5 - Surprise and Self-Knowledge
Published online by Cambridge University Press: 10 November 2009
Summary
HERE is a familiar puzzle. Ateacher announces on Monday that there will be a surprise exam on either Wednesday or Friday. Her students reason as follows. Say that the teacher's announcement is true. Then, if the exam were on Friday, we would know by Thursday, and it wouldn't be a surprise. Therefore it won't be on Friday. This means it must be on Wednesday, and since we know that, it can't then surprise us. There can't be a surprise on either of these days. The teacher's announcement is false. It contradicts itself.
The teacher gives the exam on Friday and everyone is surprised. Where did the students go wrong? This has been much discussed, and I want to discuss it once more. I want then to extend my discussion to some larger issues.
Let me bring it down to just a single student's problem. The announcement can be put as five propositions. There will be an exam on either Wednesday or Friday:
(1)W ∨ F
it will come as a surprise to this student. If, that is, it will be on Wednesday, he won't, on Tuesday, believe it will be on Wednesday. And if it will be on Friday, he won't, on Thursday, believe it will be on Friday:
(2)W ⊃ ∼BtW
(3)F ⊃ ∼BthF1
also that
(4)∼(W · F) and
(5)∼W ⊃ Bth∼W
too are a part of what was announced.
suppose that the test will come Friday. This begins a conditional proof2:
(6)F
(7)∼BthF from (6) and (3)
(8)∼W ⊃ from (6) and (4)
(9)Bth∼W from (8) and (5)
want to proceed to BthF, but (9) and (1) don't warrant that.
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- Ambiguity and Logic , pp. 79 - 96Publisher: Cambridge University PressPrint publication year: 2003