philosophy paper 3

8/28/2015

2

But, because you know it is a fair lottery, it is also rational for you to believe

that one of the tickets will win, that is, it is rational for you to believe

You know (and rationally believe) that this is equivalent to the proposition

that not-(p1 & p2 & … & p1000). Using the conjunction principle again, it is

rational to believe

But this proposition is a contradiction. So we seem compelled to reject one

of the two intuitively compelling principles of rational belief – the probability

principle or the conjunction principle

5

p1 & p2 & … & p1000 & not-(p1 & p2 & … & p1000)

either not-p1, or not-p2, …, or not-p1000

The lottery paradox illustrates the incompatibility of three commonly

desirable properties of a body of rational belief. For any two

statements x and y, we would like the following three properties to

hold.

– [Probability Principle]: x is rational to believe if it is sufficiently

probable/likely/typical.

– [Conjunction Principle:] If x is rational to believe and y is

rational to believe, then x & y is rational to believe.

– [Consistency:] x and not-y are not both rational to believe.

6

Solutions to the Lottery Paradox

• Denial of the Probability Principle

• Acceptance of the conclusion

• Graded notion of belief & Denial of the Conjunction

Principle

7

Denial of Probability Principle

• Proposal: To rationally believe that p it is not enough that p be

highly probable. Instead p must be certain.

• The probability principle is motivated by the threat of skepticism. To

insist on evidence that confers absolute certainty (probability of 1)

leads to skepticism. Any (empirical) belief admits of the possibility of

error. If requiring certainty for justified belief is setting the bar too

high, then it seems the only alternative is to retreat to high

probability (e.g. .999 probability).

• Denial of the probability principle seems to lead straight into

skepticism.

8

8/28/2015

3

Acceptance of the Conclusion

• Proposal: We can deny the assumption that rational belief is

closed under conjunction. A consequence of this move is that it

can be rational to have inconsistent beliefs. This consequence is

easier to accept if one remembers that some inconsistencies are

not discernable by normal human means.

• So on this view it is possible to have rational but incoherent

beliefs. (This is Richard Foley’s solution to the lottery paradox.)

9

Graded Notion of Belief

• Proposal: Replace the binary notion of belief with the graded notion of

belief. Instead of saying that I (categorically) believe that ticket T1 loses,

and (categorically) believe that ticket T2 loses etc., we can say that my

degree of belief in T1 losing is, say, 0.9, the same for ticket T2, etc.

• When binary beliefs are replaced by degrees of belief we cannot use

principles of deductive logic (such as the conjunction principle) any

more but instead have to use the probability calculus. Deductive logic

works only for binary beliefs. But according to the probability calculus, if

two propositions have individual degrees of belief of 0.9 it is rational to

believe the conjunction only to degree 0.81. [Note: 0.9 x 0.9 = 0.81].

10

• So the larger the number of conjuncts the smaller the degree

of belief in the conjunction. In the case of a lottery with many

tickets the degree of belief in all tickets being losers is very

small. So now it is possible to believe to degree 0.9 that T1 is

a loser but only believe to degree 0.19 that tickets T1 through

Tn are losers. No contradiction!

• So in the end we are solving the lottery paradox by denying

the conjunction principle for probabilities. This is Henry

Kyburg solution to the lottery paradox.

11

• This solution to the lottery paradox has intuitive appeal: when

pressed on any particular ticket, you are likely to say that you

are confident that this particular ticket is a loser. But the larger

the set of tickets you make a judgment on, the lower your

confidence that none of the tickets in the set is a winner.

12