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Question closure to solve the surprise test


This paper offers a new solution to the Surprise Test Paradox. The paradox arises thanks to an ingenious argument that seems to show that surprise tests are impossible. My solution to the paradox states that it relies on a questionable closure principle. This closure principle says that if one knows something and competently deduces something else, one knows the further thing. This principle has been endorsed by John Hawthorne and Timothy Williamson, among others, and I trace its motivation back to work by Alvin Goldman. I provide counterexamples to the principle and explain the flaw in the reasoning of those who defend it.

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  1. For a nearly complete list of papers written on the paradox, see (Chow 2011, p. 14).

  2. Some may wish for something stronger than this, namely an identification of a false proposition. But this cannot always be provided. For instance, consider the epistemicist solution to the sorites paradox concerning baldness. The epistemicist will say that one proposition of the form ‘if someone with i hairs is bald, then someone with \(i+1\) hairs is bald’ is false. But it is part of the epistemicist solution that we do not know which proposition this is.

  3. Thanks to an anonymous referee for this journal for pressing this worry.

  4. Thanks to an anonymous referee for this journal for pressing this worry.

  5. Thanks to an anonymous referee for this journal for pressing this worry.

  6. Thanks to an anonymous referee for this journal for pressing this worry.

  7. For examples, see e.g. Atkins and Nance (2014, p. 37) Campbell et al. (2010, p. 12), Feldman (2001, p. 64), Kvanvig (2004, p. 207), Leite (2010, p. 40), Pritchard (2002, p. 245) and Steup (2013).

  8. See, e.g Ferguson (1991, p. 293).

  9. See e.g. Levi (2000, pp. 449–451).

  10. Discussions of this reason include those by Chapman and Butler (1965, p. 424), O’Beirne (1961, p. 464) and Olin (1983, pp. 228–231).

  11. See Ayer (1973, p. 125) and Williamson (1992, p. 233) for this formulation of the surprise test paradox.

  12. It should be noted that several authors have offered arguments similar to the one that I give, although they have not been as fully fleshed out. James Cargile writes that ‘the teacher ...should ...[hold] the test not Monday but Tuesday. Then if the students try to claim to have known Monday evening the test would be Tuesday, the teacher can simply point out that similar grounds hadn’t been very reliable with respect to Monday (Cargile 1967, p. 552). See also Levy (2009, p. 140) and Williamson (1992, p. 223).

  13. Some authors, instead of using phrases like these, use phrases like ‘by similar reasoning.’ We can run an altered version of the teacher’s argument to account for this by altering (3) to (3*): If a reasoning process and those it is similar to, are unreliable, then they do not produce knowledge.

  14. It is worth noting that Reliability is somewhat controversial, in part because Timothy Williamson has used a very similar principle in an argument for a fairly controversial conclusion. In particular, he used it to argue that various mental states, such as feeling cold, are not luminous, that is, are not such that if one instantiates them, one is in a position to know one instantiates them Williamson (2000). To those who take Reliability to be implausible I offer the following challenge: do you agree that the student in the Conditional Surprise does not know on Wednesday that if he receives a test at all that week, he’ll receive it on Wednesday? If so, give me an argument for why he does not know this that does not make use of Reliability.

  15. It is less clear that Reliability will remain true if it is altered by replacing ‘knows’ with ‘rationally believes’ or ‘proves.’ This is noteworthy because economists and mathematicians often define ‘surprise test’ in terms of rational belief or proof. One nice thing about my solution to the surprise test paradox is that it can account for why mathematicians and economists tend to think the student’s reasoning is sound. In particular, in calling certain premises of the student’s argument into question, I make use of the idea that knowledge requires reliability. But if rational belief and proof do not require reliability, then I need not hold that the student’s reasoning is unsound if it talks about a surprise test defined in terms of these notions, and thus my approach is successful in solving such versions of the surprise test paradox.

  16. I will avoid endorsing a particular view about the best way to individuate belief-forming processes because I wish my argument to be compatible with multiple ways of individuating belief-forming processes.

  17. Of course, I am assuming every version of the surprise test paradox will involve multiple days on which the test can be given. Some contend that there is a one-day version of the surprise test paradox in which the teacher tells the student: ‘tomorrow you will receive a surprise test’ and the student nonetheless receives a surprise test. My solution cannot handle this formulation, but only multiple day formulations. That said, I would be in good company if I decided to declare the one-day version not a real version of the surprise test paradox. Timothy Chow, who maintains what is arguably the definitive bibliography on the surprise test paradox, notes that in updating his bibliography, ‘I decided that [the one day version] was distinct from the surprise examination paradox, and I even went back and deleted some entries that I had included in previous versions of my bibliography that were concerned only with [it]’ Chow (2011, p. 14).


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Thanks for comments to Katie Finley, Ryan Hammond, Richard Kim, Graham Leach-Krouse, Marc Moffett, Ted Shear, Jeff Speaks, Fritz Warfield, an audience at the 2013 Pacific APA and two anonymous referees.

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Correspondence to Daniel Immerman.

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Immerman, D. Question closure to solve the surprise test. Synthese 194, 4583–4596 (2017).

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  • Surprise test
  • Closure principle
  • Reliabilism