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354 logic/unexpected.p


This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

354 logic/unexpected.p

Swedish civil defense authorities announced that a civil defense drill would
be held one day the following week, but the actual day would be a surprise.
However, we can prove by induction that the drill cannot be held. Clearly,
they cannot wait until Friday, since everyone will know it will be held that
day. But if it cannot be held on Friday, then by induction it cannot be held
on Thursday, Wednesday, or indeed on any day.

What is wrong with this proof?


This problem has generated a vast literature (see below). Several
solutions of the paradox have been proposed, but as with most paradoxes
there is no consensus on which solution is the "right" one.

The earliest writers (O'Connor, Cohen, Alexander) see the announcement as
simply a statement whose utterance refutes itself. If I tell you that I
will have a surprise birthday party for you and then tell you all the
details, including the exact time and place, then I destroy the surprise,
refuting my statement that the birthday will be a surprise.

Soon, however, it was noticed that the drill could occur (say on Wednesday),
and still be a surprise. Thus the announcement is vindicated instead of
being refuted. So a puzzle remains.

One school of thought (Scriven, Shaw, Medlin, Fitch, Windt) interprets
the announcement that the drill is unexpected as saying that the date
of the drill cannot be deduced in advanced. This begs the question,
deduced from which premises? Examination of the inductive argument
shows that one of the premises used is the announcement itself, and in
particular the fact that the drill is unexpected. Thus the word
"unexpected" is defined circularly. Shaw and Medlin claim that this
circularity is illegitimate and is the source of the paradox. Fitch
uses Godelian techniques to produce a fully rigorous self-referential
announcement, and shows that the resulting proposition is
self-contradictory. However, none of these authors explain how it can
be that this illegitimate or self-contradictory announcement
nevertheless appears to be vindicated when the drill occurs. In other
words, what they have shown is that under one interpretation of "surprise"
the announcement is faulty, but their interpretation does not capture the
intuition that the drill really is a surprise when it occurs and thus
they are open to the charge that they have not captured the essence of
the paradox.

Another school of thought (Quine, Kaplan and Montague, Binkley,
Harrison, Wright and Sudbury, McClelland, Chihara, Sorenson) interprets
"surprise" in terms of "knowing" instead of "deducing." Quine claims
that the victims of the drill cannot assert that on the eve of the last
day they will "know" that the drill will occur on the next day. This
blocks the inductive argument from the start, but Quine is not very
explicit in showing what exactly is wrong with our strong intuition
that everybody will "know" on the eve of the last day that the drill
will occur on the following day. Later writers formalize the paradox
using modal logic (a logic that attempts to represent propositions
about knowing and believing) and suggest that various axioms about
knowing are at fault, e.g., the axiom that if one knows something, then
one knows that one knows it (the "KK axiom"). Sorenson, however,
formulates three ingenious variations of the paradox that are
independent of these doubtful axioms, and suggests instead that the
problem is that the announcement involves a "blindspot": a statement
that is true but which cannot be known by certain individuals even if
they are presented with the statement. This idea was foreshadowed by
O'Beirne and Binkley. Unfortunately, a full discussion of how this
blocks the paradox is beyond the scope of this summary.

Finally, there are two other approaches that deserve mention. Cargile
interprets the paradox as a game between ideally rational agents and finds
fault with the notion that ideally rational agents will arrive at the same
conclusion independently of the situation they find themselves in. Olin
interprets the paradox as an issue about justified belief: on the eve of
the last day one cannot be justified in believing BOTH that the drill will
occur on the next day AND that the drill will be a surprise even if both
statements turn out to be true; hence the argument cannot proceed and the
drill can be a surprise even on the last day.

For those who wish to read some of the literature, good papers to start with
are Bennett-Cargile and both papers of Sorenson. All of these provide
overviews of previous work and point out some errors, and so it's helpful to
read them before reading the original papers. For further reading on the
"deducibility" side, Shaw, Medlin and Fitch are good representatives. Other
papers that are definitely worth reading are Quine, Binkley, and Olin.

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,
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,
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,
C. Harrison, "The Unanticipated Examination in View of Kripke's Semantics
for Modal Logic," in Philosophical Logic, J. Davis et al (ed.), Dordrecht,
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,
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.

-- tycchow@math.mit.edu.


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