The Surprise Examination

Page last updated 03 May 2021

For readers unfamiliar with this paradox, the surprise examination puzzle can be summarized as:

A teacher announces to his student that an examination will be held on some day during the following week, and moreover that the examination will be a surprise. The student initially assumes that the teacher’s pronouncement is correct and he tries to analyze it. He first considers the case where the exam is on the last day of the week. If so, then on the previous night, he would be able to predict that the exam would occur on the following day, and the exam would not be a surprise; so the surprise exam cannot occur on the last day. But then the surprise exam cannot occur on the second to last day either, for in that case the student knows that the surprise exam cannot be on the last day, so he would be able to predict on the night before the second to last day of the week that the exam could only occur on that second to last day. Similarly, the student concludes that a surprise exam cannot occur on any other day of the week either. But when the exam occurs (perhaps on Wednesday), the student cannot be expecting it and so it is a surprise and the teacher’s pronouncement was true after all. Where is the error in the student’s reasoning?

Rather than present a long discourse on this paradox, I shall go straight into a logical analysis of it since, as Timothy Chow has said in his article PDF The Surprise Examination or Unexpected Hanging Paradox (Footnote: Timothy Y Chow, “The surprise examination or unexpected hanging paradox” The American Mathematical Monthly 105.1 (1998) pp 41-51.) there are already about 100 articles that have been written on it, but without any clear agreement as to how to deal with it. If the reader wants further background reading, I would suggest using Chow’s article initially as it gives a good overview of the different approaches that have been made to the paradox. I have not included a list of references to other articles on the paradox as Chow provides a comprehensive list in his online article as in the above link.

Many of the articles have attempted an analysis which turns out to be horrendously complicated and, as shown below, quite unnecessarily so. Chow’s analysis has the advantage that it is not overly complicated. However, I do not consider that he has provided a completely correct analysis of the paradox. He follows an analysis previously made by Fitch (Footnote: Frederic B. Fitch, “A Goedelized formulation of the prediction paradox” American Philosophical Quarterly 1.2 (1964) pp 161-164.
Fitch’s version is:
P = [[Q₁ & there is no proof of [P ⇒ Q₁]] or else
[Q₂ & there is no proof of [[P & ¬Q₁] ⇒ Q₂]]]) which gives the formal interpretation of the paradox as:

S ≡(Q₁ & ([S ⇒ Q₁] is unprovable))

or else

(Q₂ & ([S & ¬Q₁ ⇒ Q₂] is unprovable)).

where the problem is reduced to two days, and

Q₁ ≡ the exam occurs on day 1

and

Q₂ ≡ the exam occurs on day 2.

The reduction to two days is for simplification only and can be expanded to several days, though this would become rather unwieldy.

Chow notes that there is an objection to his and Fitch’s version, that:

“A related objection rests on the observation that if the teacher had not announced the exam to the class but had simply decided in secret to give a surprise exam, then no paradox would have occurred. Therefore the trouble cannot be attributed solely to the propositional content of the teacher’s announcement; the act of announcing it to the students must play a crucial role. The purely logical analysis seems to ignore this.”

But this failure to account for this factor can be easily fixed. We can note that there are at least two logic systems to be considered in this situation, the overall logic system of the teacher, and the logic system of the student. If we presume that the teacher who makes the pronouncement about the exam intends his pronouncement to be meaningful, then the day of the exam will not be unprovable by himself and will not be a surprise to him. Chow’s expression does not account for this at all.

But even if we constrict Chow’s expressions to the student’s logic system, the expression does not reflect the actual situation, since it does not include the case where the exam does not occur on the first day. But that could be the case, and for that case the negation of Q₁ applies (regardless of any related provability or implication), and yet that is not included within Chow’s statement S whereas if Q₁ has not occurred on day 1, the student can prove the negation of Q₁. (Footnote: It is mentioned, but only within the provability expression, hence it appears as not available to the student as being proved by the student’s observation. The same conclusion can be reached by noting that Chow does not explicitly state the stipulation that one of either Q₁ or Q₂ must apply, and since that must be the case, that should be explicitly stated.) Chow also includes a direct self-reference where the expression S is made to refer directly to S itself. It is shown by the analysis on this page that this is not necessary and hence that inclusion serves to confuse rather than elucidate.

I postulate that for a strictly logical analysis, the following statement is a more correct reflection of the common informal presentation of the paradox, where the following statement is taken to be the teacher’s statement to the student: (Footnote: Apologies for an error in the previous version of this section which does not affect the overall conclusion.)

T = [ Q₁ ∧ ¬P_s(Q₁) ]   XOR   [ Q₂ ∧ ¬P_s(Q₂) ]

where P_s represents “Provable in the student’s logic system” (i.e: is either an axiom or a theorem in the student’s logic system). XOR indicates the exclusive OR.

