Assignment Writing Service http://StudyHub.vip/A-Metalogical-Solution-Of-The-Surprise- 👈

1. A Metalogical Solution of the Surprise Quiz Paradox
Jan Dejnožka
March 27, 2023
Abstract
This logic puzzle has been studied for over seventy years with no consensus on its
solution. Most solutions have been based on semantic or epistemological interpretations, often
focusing on the word “surprise.” I offer a radically new, purely logical solution based on Ludwig
Wittgenstein and Bertrand Russell’s theory of tautology. The solution validates, in an empty,
tautological way, all solutions that validly follow from any interpretation (we need not even say
plausible interpretation) of the situation. Thus my solution is both metalogical and ecumenical. I
argue that Kurt Gödel is wrong and Wittgenstein and Russell are right on logical truth and
metalogical paradoxes. My solution does not depend on the specific views of Wittgenstein or
Russell, but only on the statements in question being tautological. In fact, my solution turns on
general issues of philosophy of logic, and not on any specific interpretation of the puzzle.
Keywords
surprise quiz paradox; relevance logic; analytic philosophy; Wittgenstein, Ludwig; Russell,
Bertrand; Gödel, Kurt
1. Introduction
I shall present the paradox in my own way. The exact details do not matter very much.
We can even use similar paradoxes such as the hangman (what day will the criminal be hanged?).
It is only the logical or relational structure that matters, not the instances or relata of the structure.

2. 2
And that in effect is the solution, as I shall explain. But first, I must describe the paradox.
Situation S. A teacher tells his students that he will give a surprise quiz next week, and
that the students will not know which day it will be, Monday through Friday, until he tells them
the afternoon before. A student then argues, by a sort of proof by cases, that it follows that there
will be no surprise quiz at all. For if the quiz is not given by Thursday, then it will not be a
surprise that it will be given on Friday, since that is the only day left. But if it cannot be a surprise
on Friday, then it cannot be a surprise on Thursday either. For in that case, if the quiz is not given
by Wednesday, then it will not be a surprise that it will be given on Thursday, since then that is
the only day left. And so on backwards through the days of the week. All of the students accept
this argument. Thus they are all surprised when the teacher gives the quiz on Wednesday. The
very first philosophy paper on the paradox agreed with the student (O’Connor, 1948).
2. Materials and methods
The literature is criticized based on relevance logic and the classical analytic tradition.
3. Results
The puzzle is a purely logical illusion. The teacher said that there would be a quiz and
that it would be a surprise, and those would be factually contingent events if they happened. But
he said nothing about the world that would allow anyone to predict when the quiz would be. And
that explains the surprise. The student states only empty, tautological hypothetical (if-then)
statements as consequents implied by what the teacher said. They are not even logically relevant
hypotheticals, since they are empty tautologies on a purely logical par with “P, therefore any old

3. 3
Q or not-Q,” whether P is logically related to Q or not. But any day the teacher gives the test will
be a logically contingent surprise. “If he does not give it by Thursday, it will not be a surprise if
he gives it on Friday” is indeed true, but it is only an empty tautological hypothetical consequent
of the logically contingent factual assumption that he gives the test that week at all.
4. Discussion
My insight is that every hypothetical (if-then) statement of the form “If interpretation I of
situation S is correct, then it logically follows that solution S* is correct” is an empty tautology
that says nothing about situation S, any more than “Either it’s raining or it’s not” tells us what the
weather is like (Wittgenstein, 1969, §§ T 4.46–4.4661, 5.101, 5.142–5.143, 6.1–6.1224, see
6.22).[FN1] And the same applies to the contraposition, “If solution S* is incorrect, then it
logically follows that interpretation I is incorrect.”
Now, the student in S does not give a specific semantic or epistemological interpretation
of S, but does logically interpret S as being such that a whole progressive series of implications is
logically valid, starting with “If the test is not given by Thursday, then it will not be a surprise
that the quiz will be on Friday.” That contraposits to “If it will be (or would be) a surprise that the
quiz will be on Friday, then the quiz is given (or was given) by Thursday.” That sounds odd, but
the contraposition is correct. And it would certainly be a surprise if the quiz were given a second
time on Friday! But except for the contraposition itself, the two hypothetical statements say
exactly the same thing, which is something about the logical structure of S, but nothing about
what actually happens in S.
A paper by Joseph Y. Halpern and Yoram Moses is handy for the ecumenical point, since

