Science

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Phil 2: Puzzles and Paradoxes

Prof. Sven Bernecker

University of California, Irvine

Lecture 12.2

Solutions to the

Prediction Paradox I

Proposed Solutions

• Reinterpreting the teacher‘s announcement (Olin, “Believing in

Surprises“, pp. 42-3)

• Disallowing reasoning backwards in time (Olin, “Believing in Surprises,“

pp. 41-2)

• 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)

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Prop. Solution: Reinterpreting the

Teacher’s Announcement

• 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.

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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.

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LiYuxi
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A version of the prediction paradox that does not involve reasoning backwards

in time:

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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.

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Replies:

• 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.

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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?

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KK-Thesis: If subject S knows that p, then S knows that S knows that p

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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:

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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?

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Pro 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

assumption.

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Con KK-Thesis

• 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.

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Con KK-Thesis

• 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

knowing.

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• 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

false.

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