For convenience in the following we shall at times refer to T by:

T = A  XOR  B where

A = [ Q₁ ∧ ¬P_s(Q₁) ] and

B = [ Q₂ ∧ ¬P_s(Q₂) ]

Using T we first of all follow the standard argument from the point of view of the student’s logic system, given that the statement T is a true axiomatic statement in the student’s system. First the student analyses the case if he assumes that A is provable in his system, i.e: that:

Q₁ ∧ ¬P_s(Q₁)

Clearly there is a contradiction, since then we would have that the student proves Q₁ but at the same time, he proves that he cannot prove Q₁. Hence the student concludes that his assumption that A is provable cannot apply. He then analyses the case if he assumes that the statement B is provable in his system, i.e: that:

Q₂ ∧ ¬P_s(Q₂)

and again we have a contradiction, so the statement T gives the conventional result - a contradiction. The student then concludes that the statement T cannot be a true statement in his logic system. Therefore the assumption that T can be a valid statement in the student’s logical system is incorrect.

At his point, most commentators assume that the statement T is false, and that that is the cause of the contradiction. However, that assumption also leads to a contradiction, as shown in the following. If the statement T is false, then the negation of T is necessarily true, i.e: we have that:

¬{ [ Q₁ ∧ ¬P_s(Q₁) ]   XOR   [ Q₂ ∧ ¬P_s(Q₂) ] }

or, for convenience,

¬{A  XOR  B}

which can be shown to be logically equivalent to:

[A ∧ B] ∨ [¬A ∧ ¬B]

Clearly, as before, in the student’s system A ∧ B cannot apply (since we would have the student proves both that Q₁ and ¬ Proof  Q₁, that is, that he proves Q₁ and at the same time he proves that he cannot prove Q₁.

That leaves the possibility of ¬A ∧ ¬B. Expanding this gives:

¬[ Q₁ ∧ ¬P_s(Q₁) ] ∧ ¬[ Q₂ ∧ ¬P_s(Q₂) ]

which is equivalent to:

[ ¬Q₁ ∨ P_s(Q₁) ] ∧ [ ¬Q₂ ∨ P_s(Q₂) ]

which is equivalent to:

{ ¬Q₁ ∧ [ ¬Q₂ ∨ P_s(Q₂) ] } ∨ { P_s(Q₁) ∧ [ ¬Q₂ ∨ P_s(Q₂) ] }

which is equivalent to:

{ ¬Q₁ ∧ ¬Q₂ }        ∨(1)

{ ¬Q₁ ∧ P_s(Q₂) }     ∨(2)

{ P_s(Q₁) ∧ ¬Q₂ }     ∨(3)

{ P_s(Q₁) ∧ P_s(Q₂) }    (4)

From within his system, the student has no further information on Q₁ or Q₂, , and hence he cannot prove Q₁ nor can he prove Q₂, so neither P_s(Q₁) nor P_s(Q₂) can apply, hence none of (2), (3) or (4) can apply. That leaves (1), and so, if the negation of T is true, then we have that in the student’s system it must be the case that there is no exam on either day, that is, that:

( ¬Q₁ ∧ ¬Q₂ )

is true, and this is the same as:

¬(  Q₁ ∨  Q₂ )(a)

But we also know that, regardless of whether the negation of T is true or false, it must also be the case in the teacher’s system, regardless of the truth value of the teacher’s statement T that:

(  Q₁ ∨  Q₂ )(b)

is true, since we are told that the teacher must have the exam occur on day one or day two. The two assertions (a) and (b) are mutually incompatible, and hence we have a logical contradiction.

Hence the teacher’s statement T cannot be either true or false since in both cases there is a logical contradiction. Hence the common notion that the negation of the teacher’s pronouncement must be false is completely erroneous. (Footnote: Fitch says of his analysis:
“The conclusion that has been so far reached, therefore, is that the ‘paradoxical’ prediction is paradoxical only in the rather weak sense of being self-contradictory, and that anyone uttering such a prediction is merely uttering a logically false statement.”)  (Footnote: Chow says in his analysis that:
“We can now imitate the students’ argument to show that S is logically false”.)

The Cause of the Contradiction

The cause of the contradiction is easily seen, which is that for a valid logical system, the system cannot express the notion of the provability of its own statements. In reality, when we postulate that the statement T is a true statement of the student’s logical system, we are in fact postulating that:

T_s = P_s{ [Q₁ ∧ ¬P_s(Q₁)]   XOR   [Q₂ ∧ ¬P_s(Q₂)] }

is a valid statement of the student’s logical system, where P_s is an expression of a language that is a meta-language to the logic system of the expression that it applies to. Hence we can clearly see that T_s is an illogical conflation of language where it is assumed that the meta-language expression P_s can be an expression within the sub-language which is the student’s logic system.