4. 4
they already offer four alternative interpretations. They say:
We have given four possible translations of the teacher’s statement. The first is
provably false, the second is consistent, and is true no matter which day the
teacher gives the test, the third is consistent but rules out the last day, and the
fourth is paradoxical in that it cannot be given a truth value in our semantics.
(Halpern & Moses, 1986, p. 301)
Assuming that their four alternative solutions logically follow from their four alternative
interpretations respectively, what they have really succeeded in doing is to provide four empty if-
then tautologies that can say nothing about what happens in situation S.
Likewise for every paper on the paradox that validly infers a solution of S from an
interpretation of S. And that is why my solution is ecumenical. The interpretation need not even
be plausible! And if anything, that makes my solution even more ecumenical.
Many of these interpretations show ingenuity, and enrich our understanding of semantics
and epistemology. But none of that matters to actually solving the paradox, since all such papers
are on the wrong logical level to begin with. On the right level, we see that every such paper
merely provides an empty if-then tautology (Halpern and Moses provide four) that says nothing
about what actually happens in S. And that most deeply and generally explains why the students
are surprised when the teacher gives the test on Wednesday. For that is what actually happens in
S. Even if the teacher gives the test on Friday, no if-then tautology can have anything to say
about it. That includes O’Connor’s (1948) paper, which agrees with the student’s logical

5. 5
interpretation. For anything that happens in S is a surprise relative to any such if-then tautology,
since no tautology can say anything about what happens in S in the first place.
Aristotle says that the most general explanation is the true explanation (Aristotle, 1968, p.
118, Posterior Analytics 74b10–20; see bk. 1, ch. 1, §§ 5, 24, 27; compare p. 96, Prior Analytics
66a 10–15). If so, then I am giving the true explanation.
Also recall the familiar, pre-philosophical relevantist insight that in some sense, the
conclusion of a logically valid argument is contained in its premisses; and that this is why a
logically valid argument says nothing about the world. Modern classical (Frege-Russell style)
logic has what seems to be the deepest and most general (and purely extensional) version of this,
that the truth-grounds of a valid conclusion are contained in the truth-grounds of the premisses
(Wittgenstein, 1969, §§ T 5.101–5.123).
But let the interpretations keep coming! Even if they cannot solve the paradox in the final
way my solution does, they can still enrich our understanding of semantics and epistemology.
In fact, I can assist the interpretations with another general point. Namely, regardless of
the specific semantic or epistemological interpretation of S, S keeps changing with respect to its
ostensibly correct logical interpretation. It changes five times. First, there is the original S. Then
the hypothetical about Friday is added to S as an ostensibly correct logical interpretation of S.
Then the hypothetical about Thursday is added, and so on down the line. Yet none of these
merely hypothetical additions, if logically valid, can say anything about what actually happens in
S, and in that sense S remains the same situation, and the quiz remains a surprise regardless of
what day it is given. In fact, all the hypotheticals, valid or not, are already included in S, as
statements by the student. (Thus I mean by “original S,” S as stated by the teacher.) And this

6. 6
other general point really just restates the same general point as my solution. It merely restates
the empty logical structure. And in that sense, my assistance is no help at all. But why should it
be, and how could it be? The specific solutions are not solutions in the first place! And pointing
that out is my real assistance. This assistance is limited to papers that interpret S and validly
derive a solution, but hopefully that includes all or nearly all of them.
In the sense in which tautologies say nothing, this puzzle is much ado about nothing.
Wittgenstein aptly sums it up for the surprise quiz paradox when he says “there can never
be surprises in logic” (Wittgenstein, 1969, § T 6.1251, his emphasis; see T 6.1261). Perhaps he
does not mean exactly the same thing as in the quiz, but it is essentially related. Thus, so much
for the semantic or epistemological interpretation of the term “surprise” in the surprise quiz!
Even synthetic a priori hypothetical statements, such as “If this apple is red, then it has a
color,” tell us nothing about the world. Compare Wittgenstein (1969, § T 6.3751) on colors. Thus
even if an interpretation implies a solution synthetic a priori, my solution still applies.
5. Objections
One might object that a complete solution would specify an interpretation of the key
terms, whatever they may be, in the description of S. My reply is that is not necessary to solve S,
and there is no difference between a solution and a complete solution. Also, the key terms,
whatever they may be, are doubtless determinables which have indefinitely many interpretations
as reasonable determinates, and there is no principled way to decide which is best, much less the
“true” one. And no doubt that is why the more specific literature is all over the map with no
consensus, though very understandably there have been two main avenues, semantics and

