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14.2LotteryPuzzle.pdf

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Phil 2: Puzzles and Paradoxes

Prof. Sven Bernecker

University of California, Irvine

Lecture 14.2

Lottery Puzzle

Lottery Puzzle

• There are knowledge versions of the lottery paradox (I refer to them

as Lottery Puzzles to distinguish them from the Lottery Paradox)

that are not about actual lotteries. These lottery puzzles operate with

the principle of closure under known implications (instead of the

conjunction principle) and with fallibilism (instead of the probability

principle).

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“Recently … there has been a flurry of debates on the epistemic status

of lottery statements [i.e., lottery propositions], in the course of which

examples have been produced that may have significance for the

paradox and for closure” (Olin, A Case Against Closure, p. 242).

Two Epistemic Principles

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Fallibilism: If it is both highly probable and true that p, then (ceteris

paribus) S knows that p.

Closure under known implication: If S knows that p and if S

knows that p implies q, then S knows that q.

Fallibilism

• A subject can know something even though it could have been

false. This is not the same as saying that the subject can know

something that is false. Only truths can be known. (see lecture

11.3, slide #4).

• Fallibilism claims that a belief held on the basis of some evidence

can be knowledge even though the subject could have held that

belief on the basis of the same evidence in circumstances where

the belief is false. So the evidence that is sufficient for knowledge

in certain circumstances need not preclude the possibility of

error.

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Closure

Example: Sally knows that she is petting a dog. She also knows that

“being a dog implies being a mammal". Sally knows that she is petting a

mammal.

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Closure under known implication: If S knows that p and if S knows

that p implies q, then S knows that q.

The intuitive appeal of closure lies in the sense that in a valid argument,

the epistemic status of the premises extends to the conclusion” (Olin, A

Case Against Closure, p. 236)

Closure under known implication is weaker than:

Closure under implication: If S knows that p and if p implies q,

then S knows that q.

Closure under implication is implausible because it requires S to be

logically omniscient.

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Closure under known implication: If S knows that p and if S knows

that p implies q, then S knows that q.

• Note that the closure principle leaves it open how S knows

that q. That’s why the closure principle is different from this

principle:

– Transmission of knowledge: If S knows that p, and

knows that p implies q, then S knows q on the basis of p.

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Closure under known implication: If S knows that p and if S knows

that p implies q, then S knows that q.

Counterexample to Transmission of Knowledge (Olin, p. 237-8):

• Suppose I know

p: Cynthia will watch TV the entire evening on Wednesday

• based on her announced intention to do so. It is clear to me that p

implies:

q: There will not be a power failure on Wednesday evening.

• Transmission fails because my evidence that Cynthia will watch TV

on Wednesday evening -- her announced intention -- is clearly not

evidence that there will not be a power failure on Wednesday

evening. Since the evidence for p is not evidence for q I cannot

come to know q on the basis of knowing p.

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Examples of Lottery Puzzle

1) S knows that S won’t have enough money to go on a safari this year.

2) If S knows that S won’t have enough money to go on a safari this

year, then S is in a position to know that S will not win a major prize in

a lottery this year.

C) Therefore, S is in a position to know that S will not win a major prize in

a lottery this year.

Intuitively (1) and (2) are true, but in spite of the fact that (C) follows from

(1) and (2), intuitively (C) is not true.

Closure principle:

1) If S knows that p and

2) S knows that p implies q

C) Then, S knows that q 9

Another example:

1)I know that my car is now parked in Mesa Parking Structure

2)If I know that my car is now parked in the Mesa Parking Structure,

then I am in a position to know that my car has not been stolen in the

last few minutes.

C) Therefore, I am in a position to know that my car has not been

stolen in the last few minutes.

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My epistemic situation with respect to

parking my car is like my epistemic situation

with respect to a lottery. When I park my car

in an area with an appreciable rate of auto

theft, I enter a lottery in which cars are

picked, essentially at random, to be stolen

and driven away. Having my car stolen is the

unfortunate counterpart to winning the lottery.

Structure of Lottery Puzzle

Closure under known implications: If S knows that p and if S knows that p

implies q, then S knows that q.

p = ordinary proposition

q = lottery proposition

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“There is ... an ordinary proposition, a proposition of the sort that we ordinarily

take ourselves to know. There is, on the other hand, a lottery proposition, a

proposition of the sort that, while highly likely, is a proposition that we would be

intuitively disinclined to take ourselves to know. And ... the ordinary proposition

entails the lottery proposition” (Hawthorne, Knowledge and Lotteries, p. 5).

• Why ordinary propositions but not lottery propositions are knowable?

• Many philosophers define knowledge by means of this necessary

condition:

• If “p“ stands for an ordinary proposition, this knolwedge condition is

satisfied; but if “p“ stands for a lottery proposition it is not satisfied.

– If my car were stolen, I would still believe that it is not.

– If my ticket were going to win, I would still believe that it‘s a losing

ticket.

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S knows that p only if: if p were false, S would not believe that p

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Solutions to Lottery Puzzle

• Denial of knowledge of ordinary propositions (skepticism)

• Acceptance of the conclusion

• Denial of closure under known implications

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