Phil 2: Puzzles and Paradoxes
Prof. Sven Bernecker
University of California, Irvine
Solutions to the
Prediction Paradox I
• Reinterpreting the teacher‘s announcement (Olin, “Believing in
Surprises“, pp. 42-3)
• Disallowing reasoning backwards in time (Olin, “Believing in Surprises,“
• Denying knowledge of future contingents
• Rejecting the KK-thesis (Olin, “Believing in Surprises,“ pp. 46-51)
• Denying knowledge of the teacher‘s announcement (Olin, “Believing in
Surprises,“ pp. 51-7)
Prop. Solution: Reinterpreting the
• Proposal: the surprise component of the teacher’s announcement should
be understood as implicitly qualified. It claims that the exam will be a
surprise unless it takes place on Friday.
• The student rules out Friday as the day where the exam is held because
then it wouldn’t be a surprise. But on the proposed interpretation of the
teacher’s announcement the exam need not be a surprise if it’s held on
Friday. So the paradox disappears.
• Reply: There is no reason to suppose that the teacher’s announcement
must be understood as qualified.
Prop. Solution: Disallow Reasoning
Backwards in Time
• Proposal: the student reasons backwards — starting with Friday and
working his way to Monday. When the student rules out Friday she
assumes that she already knows that no exam has been held on the
first four days of the week. But this is something she can only know on
Thursday evening. Hence the student begins with a piece of knowledge
she cannot have at the beginning of the week.
• Reply: the statement that the exam is held on Friday is only an
assumption. To know that “If p, then …” you do not need to know that p.
A version of the prediction paradox that does not involve reasoning backwards
Of five students, Art, Bob, Carl, Don, and Eric one is to be given an exam. The students
are lined up in alphabetical order, so that each can see the backs of those before him. The
teacher has four silver stars and one gold star which will be placed on the student’s backs.
The gold star designates the student who will be examined. The teacher tells the students
this and also informs them that the designated student will have no reason to believe that
he is the designated student until after the students break formation.
The students reason: if Eric is the designated student, then he will see four silver stars
ahead of him and from this be able to infer that he is the designated student. So Eric
cannot be the designated student. But if Don is the designated student, then since he will
see three silver stars ahead of him and will realize that Eric is not the designated student,
he will be able to infer that he is the designated student. So Don cannot be the designated
student. And so on.
The students conclude that the specified exam cannot be given. They then break formation
and Carl finds the golden star on his back. (Olin, “Believing in Surprises,” p. 42).
Prop. Solution: Denying Knowledge of
Future Contingents • Proposal: presumably the teacher is free to choose the day of the exam. If
the statement “the exam will be on Wednesday” is true it a contingent
rather than a necessary truth. In lecture 6.3, slides #18 we saw that
Aristotle claims that future contingents are neither true nor false. (Aristotle
felt compelled to make this move to disprove fatalism.) In lecture 11.3, slide
#4 we saw that knowledge implies truth (factivity of knowledge). Whatever
is known is true. But if future contingents are neither true nor false, then the
student cannot know that the exam will be on Wednesday. Similarly she
cannot know that the exam will be on any of the other days. Thus, the
teacher’s announcement cannot be known.
• It is not clear that future contingents lack truth values (are
neither true nor false).
• The prediction paradox is not about freedom of will. We can
replace the teacher with a deterministic machine (a robot)
and get the same paradoxical conclusion.
Prop. Solution: Rejecting KK-Thesis
• Proposal: the student’s argument to the effect that a surprise exam
is impossible relies on a flawed assumption, namely the KK-thesis.
• Knowing that p is first-order knowledge. Knowing that you know that
p is second-order knowledge.
• Given the KK-thesis, you cannot know that p if you have doubts that
you know that p. You need conclusive reasons to know that p.
• How does the student’s argument rest on the KK-thesis?
KK-Thesis: If subject S knows that p, then S knows that S knows that p
How does the student’s argument rest on the KK-thesis?
This is the first step of the students argument:
So far there is no need for the KK-thesis. But now consider the next two steps:
First Step: If the only exam of the week is held on Friday, then on Thursday
evening the student will know that an examination will be held on Friday.
Second Step: If the only exam of the week is held on Thursday, then on
Wednesday evening the student will know what she knew in the First Step,
and thus know that the examination will be held on Thursday.
Third Step: If the only exam of the week is held on Wednesday, then on
Tuesday evening, the student will know what she knew in the First and
Second Step, and thus know that the exam will be held on Wednesday.
• Each step of the argument, after the first, requires that the
student have knowledge of the preceding steps. This is where
the KK-thesis is involved.
• The student’s argument assumes not only that she knows that
there will be an exam and that it was not held on the
subsequent days but it also assumes that student knows that
she knows this.
• What are the arguments that speak for/against the KK-thesis?
• Argument: All those circumstances which would justify you in
saying “I know” also justify you in saying “I know that I know”
• Rebuttal: The problem with this argument is that it assumes
that to know something you must be justified in saying that
you know it. Critics of the KK-thesis simply deny this
• Argument: Because the KK-thesis demands that to know
something you have to be sure that you know it and because
you can never be sure that you know something, the KK-
thesis leads to skepticism.
• Rebuttal: Given the KK-thesis, the truth-conditions of Kp and
KKp are the same. KKp is not any more demanding than Kp.
• Argument: The KK-thesis entails an infinite regress. If knowing
implies knowing that you know, knowing that you know implies
knowing that you know that you know, etc.
• Rebuttal: The infinite regress entailed by the KK-thesis is harmless.
Given that the conditions for knowing and the conditions for knowing
that you know are the same (see previous slide), knowing that you
know does not entail information in excess of that provided by
knowing. Knowing that one knows is a gratuitous by-product of
• Upshot: since there are no decisive arguments for or
against the KK-thesis, it is prudent to not rest the solution to
the prediction paradox on the claim that the KK-thesis is