7. 7
epistemology.
One might object that only a specific interpretation can give a specific solution. My reply
is that the objection has been asked and answered by this whole paper. And good luck on finding
the “true” interpretation.
One might object that a specific interpretation is not just a tautological hypothetical
statement, but a modus ponens with the tautological hypothetical as the major premiss and the
specific interpretation as the minor premiss. My reply is that this only postpones the problem. For
modus ponens is an empty tautological argument form that says nothing about what is actually
happening. And good luck on finding the “true” minor premiss, which will be an unnecessary
solution even if it is found.
My solution can be extended to all logic puzzles where rival solutions are logically
derived from rival interpretations with no logical way to determine the “true” interpretation.
My solution cuts the Gordian knot of semantics and epistemology, though only with
respect to such puzzles.
Russell says, “For the moment, I do not know how to define ‘tautology’.... in spite of
feeling thoroughly familiar with the characteristic” (Russell 1919: 205). Of course, not all
notions can be defined. But we can define a formal tautology as a purely formal statement that is
true in virtue of its logical form. And the notion of logical form is easy to define, since it is
syntactical. To arrive at the logical form of a statement, i.e. at a purely formal statement, we
replace all of its constants with variables, so that only variables and logical expressions remain
(Russell 1919: 198–199). We may say with Wittgenstein that a formal tautology, or formal
statement that is true in virtue of its logical form, is a formal statement that is true on every row

8. 8
of its truth-table (Wittgenstein, 1969, §§ T 4.31–4.46). And that defines truth in virtue of form.
And the truth-table for “P, therefore Q or not-Q,” shows purely extensional relevant truth-ground
containment entailment of the consequent by the antecedent, even though there is no intensional
logical relation between P and Q at all.
We come even closer to the student’s empty hypothetical consequents when we look at
“(Teacher:) factual P and factual Q, therefore (student:) if P, then (if Q, then P).” In situation S,
the teacher is factually saying, “There will be a quiz next week and it will be a surprise.” And the
student is correctly but uninformatively inferring the empty tautological consequent, “For any x,
if x is a quiz next week and if x will be a surprise, then (it is tautological that) if x is not given by
the next to the last day of that week, then (it will not be a surprise that) it will be given on the last
day (since that is logically the only day left).” And that consequent tells us nothing about when
the quiz will be given, since it says nothing about the world at all. The consequent can also be
stated using an exclusive disjunction of the five days of the class week; it makes no difference.
Note that any interpretation of “surprise” can be used, as long as it makes the teacher’s and
student’s statements true, and the student’s statement intensionally tautological (a priori). And if
an interpretation of “surprise” did not do both of those things, then the student’s argument would
not be logically valid. Thus its logical validity comes at the price of its being empty.
This is not to say that all logically valid arguments are empty and uninformative, but only
that the student’s is. The famous counterexample to that ancient general thesis remains correct. If
a priest says “The first person I heard in confession was a murderer,” and then the guest speaker
arrives and says “I was the first person this priest heard in confession,” then the congregation
learns something very new! But the reason lies in the conversational situation, not in the truth-

9. 9
grounds. For the containment of the truth-grounds of the shocking conclusion in the truth-
grounds of the two premisses, one stated by the priest, and the other by the guest speaker, can
even be diagrammed by a Venn diagram (an early kind of truth-table).
It is obvious from the Tractatus and Russell’s introduction to it that both Wittgenstein
and Russell are extensional truth-ground containment entailment relevance logicians. They speak
of following from repeatedly, and even of containment. The more recent Anderson-Belnap style
relevantist tradition overlooks this because it focuses on intensional relevance. Not only that, but
extensional relevance is deeper and more general. It is implicit even in Aristotle’s syllogistic.
6. Metalogic
In the broad sense of reasoning puzzles, not all logic puzzles live by logical form alone,
but also by specific content. And those that require specific content for their solution generally if
not always have specific content that is there to be the solution. That is the ‘puzzle version’ of
Hegel’s thesis that the real is the rational and the rational is the real, that is, the thesis that every
intelligible question has a determinate answer. That is dicey with respect to apparently insoluble
mathematical puzzles. No doubt purely mathematical puzzles live by mathematical form alone,
and even by logical form alone, if logicism is correct. But then there is Gödel’s first
incompleteness theorem, which applies to many, though not all, mathematical and logical
systems. The theorem shows that in certain systems, not every true statement can be proved. The
exact general scope of the theorem is itself a logic puzzle that may be insoluble. For the theorem
is always relative to a system; one must always construct a proof to show whether the theorem
applies. Also, sometimes a statement that cannot be proved in one system can be proved in

