Rewrite of a paper first presented in a graduate seminar at Temple University, Fall 1969.

©2000 Edward G. Rozycki

edited 8/16/11

**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:

Assume classes to be held only from Monday through Friday.

a. A teacher says he will give a surprise test the next week;

If

b. the test is not given by Thursday night,

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

f. the test has not been given by Wednesday night,

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

l. the test has not been given by Tuesday night,

The reader can see than an inductive proof is in the offing. Similar argumentation will exclude all the days of the (school) week as possibilities for giving a surprise test. Thus, it seems, a paradoxical situation arises: if one announces a surprise test, one cannot give one. This is indeed strange considering that any unannounced test is (considered to be) a surprise test. Also, by saying merely, "I will give a test sometime next week" without specifying the day of the week, one can still give a test that is something of a surprise.
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

a. the argument from which the paradox derives is inconsistent, i.e. there is no paradox;

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

1. surprise

2. surprise_{2
}
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:

a. If a test is preannounced, it is a surprise test as long as (if and only if) there is more than just one possible day for giving it. This is a surprise_{1
}
test. Neither of the following are announcements of surprise tests:

1. "I'll give a test next Wednesday."

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:

If the test has not been given by Thursday, then a surprise test cannot be given on Friday.

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:

a. Consider the following conditions:

a/: There is (remains) only one possible date for a test.

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.

1. not-d/ (implying c/) holds.*

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.

1. On Thursday night, a/ holds. Thus not-S. Only Friday remains for the test, so it is no surprise test.

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:

1. You can know the fifth card is an ace before turning it over if and only if four cards have been turned over and none of them is the ace.

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

a. Thursday is not the last day for a surprise test"

But this contradicts a/, or by asserting that

b. Thursday's test is no surprise test.

from which follow:

b'. Wednesday is the last day for a surprise test,

and the paradox again raises its head, or

b''. b/ is contradicted.

It is here that the deficiencies of our definition of "surprise test" show up most sorely. Paraphrasing b/, a surprise test is a test for which remains yet an alternative date on which to give it. Except for the ponderousness of the locution, there seems to be no difficulty in saying, "Tomorrow is the last day for a test for which there remains yet an alternative to give it on."

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

1. surprise_{1
}
test -- a test given when yet an alternative date is possible;

Thursday's test is still a surprise
2. surprise_{2
}
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 surprise_{1}) 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 surprise_{1
}
tests. On Tuesday night, one cannot exclude Thursday as a day for given a surprise test (vacillate as one may between surprise_{1
}
and surprise_{2}), 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 surprise_{1
}
and surprise_{2
}
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 surprise_{2
}
test.

Those students who deduce that a surprise test ( a surprise_{1
}
test) cannot be given on Friday will, indeed, be surprised when the teacher gives a test on Friday. This, again, is a surprise_{2
}
test.

Even if the students realize the ambiguity, they are not helped much. The definition of a surprise_{1
}
test excludes Friday, but surprise_{2
}
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.

-------------------

*Thanks to *Pansophist *