (perfectly-reasoning, perfectly self-aware, credulous) students.
Note:SOLUZIINE VENTILATA...MA NSODDISFACENTE

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it is highly implausible that one cannot give a surprise quiz to an ideal student
Note:INFATTI

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If the ideal student reasons to the conclusion that the surprise quiz cannot be given on any day, then the student will, after all, be surprised
Note:E POI LA CONTRADDIZ RESTA

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a perfect reasoner cannot be perfectly credulous.
Note:FORSE LE PREMESSE NN SONO FALSE MA CONTRADDITTORIE

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They are, however, reasonably credulous: they are prima facie disposed to accept the professor’s statements about the conduct of the class, unless and until they have good reasons for doubting those statements. This looks like enough to get the paradox going.
Note:MA IL PARADOSSO PUÓ ESSERE RILASSATO E RESO PIÙ CREDIBILE ELIMINANDO IL RISCHIO DI CONTRADDIZIONE

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6.3 What Is Surprise?
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thinking in terms of vague,
Note:L ORIGINE DI MOLTI PARADOSSI

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You are surprised by an event when you previously had zero credence that it would occur.
Note:PRIMA DEF DI SORPRESA

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You are surprised by an event when you previously assigned it less than 100% credence.
Note:SECONDA

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You are surprised by an event when you previously considered it sufficiently unlikely, that is, your credence that it would occur fell below some threshold, t.
Note:TERZA E PIÙ PLAUSIBILE

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“surprise threshold”
Note:CONCETTO

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This is both vague and context-dependent.
Note:VAGHEZZA NN ELIMINATA

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the teacher’s initial announcement, strictly speaking, lacks propositional content.
Note:CONSEGUENZA DELLA VAGHEZZA

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nor is this vagueness the source of the paradox. The teacher’s announcement has sufficient meaning to be aptly made,
Note:MA QUI IL PARADOSSO CONTINUA A REGGERE

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we aptly use vague language all the time in ordinary life.
Note:COME LA VAGHEZZA CHE REGGE NEL LINGUAGGIO COMUNE

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6.4 Quiz Comes if and Only if Surprising
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If the quiz has not come by the end of class on Thursday,
Note:COSA PENSA LO STUDENTE IN QS CONDIZIONI?

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The professor is committed to giving a quiz,
Note:1

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The professor is committed to giving a quiz if and only if it will be a surprise.
Note:2

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Case (ii), however, generates a (sub-)paradox, which is the subject of the present section;
Note:SUB PARADOX

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6.4.1 Self-Undermining Beliefs with a Vague Surprise Threshold
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Thursday night, one student says, “Well, it looks like the quiz is coming tomorrow.” Another says, “But wait. Since you just predicted that, that means it won’t be a surprise. Since he has no interest in giving a non-surprise quiz, the prof won’t give a quiz at all.” A third replies, “But wait. Since you just predicted no quiz, that means that it would be a surprise if he gave one.
Note:IL SUB PARADOSSO...DESCRIZIONE

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A fourth replies, “But wait . . .”
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If the students think there will be a quiz, they can infer that there won’t.
Note:IN SINTESI

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What can the students rationally do?
Note:LA DOMANDA.....

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Answer: assign an intermediate degree of belief.
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assign a degree of belief such that, if a quiz is given, it will be a borderline case of a surprise.
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Fig. 6.1 Optimal credence with a vague surprise threshold
Note:CI CREDENZA INIZIALE: CHE PROB CI SONO CHE LO DARÀ?CF CREDENZA INFERITA: DATA CI INFERISCO CHE LO DARÀ ALLA PROB XCI E CF SI INSEGUONO.L UNICA CREDENZA STABILE È QIELLA SULLA BOSETTRICE

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6.4.2 Self-Undermining Beliefs with a Precise Threshold
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if your credence in some event is exactly 60% or below, then the event will count as “surprising”
Note:SUPPONIAMO DI ESSERE PRECISI

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if your credence is 60%, it should be much more than 60%. There is no stable credence.
Note:NN C È STABILITÀ

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For example, imagine a series of one thousand cups of water, cup 1 through cup 1,000, each having a very slightly higher physical temperature than the last. You try feeling the water in each cup, in sequence. For any two adjacent cups, you can’t tell which one feels warmer to you. But by the end, you can tell that cup 1,000 feels much warmer than cup 1. This shows that there can be temperature sensations so similar that one cannot introspectively distinguish them, even though one is in fact warmer-feeling than the other.
Note:SULLA CONTINUITÀ DEGLI STATI MENTALI

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A credence of 0.6 would be introspectively indistinguishable from a credence of 0.600001 or 0.599999.
Note:LA CONTINUITÁ DEGLI STATI MENTALI NN ESISTE....QUINDI LA SOGLIA PRECISA È IMPOSSIBILE

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if the quiz were on Friday, it would be a borderline case of a surprise (if “surprise” is vague),
Note:CONCLUSIONE

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6.4.3 The Rest of the Week
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6.5 Quiz Comes, with or without Surprise
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The professor is committed to giving a quiz, whether or not it will be a surprise.
Note:CONSIDERIAMO QS EVENTUALITÀ

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the professor would prefer the quiz to be a surprise,
Note:PRECISAZIONE

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6.5.1 No Friday Surprise
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the students should adopt a very high credence that a quiz will come on Friday. Not 100%, because there is always a chance that the professor will have forgotten,
Note:GIOVEDÌ SERA E ANCORA NESSUN QUIZ

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the quiz very probably will not be on Friday.
Note:IL PENSIERO DELLA DOMENICA SERA

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6.5.2 Borderline Thursday Surprise
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he’ll probably give it Thursday.
Note:MERCOLEDÍ SERA...CONSIDERANDO CHE VENE SAREBBE POCO SORPRENDENTE...

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Since you just made that prediction, it wouldn’t be a surprise on Thursday
Note:MA....SI RIPETE LA SITAUZIONE PRECEDENTE IL GIOVEDÌ ANZICHÈ VENERDÌ

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Since you just made that prediction, it would be a surprise tomorrow.
Note:INFATTI

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we have a self-undermining belief system.
Note:ANCORA COME PRIMA

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The solution is to assign a credence at about the threshold.
Note:LO STUDENTE RAZIONALE

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if the quiz occurred on Thursday, it would be a borderline case of a surprise,
Note:QUINDI

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6.6 Surprising as Not-Most-Expected
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There is a version of the story where it would be impossible to give a surprise quiz. Suppose we define an “expected” event as one that was antecedently considered the most likely of the alternatives
Note:UN CASO OSTICO

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On this definition of “surprise quiz”, the professor cannot follow through on his promise
Note:PROMESSA IMPOSSIBILE

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He cannot do it on Friday, since on Thursday night, trivially, the students would consider Friday the most likely day
Note:ESEMPIO

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So the professor can’t give a surprise quiz. But this is not paradoxical. This is just a result of the peculiar definition of “surprise”,
NESSUN PARADOSSO