10. 10
another. All we need is one proof!
Again, Wittgenstein says “there can never be surprises in logic” (Wittgenstein 1929, T
6.1251). Of course, he was pre-Gödel. Puzzles about Gödelian metalogic are beyond the scope of
this paper, as are the later Wittgenstein’s remarks on Gödel (Wittgenstein, 1972, pp. 50–54, 174,
176–177). But I cannot resist a jab. Is “This statement is not provable,” which is false if it is
provable, really a well-formed formula of Whitehead and Russell’s Principia Mathematica?
The closest I can get is “-|P”, or perhaps “|(-|P)”, but they are not self-reflexive. To get that,
we would have to define “P” as meaning “-|P”, and that would be circular to say the least. In
fact, if Wittgenstein is right that every logical truth has its own proof via its truth table’s being
true on every row, then Gödel is just wrong about the system of logical truths, or should I say the
system of logical truths proved by their truth-tables. Note that “This statement is not provable”
fails to be a logical truth precisely because it is false if it is provable. Thus it is not true on every
row of its truth-table. At least this is so if it is not logically necessary that it is unprovable; but I
see no reason to suppose that is the case. All that is given is that it is true if and only if it is
unprovable, and false if and only if it is provable. And if that makes the truth-table indeterminate,
then it is still not true on every row.
Gödel is praised for going metalinguistic and discovering statements that cannot be
proved relative to a system. But really he should be condemned for going metalinguistic and
thereby smuggling in paradoxical statements. This is not Gödel’s Eureka moment, but Gödel’s
Trick.
Gödel’s first incompleteness theorem was published in 1931. Russell wrote in 1938:

11. 11
It seems clear that there must be some way of defining logic otherwise than in
relation to a particular logical language. The fundamental characteristic of logic,
obviously, is that which is indicated when we say that logical propositions are true
in virtue of their form. The question of demonstrability cannot enter in, since
every proposition which, in one system, is deduced from the premisses, might, in
another system, be itself taken as a premiss.... All the propositions that are
demonstrable in any admissible logical system must share with the premisses the
property of being true in virtue of their form; and all propositions which are true
in virtue of their form ought to be included in any adequate logic. It seems to me
that [propositions] either do, or do not, have the characteristic of formal truth
which characterizes logic, and that in the former event every logic must include
them, while in the latter every logic must exclude them. (Russell, 1964, p. xii)
Gödel overlooks this basic point. Namely, a logical truth is true in all logically possible worlds.
And that is just what a truth-table represents by being true on all rows. It is also why a logical
truth cannot tell us what the actual world is like. That includes both what day a quiz is given, and
specifying what language or logic system we use to express a logical truth. For both are logically
contingent. Metalinguistically “specifying” logic relative to a language or system therefore
cannot change what is logically necessary and what is not. And perhaps that is the sense in which
there can never be surprises in logic. Yes, Gödel can claim that it is his results that are logically
necessary, and in that sense logically unsurprising. But on the face of it, the burden is on Gödel to
show why Russell is wrong.[FN2]

12. 12
Metalanguages can be used to prove things about logic. But by their very nature, they can
also introduce paradoxical statements that cannot be part of logic. A logically paradoxical
statement cannot be part of logic because it cannot be a logically necessary truth. It cannot even
be straightforwardly true! We can either introduce some general restriction, or we can regard
such paradoxical statements as individually self-defeating. Per Aristotle, the most general
restriction would be the true explanation. But any restriction is only meant to exclude the
problem cases exactly, and we might not get the general restriction right. In particular, Gödel’s
paradox and a general restriction against self-reference would be much like Russell’s paradox
and Russell’s theory of types, which excludes infinitely many innocent statements about self-
referential classes along with the paradoxical ones. Thus in both Gödel’s case and Russell’s, the
safest, i.e. least Draconian or Procrustean solution, would be to regard any paradoxical
statements as individually self-defeating, and as somehow wrongly introduced into logic by our
own naive misconceptions of metalanguage or classes. For there is nothing like a sordid
counterexample to slay a beautiful restriction. We can always introduce arbitrary restrictions, and
even make them definitional. But arbitrariness does not advance the analysis, and any suggested
restrictions should not be picked out of a hat, or simply because they exclude paradoxes.
Frege, the early Wittgenstein, the early Russell, and Gödel alike locate logical truth in a
timelessly eternal Platonic heaven. Yet it is logically contingent that we define a certain logic a
certain way. And why need Gödel’s diagonal method to prove there are statements that cannot be
proved? If a logic is rich enough to allow self-referring statements to be well formed formulae,
then why not simply look for the self-defeating ones? For they have been well known since
ancient times, when a Cretan said that all Cretans are liars. Gödel is flying a kite to prove the

