1
A threshold turns credence into acceptance
Accept a claim when its credence is at least some t below 1. The rule is simple and has many defenders.
Lesson · Knowledge and Evidence · Evidence and belief
The Lottery Paradox
If to believe a claim is to give it a credence at or above a threshold, then in a fair lottery with enough tickets you believe of each ticket that it loses. The claims cannot all be true, because one ticket wins. Belief at a threshold is therefore not closed under conjunction, and raising the threshold does not help.
n tickets, one winner: P(loses) = 1 − 1/n
accept a claim when its P is at least t
all n accepted: they cannot all be true
belief at a threshold is not closed under ∧
Each ticket loses; one wins
Core ideas
What this lesson establishes
The credences in a lottery are perfectly coherent. The trouble is what happens when belief is read off credences by a threshold.
2
Closure under conjunction is a second principle
If you accept A and accept B, you accept A and B. It is what lets a reasoner put premises together.
3
The two cannot both hold in a lottery
Accepting each claim that a ticket loses and accepting their conjunction would be accepting that no ticket wins, which is impossible.
The material
A hundred tickets, one winner
A threshold, a lottery, and the set of claims it accepts.
A fair lottery sells 100 tickets and draws one winner. For each ticket there is a claim, this ticket loses. Every one of those claims has credence 99/100: 99 of the 100 equally likely outcomes make it true.
Definition · Threshold rule
To accept a claim at threshold t is for its credence to be at least t, where t is below 1. A reasoner who believes what she accepts has the whole set of accepted claims as her beliefs.
Set t = 99/100. Each of the 100 claims has credence exactly 99/100, so each is accepted, and the lab reports 100 of 100. Now put the claims together.
- Each claim “ticket i loses” has credence 99/100, so by the threshold it is accepted.
- The lottery has a winner, so not every ticket loses: some claim “ticket i loses” is false.
- So the 100 accepted claims cannot all be true, and their conjunction, which says that every ticket loses, has probability 0.
The lab reports the accepted set as inconsistent and the probability that all the accepted claims are true as 0. No ticket survives, because it would have to be the winner and then the claim about it would be false. This is the lottery paradox.
Two principles are in play, and one fact. The principles: accept a claim when it is probable enough, and accept the conjunction of what you accept. The fact: the lottery has a winner. With 100 tickets at threshold 99/100 the three cannot stand together, and since the fact is not negotiable, the exits are ways of giving up a principle or restricting it.
- Give up the threshold rule. Belief is more than high credence, and no credence short of 1 makes it rational to believe that this ticket loses.
- Keep the threshold and give up closure. One may rationally accept each claim and not their conjunction, so belief is not closed under and; the price is holding a set of beliefs one knows are not all true.
- Restrict the threshold rule. Statistical evidence alone, however strong, is not the kind that supports belief, so “ticket i loses” is never accepted. This keeps the threshold for evidence of other kinds and owes an account of the difference.
Each exit is a position with defenders, and each has a price. The first needs a theory of what belief is, the second gives up an inference used whenever premises are combined, and the third leaves open why the same worry does not arise for a claim supported by an ordinary statistic.
A higher threshold does not help
Raise the threshold to 999/1000. With 100 tickets each claim has credence 99/100, which is below it, so nothing is accepted and the paradox is gone. But the paradox returns with 1,000 tickets: each claim now has credence 999/1000, all are accepted, and the conjunction is again impossible.
For any threshold below 1 there is a lottery large enough to reach it, so no threshold avoids the problem. Setting the threshold at 1 avoids it by accepting only what is certain, which is a different and much narrower account of belief.
The credences here are coherent in the sense of the earlier lesson on the Dutch book: they add up correctly and no book exists. The incoherence appears only in the acceptances, and the lab can show that and cannot say which exit to take.
Interactive
A threshold for belief, and a book
what a threshold accepts, whether it can all be true, and what incoherence costs
Method
Running the paradox
From a fair lottery to the failure of closure.
1
Compute the credence of each claim
In a fair lottery of n tickets, the claim that ticket i loses has credence 1 − 1/n.
2
Apply the threshold
Accept the claim when its credence is at least t. All n claims share one credence, so all are accepted or none are.
3
Form the conjunction
If all n are accepted, their conjunction says that no ticket wins.
4
Compare the conjunction with the facts
The lottery has exactly one winner, so the conjunction has probability 0 and the accepted set has no model.
5
Name the exit
Reject the threshold rule, reject closure under conjunction, or restrict the threshold so that statistical evidence alone accepts nothing, and say what each costs.
Tickets, thresholds and conjunctions
Score 0 / 0Choose the answer you can justify.
100 tickets at threshold 99/100
One ticket wins, drawn at random.
credence in each ‘ticket i loses’: 99/100
threshold: 99/100
claims accepted: 100 of 100
conjunction (every ticket loses): probability 0
accepted set: inconsistent
Each acceptance is reasonable on its own terms, and the set of all of them cannot be true. If belief is closed under conjunction, one of the steps must go.
Common mistakes
1
Assuming rational belief is closed under conjunctionIt seems obvious that someone who believes A and believes B should believe both together. In the lottery that principle, with the threshold rule, makes a person believe that no ticket wins while knowing that one does. Either the principle or the threshold has to give, and the lab cannot say which.
2
Thinking a bigger threshold fixes itAt 999/1000 the 100-ticket lottery accepts nothing, and the 1,000-ticket lottery accepts everything and fails again. A threshold below 1 is always reached by a large enough lottery, so it only moves the example.
3
Concluding that the credences are incoherentThe credences are 99/100 for each of 100 losing claims and 1/100 for each ticket winning, and they obey every rule of the earlier lesson. The failure is in moving from credence to belief, not in the credences.
Check your understanding
✓
Finish when you can show that the accepted set has no model.
You should be able to apply a threshold to every claim in a fair lottery, state how many are accepted, show that their conjunction has probability 0, and say which principle a given reply gives up or restricts.
Note: The paradox is about belief, not about lotteries. Any claim supported by a large enough statistical base can play the role of a ticket.