From: tycchow@riesz.mit.edu (Timothy Y. Chow)
Subject: FAQL: unexpected.p
Organization: None.  This saves me from writing a disclaimer.
Date: Thu, 13 Aug 92 04:06:07 GMT

I think the FAQL entry about the unexpected drill paradox should be revised.
I quote the current solution below:

<The resolution of this paradox is as follows.  The argument that
<the drill could not be held at all uses the following inductive step:
<
<      We have shown inductively that the element of surprise would 
<      be lost if the drill was held on the n+1 th day, or any later
<      day.  Therefore, the nth day is the last day that the drill
<      can be held.  But if it is held on the nth day, the public,
<      having realized beforehand that this was the last possible
<      day, will not be surprised.  Therefore, if the authorities
<      want to maintain the element of surprise, they cannot hold
<      the drill on the nth day.
<
<But the problem is that this induction assumes that certain 
<assumptions are certain when in fact they are only probable.
<Let's follow this through step by step.  Assume the week runs
<from Sunday to Saturday.  Would the authorities hold the drill
<on Saturday? Perhaps they would not consider it.  But is the 
<public sure that they won't hold it on Saturday? No.  Because
<the authorities are human, and just because your reasoning
<tells you that they shouldn't hold the drill on Saturday, doesn't
<mean they won't.  The public can only say that it's very unlikely 
<that they'll hold it on Saturday.
<
<Now, the induction step says that since the government won't
<hold the drill on Saturday, they won't hold it on Friday either,
<because if they did, the public wouldn't be surprised.  But 
<the truth is that the public would be a little surprised, because
<they're not absolutely sure that it won't be on Saturday.  And that
<small element of surprise might be all the government wants.  So all
<you can say is that it's pretty unlikely that the drill would be held
<Friday, but certainly not impossible.  
<
<If they hold the drill on Thurday, there will be a considerable element
<of surprise, because the public wouldn't consider it at all impossible 
<for it to be Friday or Saturday.  So even if the government wasn't 
<satisfied with the degree of surprise on Friday, they might be 
<satisfied with the degree of surprise on Thurday.
<
<And so on.  The probability of having the drill increases as you 
<move backwards through the week, because the element of surprise
<increases.

There are at least two broad schools of thought on this paradox, one of
which interprets "unexpected" epistemologically or psychologically (i.e.,
in terms of knowledge or feelings of surprise) and the other of which
interprets "unexpected" logically (i.e., day x is unexpected day = cannot
produce a logical proof that x is the day).

Currently the FAQL discusses only the former view (advocated notably by
Quine and Olin) and ignores the latter.  In fact, the list of references
does not even mention most of the work done on the logical versions of the
paradox (with the exception of Medlin and perhaps Shaw).

I think the FAQL should at least point out that there have been attempts to
formalize the paradox, and should include these other references.  I have
included below an expanded and corrected version of the FAQL reference list.
Page numbers have been added where I have been able, and some typos have
been fixed.

Furthermore the list is too unwieldy for any but the most fanatical to
wade through, so perhaps some suggestions for which ones are most worth
reading are in order.  At a minimum I would recommend Quine, Shaw,
Bennett-Cargile, Olin, and Sorenson.  For those with some logic
background, Medlin and Fitch are good representatives of the logic
papers.  In general, the papers after 1970 are more illuminating than
the earlier ones (with the exceptions noted above), and after reading
one or two of the logic papers you probably don't need to read the rest.


D. O'Connor, "Pragmatic Paradoxes," Mind 57:358-9, 1948.
L. Cohen, "Mr. O'Connor's 'Pragmatic Paradoxes,'" Mind 59:85-7, 1950.
P. Alexander, "Pragmatic Paradoxes," Mind 59:536-8, 1950.
M. Scriven, "Paradoxical Announcements," Mind 60:403-7, 1951.
D. O'Connor, "Pragmatic Paradoxes and Fugitive Propositions," Mind 60:536-8,
 1951
P. Weiss, "The Prediction Paradox," Mind 61:265ff, 1952.
W. Quine, "On A So-Called Paradox," Mind 62:65-7, 1953.
R. Shaw, "The Paradox of the Unexpected Examination," Mind 67:382-4, 1958.
A. Lyon, "The Prediction Paradox," Mind 68:510-7, 1959.
D. Kaplan and R. Montague, "A Paradox Regained," Notre Dame J Formal Logic
 1:79-90, 1960.
G. Nerlich, "Unexpected Examinations and Unprovable Statements," Mind
 70:503-13, 1961.
M. Gardner, "A New Prediction Paradox," Brit J Phil Sci 13:51, 1962.
K. Popper, "A Comment on the New Prediction Paradox," Brit J Phil Sci 13:51,
 1962.
B. Medlin, "The Unexpected Examination," Am Phil Q 1:66-72, 1964.
F. Fitch, "A Goedelized Formulation of the Prediction Paradox," Am Phil Q
 1:161-4, 1964.
R. Sharpe, "The Unexpected Examination," Mind 74:255, 1965.
J. Chapman & R. Butler, "On Quine's So-Called 'Paradox,'" Mind 74:424-5, 1965.
J. Bennett and J. Cargile, Reviews, J Symb Logic 30:101-3, 1965.
J. Schoenberg, "A Note on the Logical Fallacy in the Paradox of the
 Unexpected Examination," Mind 75:125-7, 1966.
J. Wright, "The Surprise Exam: Prediction on the Last Day Uncertain," Mind
 76:115-7, 1967.
J. Cargile, "The Surprise Test Paradox," J Phil 64:550-63, 1967.
R. Binkley, "The Surprise Examination in Modal Logic," J Phil 65:127-36,
 1968.
C. Harrison, "The Unanticipated Examination in View of Kripke's Semantics
 for Modal Logic," in Philosophical Logic, J. Davis et al (ed.), Dordrecht,
 1969.
P. Windt, "The Liar in the Prediction Paradox," Am Phil Q 10:65-8, 1973.
A. Ayer, "On a Supposed Antinomy," Mind 82:125-6, 1973.
M. Edman, "The Prediction Paradox," Theoria 40:166-75, 1974.
J. McClelland & C. Chihara, "The Surprise Examination Paradox," J Phil Logic
 4:71-89, 1975.
C. Wright and A. Sudbury, "The Paradox of the Unexpected Examination,"
 Aust J Phil 55:41-58, 1977.
I. Kvart, "The Paradox of the Surprise Examination," Logique et Analyse
 337-344, 1978.
R. Sorenson, "Recalcitrant Versions of the Prediction Paradox," Aust J Phil
 69:355-62, 1982.
D. Olin, "The Prediction Paradox Resolved," Phil Stud 44:225-33, 1983.
R. Sorenson, "Conditional Blindspots and the Knowledge Squeeze: A Solution to
 the Prediction Paradox," Aust J Phil 62:126-35, 1984.
C. Chihara, "Olin, Quine and the Surprise Examination," Phil Stud 47:191-9,
 1985.
R. Kirkham, "The Two Paradoxes of the Unexpected Hanging," Phil Stud
 49:19-26, 1986.
D. Olin, "The Prediction Paradox: Resolving Recalcitrant Variations," Aust J
 Phil 64:181-9, 1986.
C. Janaway, "Knowing About Surprises: A Supposed Antinomy Revisited," Mind
 98:391-410, 1989.

-- 
Tim Chow     tycchow@math.mit.edu
Where a calculator on the ENIAC is equipped with 18,000 vacuum tubes and weighs
30 tons, computers in the future may have only 1,000 vacuum tubes and weigh
only 1 1/2 tons.                               ---Popular Mechanics, March 1949