13. 13
existence of wind.
Of course, my own solution of the surprise quiz paradox is metalogical. It says of all
solutions that if they are logically derived from an interpretation, then their very derivation is
tautological and says nothing about the world, including when the quiz will be.[FN3] And if my
solution is tautological and says nothing either, and even says of itself that it says nothing, that is
perfectly fine. In fact, that is the whole point. And it is logically innocent; it gives rise to no
paradox. But my solution is an empty, tautological interpretation of solutions of situation S, not
of S itself. And that should be no surprise.[FN4]
In any case, there is no doubt that the student gives a series of hypothetical statements that
are either true or false. If they are logically true, as the student and O’Connor (1948) believe, then
they are tautological and say nothing about when the quiz will be. If they are contingently true,
then they are not a logically necessary solution. And if they are false, then they are a false
interpretation of the logic of situation S. Thus the student cannot win.
It is irrelevant to the argument of this paper, but I wish to note that the word “surprise” is
covertly, if not overtly, indexical. For its applicability is relative to the cognizer. Just like the
indexicality of “this,” “here,” and “now,” its indexicality is secondary; it depends on the primary
indexicality of “I” and the personal pronouns. Who is surprised? I am!
The same fact can surprise one person but not another. There is a difference between
being novelly informative (learned for the first time) and factually informative. It is factually
informative that the Morning Star and the Evening Star are the same planet Venus, to everyone
and always; but it is news only to those who have not yet heard it. Of course, not everything we
learn for the first time surprises us. The puzzle is not about surprise in the emotional sense, nor

14. 14
even about surprise in the sense of being unexpected. If anything, the puzzle is about learning
something for the first time. And on my view, even that is a red herring. More precisely, the
supposed need to interpret the word “surprise,” or anything else about the puzzle, in order to
solve it is a red herring. For every interpretation and its logically consequent solution of the
puzzle comprise a tautology. And tautologies, and hence the puzzle itself, are not about anything.
Thinking otherwise is the ultimate red herring. For it is plain from the vast and conflicting
literature, if not from long study of the puzzle itself, that there is no one “true” interpretation.
7. Conclusion
The earlier writers fell into the trap, bewitched by the logically illusory picture that some
specific interpretation can explain the surprise. The truth is that no specific interpretation can
explain the surprise—and that explains the surprise.
In other words, this is essentially the same as if the teacher said nothing at all, and simply
gives an unannounced pop quiz. The only difference is that in the one case he utters statements
that say or imply nothing factual about when the quiz will be, and in the other he literally says
nothing. And what logical difference do those logically contingent facts make to the puzzle?
Compare what I call the rain puzzle. The teacher says, “Each day next week, either it will
be raining or it will not, but you can expect it to rain on one day.” The students are surprised
when it rains on Wednesday. For the teacher’s statement has led them into the trap of thinking
that if it has not rained by Thursday, they can expect rain on Friday, and so on backwards through
the week. But rain is a logically contingent event, and logically might not happen at all, however
much a meteorologist, or the teacher, might empirically predict it, or even just sincerely say it

