When rationality contradicts itself — Makinson, 1965
"I have endeavored to ensure the accuracy of every statement in this book. Each claim has been carefully researched and verified. I am confident in the truth of each individual assertion..."
"...however, I acknowledge that, given human fallibility, there are almost certainly some errors contained herein."
If you believe P₁ AND P₂ AND ... AND Pₙ are all true,
you should also believe their conjunction (P₁ ∧ P₂ ∧ ... ∧ Pₙ) is true!
But the preface says "not all are true" — the negation of that conjunction!
Both beliefs are individually rational, but together they're contradictory!
The Preface Paradox challenges a fundamental principle of rationality called the Conjunction Principle: if you rationally believe A, and rationally believe B, then you should rationally believe "A and B."
But the author's situation shows this leads to contradiction:
Henry Kyburg argued that rational belief doesn't "agglomerate" — believing A and believing B doesn't require believing (A ∧ B). We can tolerate "inconsistent beliefs" without believing explicit contradictions.
Use degrees of belief (credences) instead of binary belief. You can assign 99% credence to each statement AND 95% credence to "there's an error" — no contradiction, just probability theory.
When focused on statement P₁, you believe it. When writing the preface, you're in a different "context" evaluating the whole. Belief is fragmented across contexts.
Maybe 99% confidence isn't enough for "full belief." If you require near-certainty for belief, and account for the conjunction's lower probability, the paradox dissolves.
The paradox was introduced by David Makinson in his 1965 paper "The Paradox of the Preface" published in Analysis.
Makinson was inspired by Raymond Wilder's 1952 textbook Introduction to the Foundations of Mathematics, which included such a preface — and indeed, the 1982 reprint contained three pages of errata!
The paradox is closely related to the Lottery Paradox (also proposed in the 1960s), which similarly shows how high-probability individual beliefs can combine into certain falsehood.