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Section 1 -- The Paradox
It has been argued that a paradox arises if someone, usually a teacher, says that he will give a surprise test, i.e. a test whose future date is not specified. The argument is as follows:
a. A teacher says he will give a surprise test the next week;
If
c. it has to be on Friday,
d. But then it wouldn't be a surprise, since trivially, a test which is not a surprise is not a surprise test.
e. Thus, a surprise test cannot be given on Friday.
If
g. it has to be on either Thursday or Friday.
h. But, from e, a surprise test cannot be given on Friday.
i. Thus, it must be given on Thursday.
j. But, d.
k. So a surprise test cannot be given on Thursday.
If
m. it has to be on either Wednesday, Thursday or Friday.
n. But by e and k, exclude Thursday and Friday, so ...
Section 2:-- Two Meanings of "Surprise:"
I will show that
b. the are two (possibly more) meanings of 'surprise test" both of which are consistent with the definition given in section 1, but which obfuscate the simple logical error of the argument when used inconsistently to explicate a particular premise. These two meanings are
2. surprise2 test -- an unexpected (for any reason whatsoever) test.
Section 3: Breaking the Inductive Chain
My basic argument in response to the claim of paradox is this:
2. "I'll give a test any Wednesday of next week."
b. On Thursday night, if the test has not be given, a. is not the case, since only Friday is left. Thus a test on the next day cannot be a surprise test. This is merely a restatement of b., c. and d. from section 1.
c. But the generalized conclusion (which the paradox requires) has not been shown, i.e. section 1.e., A surprise test cannot be given on Friday. The strongest conclusion possible is:
d. On Wednesday night, assuming the test not yet to have been given, there are yet the possibilities of Thursday and Friday. But Friday cannot be excluded by introducing 1e as a premise, since this premise, "A test cannot be given on Friday," depends on conditions that hold only on Thursday night and which are incompatible with those obtaining on Wednesday night. Thus we cannot conclude that the test must (as the only day left) be given on Thursday.
e. Consideration of d. breaks down the induction.
To detail section 3:
b/: There are (remain) more than one possible date for a test.
c/: A test is preannounced.
d/: A test is unannounced.
(Note: a --> not-b; c --> not-d)
1. d/ is sufficient for a test to be a surprise test, i.e. for S to obtain.
2. c/ is neither necessary nor sufficient for S.
3. (c/ AND a/) entails not-S.
4. (c/ AND b/) is by definition S.
b. To begin: we exclude unannounced tests.
2. Thus, either the preannounced test has only one, or more than one day on which it can be given. a/ becomes necessary and sufficient for not-S and b/ becomes necessary and sufficient for S.
c. Let us reconsider the paradox.
2. On Wednesday night, b/ holds. Thus S. The test given Thursday is still a surprise test, since Friday remains as an alternative.
3. It is here that the paradox requires that Friday be excluded. But that is possible only when a/ holds. But from 2., b/ holds; and, b/ entails not-a/. Any conclusion derived by introducing the negation of an assumption as a premise, is trivial.
d. Imagine five cards dealt face down on a table. The dealer tells you that one and only one of them is an ace. You may turn the cards over one at a time from left to right. The following conditions are obvious:
2. If you have turned over only three cards, there is no reason to rule out the fifth card on the basis of the immediately preceding consideration, 1.
3. From 2., we conclude that the fourth card is not necessarily the ace.
Section 5: Responding to Objections
An objection may be raised. b/, There are (remain) more than one possible dates for a test, holds for only four of the five days of the week. On Wednesday night we know that tomorrow is the last possible day for a surprise test, i.e. a test for which b/ holds. Therefore, tomorrow's test, from the vantage point of Wednesday night (assuming no test to have been given yet) is not a surprise. It is strange that we can talk about "the last day for a surprise test" since "the last day" asserts a/ and "surprise test" asserts b/, which are contradictory.
We might want to avoid the self-contradiction of "Thursday is the last day for a surprise test" by asserting that
But this contradicts a/, or by asserting that
from which follow:
and the paradox again raises its head, or
But if tomorrow is the last date for such a test, and such a test is to be given, then it is no surprise, and thus no surprise test.
The solution is clear. Invoke the distinction between
2. surprise2 test -- an unexpected (for any reason whatsoever) test.
This is not necessarily to say that a/, There remains only one possible date for a test, holds. We did not define a/ and b/ to apply to tests which have had conditions of occurrence place on them, e.g. a/ and b/. We should not be surprised if, when we attempt self-reflexive definition, e.g. to attempt to define "a surprise test which is a surprise" in terms of "a test which is a surprise" (both meanings being surprise1) that we arrive at self-contradictory or paradoxical statements. (The reader is reminded of Russell's definition of the village barber as the man in the village who shaves every man who doesn't shave himself.)
The paradox is not the one of the impossibility of giving surprise1 tests. On Tuesday night, one cannot exclude Thursday as a day for given a surprise test (vacillate as one may between surprise1 and surprise2), even though on Wednesday night we may be able to make the statement, "Tomorrow is the last day for a surprise test." It just so happens that on Tuesday night, b/ holds non-self-contradictorily for tests which satisfy condition b/. As in section 4c, no induction can be started.
Section 6: To Conclude
If a teacher announces a surprise test, then explains it to mean a test the date of which he will not specify (nor which the student will be able to specify), both the definitions of surprise1 and surprise2 are compatible with the teachers explanation. The teacher, knowing his students to be philosophical enthusiasts, may well use the ambiguous terms "surprise test" and surprise those students who, relying on the argument of the paradox, decided it was impossible to give such a test. The teacher's surprise test is a surprise2 test.
Those students who deduce that a surprise test ( a surprise1 test) cannot be given on Friday will, indeed, be surprised when the teacher gives a test on Friday. This, again, is a surprise2 test.
Even if the students realize the ambiguity, they are not helped much. The definition of a surprise1 test excludes Friday, but surprise2 test does not. In effect, being told they will be given a surprise test ends up conveying as much information as being told they will be given a test at some indefinite time. But even if the teacher were to fix the meaning of "surprise test," which is much to expect of teachers who give surprise tests, nothing but faulty logic could raise the ghost of the paradox exorcised earlier.
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*Thanks to Pansophist