15. 15
will happen. And a quiz is a logically contingent event too. The teacher might die or forget to
give the quiz, or whatever, however much the students might empirically predict a quiz from the
teacher’s plan, his sincere statement, or whatever. The only difference is that in the actual world,
rain generally has an efficient cause, while quizzes generally have an agent cause. And what
logical difference do those logically contingent facts make to the two puzzles?
It makes no difference even if we stipulate that the quiz will be given. For that cannot
change the quiz’s status as a logically contingent event. That there would be a surprise quiz was
the only factual statement the teacher asserted; the rest was implicit tautology.
In an older lexicon, the surprise quiz paradox is categorially confused, where the two
categories are empty tautological statements and logically contingent statements. And in the
classical analytic tradition, the surprise quiz paradox presents nothing new or special. For the
surprise quiz paradox is neither semantic nor epistemic as such, but merely an instance of a
paradox of strict implication, Namely, any old P (the statement that there will be a surprise quiz)
strictly implies any old logically necessary statement (the student’s hypothetical tautologies).
Notes
FN1. A colleague thinks that “Either it’s raining or it’s not” says something about the weather.
But Wittgenstein says, “I know nothing about the weather when I know that it is either raining or
not raining” (Wittgenstein 1919, T 4.461). And the burden is on my colleague to explain how or
why Wittgenstein could be wrong. For Wittgenstein is right on the face of it.
If the statement is about anything, it is about what Wittgenstein and Russell call its
logical form. For its logical form grounds its truth. The weather is simply irrelevant to the truth

16. 16
or falsehood of the statement. You can change the weather all you want and the statement will
remain true. For it is a logically necessary truth, true in all possible worlds (and in all possible
weathers). Only the statement’s logical form is relevant to its truth. It is true in virtue of its form,
regardless of whether it is about its form. But we need not reach the question of aboutness here.
FN2. Russell wrote in 1963, “The followers of Gödel had almost persuaded me that the [years]
spent on Principia had been wasted and that the book had better been forgotten” (Russell, 1985,
p. 672). The key word is “almost.” Russell is on strong ground in the passage I just quoted.
FN3. A second colleague is unaware that if Q can be logically derived from P, then the statement
“If P, then Q” is a tautology. For if Q can be logically derived from P, then in the truth-table, all
truth-grounds of P are also truth-grounds of Q.
FN4. A third colleague kindly says, “The author argues for a distinctive solution to the Surprise
Quiz Paradox [that is] bold and iconoclastic.” Unfortunately, colleague #3 then completely
misconstrues my distinctive solution, saying “What allows the students to be surprised by a quiz,
despite their brilliant backwards-induction argument against this possibility, is the hidden
tautological structure of the teacher's announcement.... I think that the central argument rests on
three false assumptions.” But that was not my central argument at all. I never said the teacher’s
announcement was tautological, either overtly or covertly. See below for what I did find to be
tautological.
Colleague #3 continues, “First, I understand the author to be claiming that the exact
meaning or interpretation of a surprise quiz is not really important, that any interpretation... will
do. But this isn't the case.” Colleague #3 then argues that a certain psychological interpretation
fails to make the puzzle even “seem to make sense” as a paradox (seeming contradiction), [that

17. 17
is, does not even get the paradox off the ground,] while a certain epistemological interpretation
does. I shall not explain those interpretations because it does not matter to my argument. In fact,
colleague #3 overlooks that this is my own position! I wholly granted that some interpretations
are better than others. In fact, I am happy to grant that on colleague #3’s level of understanding,
the psychological interpretation does not work, and the epistemological interpretation does, just
as they say. But our colleague overlooks that on my (deeper) level of understanding, this does not
even matter. It is simply and wholly irrelevant. And as Aristotle says, it is the deepest and most
general level of explanation that truly explains.
Second, colleague #3 says “the teacher's announcement (on the epistemological
interpretation) is not tautological. Saying ‘Next week, you will have a surprise quiz’ is not at all
equivalent to saying ‘Next week, you either will or will not have a quiz’. Rather, for the simple
reason that it might not be true—i.e., for the simple reason that the teacher's promise may turn
out to be false—it is non -tautological. As a result, the paradox... does not derive from the
supposedly hidden tautological nature of the announcement. Instead, it derives from the
(apparent) contradiction between the students' seemingly airtight backwards-induction argument,
which concludes that a surprise quiz (in the epistemological sense of surprise) is impossible, and
our strong intuition that a surprise quiz (in the epistemological sense of surprise) is perfectly
possible.”
Colleague #3 overlooks that I never said the teacher’s announcement is tautological. I
said that every hypothetical (if-then) statement whose antecedent is an interpretation of the
puzzle and whose consequent is a solution that logically follows from that interpretation is
tautological. That is nothing like the teacher’s announcement, and I profess astonishment at