Conflation of levels of language is probably the commonest cause of paradoxical self-referential expressions, and in most cases it results in a statement that appears to directly reference itself. In some cases, however, as in this case, the self-reference is not a case of the statement appearing to directly reference itself. In this case, assuming that T_s is a valid statement of the system forces the system to be referencing itself, and thereby referencing all statements of the system, including the statement itself. Note that that a valid logical system cannot refer to itself in such a manner. (Footnote: Note that one often encounters the erroneous unproven belief that there can be a valid logic system where there can be an expression such as P_s within the system S itself. But an actual expression that fulfills the intended content of P_s would be an expression that states that there exists a sequence of statements of S all of which are either axioms of S or a consequence of previous statements in the sequence, and where the last expression in the sequence is the expression that P_s refers to. The requirement that it must reference axioms of S, at least one of which must include a quantifier over all formulas of the system, entails that any expression that would fulfill the intended content of P_s would thereby reference itself.) Cases such as this are more subtle since there does not appear to be any direct self-reference, and this probably explains why this paradox is so widely misunderstood.

On the other hand, we can consider the teacher’s statement T as a statement within the teacher’s logic system, where it is a meta-linguistic expression where the student’s logic system is a sub-language to the teacher’s logic system. In terms of the strict logical analysis above, the teacher’s statement T is a valid true statement in the teacher’s logic system, and the paradox only arises if one assumes that this statement can be a valid statement within the student’s logic system.

The real world situation

Of course, the reality is that humans do not act as strict logic systems that simply reject all statements that are contradictory. Instead, humans almost invariably assume that every statement addressed to them is intended to have some sort of valid meaning. So the student will attempt to make a best guess at whether the teacher intended the statement to be meaningful, whether the teacher intended it to be true and what the intention of the teacher was when he made the statement. This means that the question of whether the teacher’s statement is meaningful or true in the real world situation has no singular correct answer. In reality, most students would assume from the teacher’s statement that it is true that there will be an exam on one of the two days, and if the exam did not occur on the first day, then they would not be surprised by its occurrence on the second day. And on the morning of the first day, they would consider that there is a 50% chance of the exam occurring on that day, and hence would not be completely surprised if it occurred on that day. One can of course continue to argue the fine points of possible interpretations almost endlessly but it is not intended to do so here.

The application of Gödel coding

Note that the addition by Fitch and Chow of Gödel numbering and associated material is completely extraneous and irrelevant and only serves to introduce further confusion. After all, if the original pronouncement is contradictory for the student, then adding Gödel numbering and associated material to purportedly show that the pronouncement is contradictory actually shows nothing at all.

However, both Fitch and Chow make claims in their application of Gödel coding which should not pass without mention. Chow rejects the notion that by applying logical analysis we can prevent any self-reference to evade paradoxes, since, he claims, “it is possible in mathematics to formalize certain kinds of self-referential statements.” He supports this viewpoint by appealing to Gödel’s proof of incompleteness and similar proofs. (Footnote: These proofs can all be shown to be flawed see The Flaw in Gödel’s proof of his Incompleteness theorem and Errors in Incompleteness Proofs.) In his article he refers to his analysis of the paradox which incorporates Gödel coding and which results in a contradiction. But then Chow states:

“… although self-reference is not illegitimate in all circumstances, it is illegitimate here because this particular self-referential statement is self-contradictory.”

Here Chow appears to be claiming that it is acceptable to use any theory which allows self-reference, provided that we don’t uncover any contradictory statements in it. But it is well known that simply because no-one has discovered any contradictory statement in a given theory, that doesn’t mean that there cannot be any contradictory statements in that theory.

It would appear that Chow is claiming that since incompleteness proofs include self-reference and because no-one will admit to there being any contradictions involved in such proofs, then self-reference is sometimes acceptable in logical systems. It is surprising how people can continue to engage in convoluted attempts to explain away conundrums rather than continue to search for the root cause of the conundrum.

A simple analysis shows that Fitch and Chow both completely misunderstand the principle behind Gödel coding and its application. The principle is that given a relationship R between expressions of a formal system, then one can create a purely number-theoretic relation R_F that corresponds precisely to that relation R, where each expression referred to by R is coded by Gödel numbering into a specific number, where the number-theoretic relation R_F is a relation between those Gödel numbers. The Gödel numbering function therefore belongs to a language that is a meta-language to both the formal system in question and to the number-theoretic language system that expresses the coded relations. For more on this, see The Gödel numbering system in A Step by Step Guide to Gödel’s Incompleteness Proof or for a simpler version, see Gödel’s special numbering system.

Fitch and Chow simply throw in some Gödel coding and some number-theoretic relations into their original expressions, but they also leave some parts of that expression uncoded, thus creating a mish-mash of conflation of part meta-language, part coded number-theoretic system and part original expression. The result is a nonsensical absurdity. The root of this failure to understand the necessity for a clear delineation of what is coded and what is not coded can perhaps be laid at the foot of Gödel’s original incompleteness paper, where there is an illogical conflation of levels of language, caused by the unproven illogical assumption that the formal system itself can express the Gödel numbering function, whereas it is a function that is defined in a language that is a meta-language to the formal system. For more on this see The Flaw in Gödel’s proof of his Incompleteness theorem.

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