1
Independent claims multiply
If the claims are independent, the probability that all are true is the product of their probabilities. A hundred factors of 99/100 multiply to far less than any one of them.
Lesson · Knowledge and Evidence · Evidence and belief
The Preface Paradox
An author believes each sentence of a book and writes in the preface that errors remain. With claims independent and each at 99 in 100, a threshold accepts every one, the accepted set is consistent, and the probability that all are true is about 0.37. Closure under conjunction fails here by probability, not by logic.
100 claims, each at 99/100, all accepted
they can all be true: the set is consistent
P(all true) is the product: about 0.37
believe each, doubt the whole
A careful book, an honest preface
Core ideas
What this lesson establishes
The lottery paradox fails by logic: one claim is certainly false. The preface fails more quietly, and the difference is the lesson.
2
Consistent is not probable
A set of claims can all be true without it being likely that they are. Consistency is a matter of logic and probability of how much risk accumulates.
3
Risk accumulates with length
Each claim added to a book, or each premise added to an argument, adds a little chance of error, and the little chances add up.
The material
Believe each sentence, doubt the book
A threshold, a hundred independent claims, and what the product says.
An author has checked a hundred claims for a book. Each is supported strongly enough to have credence 99/100, and the checks are independent: an error in one tells her nothing about another. At a threshold of 95/100 she accepts every claim, and the lab reports 100 of 100.
Unlike the lottery, nothing logical ties the claims together. They could all be true, and the lab reports the accepted set as consistent. The question is how likely that is.
Definition · Independent claims
Claims are independent when the probability of two of them together is the product of their separate probabilities, and likewise for any number. For n claims each at probability p, the probability that all are true is p multiplied by itself n times.
P(all 100 true) = (99/100)¹⁰⁰, about 0.37
The lab prints the exact fraction. The conjunction of the hundred claims is true with probability about 0.37, well under the threshold of 95/100, so she does not accept it. She accepts every claim, and she does not accept the claim that all of them are true. That is the preface paradox: she may write that she is confident of each sentence and that some error remains.
The two paradoxes differ in kind. In the lottery, closure under conjunction fails because the conjunction is certainly false. In the preface the conjunction is possible and merely improbable. So the lottery shows that a threshold can accept an inconsistent set, while the preface shows that it can accept a consistent set whose conjunction it rejects.
Why it matters outside books
A valid argument preserves truth, not probability. If a conclusion follows from a hundred premises, each at 99/100, and the premises are independent, the conclusion is at least as probable as their conjunction, which is about 0.37. A long chain of reasoning in which every step is believable can end in a conclusion that is not. The principle that if you accept each premise you accept the conclusion fails for exactly this reason.
Shorten the book and the effect shrinks. Ten claims at 9/10 give a conjunction of about 0.35, which the lab prints as 3486784401/10000000000: milder, but still far below the claims it is made of. Only at certainty does it vanish: a hundred claims at probability 1 have a conjunction of probability 1. Closure under conjunction holds for what is certain and is at risk for everything else.
The lab assumes independence, and real claims in a book are more often positively related, as when they rest on one source. That raises the probability that all are true, but not to 1, and the paradox remains for any set with enough independent risk.
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 preface
From independent claims to a rejected conjunction.
1
State each claim’s probability
All the claims are given the same probability here, and the claims are independent.
2
Apply the threshold
Accept the claims whose probability is at least t. If one is accepted they all are.
3
Check consistency
Nothing ties independent claims together, so they can all be true. The accepted set has a model.
4
Multiply for the conjunction
The probability that all are true is the common probability multiplied by itself once for each claim. Compare it with t.
5
Say what is given up
If the product is below t, the reasoner accepts every claim and not their conjunction, so closure fails without any inconsistency.
Products and conjunctions
Score 0 / 0Choose the answer you can justify.
A hundred claims at 99/100
Independent, with a threshold of 95/100.
each claim: 99/100
threshold: 95/100
claims accepted: 100 of 100
accepted set: consistent
P(all true) = (99/100)¹⁰⁰: about 0.37
Nothing in the claims prevents all of them being true, and yet the conjunction is well below the threshold. Believing every sentence and doubting the book is not a contradiction.
Common mistakes
1
Thinking an author who believes each sentence must believe the book has no errorsBelieving each claim at 99/100 is compatible with giving the conjunction credence about 0.37, because the probability of all of them is the product and not the common value. The preface states exactly that: no particular sentence is doubted, and the whole is expected to contain a mistake.
2
Treating the preface as the lottery againThe lottery set is inconsistent, so one accepted claim is certainly false. The preface set is consistent, and every accepted claim might be true. They share a failure of closure and differ in why.
3
Dropping belief in each claim because the whole is doubtfulThe doubt about the book is a doubt about the conjunction. No sentence has become less probable, and withdrawing from every claim at 99/100 would give up a hundred well-supported beliefs to avoid an inconsistency the set does not contain.
Check your understanding
✓
Finish when you can say why the set is consistent and the conjunction is not accepted.
You should be able to apply a threshold to independent claims, state that the accepted set has a model, compute the probability that all are true as a product, and contrast the failure of closure here with its failure in the lottery.
Note: Whether the author should believe the book is error-free, doubt it, or suspend judgement is contested; the lab shows only that the three cannot all be derived from the threshold rule together.