18. 18
colleague #3’s conflating any such hypothetical with the teacher’s announcement, or, for that
matter, with the student’s backward-induction argument. For any interpretation is not those two
things, but is of or about those things. And those things are not the solution: instead, they jointly
constitute the puzzle that needs solution.
Third, colleague #3 says that I say “the student’s backwards-induction argument starts
with the proposition ‘If the test is not given by Thursday, then it will not be a surprise that the
quiz will be on Friday’ and that this proposition contraposits to something odd: ‘If it will be (or
would be) a surprise that the quiz will be on Friday, then the quiz is given (or was given) by
Thursday’. But the oddness of the contrapositive can be removed if we split the original
proposition into two propositions: P1: ‘If the quiz is not given by Thursday, then the students can
justifiably predict as of Thursday that the quiz will be given on Friday’, and P2: ‘If the students
can justifiably predict as of Thursday that the quiz will be given on Friday, then the quiz on
Friday will not be a surprise’. [But the] contrapositive of P1: ‘If the students cannot justifiably
predict as of Thursday that the quiz will be given on Friday, then the quiz must already have been
given by Thursday’, is not odd (regardless of whether it is true). And neither is the contrapositive
of P2: ‘If the quiz on Friday is a surprise, then the students could not justifiably predict as of
Thursday that it would be given on Friday’.”
Colleague #3 overlooks that I never said or implied the contrapositive is odd, much less
that its oddity is the basis of the surprise of when the quiz is given. I only used the fact that the
contrapositive is logically equivalent to the original statement, that is, that the original statement
logically can be true if and only if its contrapositive is true. Like Socrates, “I am just following
the argument wherever, like the wind, it may lead.” Even worse, P1 and P2 are logically

19. 19
equivalent. They are different but distinct only in reason. For when the quiz is given will be a
surprise if and only if its prediction is not justifiable. Thus colleague #3 is not seeing that my
logical equivalence (of the original statement and its contrapositive) is the logically
unquestionable basis of my argument, and is not seeing that their own logical equivalence (of P1
and P2) destroys their own criticism in part, the other part being that any oddity of the
contrapositive has nothing to do with my argument. Instead, I am using the contrapositive to help
explain the oddity of the surprise quiz paradox. I am explaining why on any interpretation,
however good or bad it is in seeming to make the paradox make sense and get off the ground, the
paradox still does not even seem to make sense, that is, still does not get off the ground, on a
deeper level of purely logical tautological structure. And the contrapositive is part and parcel of
showing that.
And I still leave logical room for a solution on colleague #3 and everyone else’s lower
level, if and only if that solution is known to be the correct solution. But the whole literature cries
out that this will not be forthcoming. And ironically, “A lower level solution known to be the
correct one is a lower level solution known to be the correct one” is just another empty tautology.
I am reduced to a Diogenes looking with my lamp in broad daylight among our colleagues for a
lower level solution that is known to be the correct one. If such a solution existed, it would be the
Achilles’ heel of my own. But no such poisoned arrow will be forthcoming. And even if there
were, perhaps per impossibile, a lower level solution known to be the correct one, my solution
would still be deeper and more general, and therefore would still be the true solution.
Conflict of interest statement

20. 20
On behalf of all authors, the corresponding author states that there is no conflict of interest. There
are no other authors, and no conflicts of interest of any kind.
References
Aristotle. (1968). The Basic Works of Aristotle. Ed. by Richard McKeon. New York: Random
House.
Halpern, Joseph Y. & Moses, Yoram. (1986). Taken by Surprise: The Paradox of the Surprise
Test Revisited. Journal of Philosophical Logic 15, 281–304.
O’Connor, D. J. (1948). Pragmatic paradoxes. Mind 57, 358–359.
Russell, Bertrand. (1985). Autobiography. One vol. ed. London: Unwin.
Russell, Bertrand. (1964). The Principles of Mathematics. 2d. ed. New York: W. W. Norton &
Company.
Russell, Bertrand. (1919). Introduction to Mathematical Philosophy. London: Allen & Unwin.
Wittgenstein, Ludwig. (1972). Remarks on the Foundations of Mathematics. Trans. by G. E. M.
Anscombe. Cambridge (MA): The M.I.T. Press.
Wittgenstein, Ludwig. (1969). Tractatus Logico-Philosophicus. Trans. by D. F. Pears and B. F.
McGuinness. London: Routledge and Kegan Paul.