Belief, Acceptance, and the Lottery Paradox

We both believe things outright and hold them to degrees, and the lottery and preface paradoxes expose a deep tension between these: plausible principles linking high probability to belief, together with the demand that belief be consistent and closed under conjunction, lead to contradiction.

Definition

This topic concerns how outright belief relates to degrees of belief, and the lottery and preface paradoxes that show a high-probability threshold for belief cannot be combined with the requirements that rational belief be logically consistent and closed under conjunction.

Scope

This topic covers the relation between graded credence and categorical (full) belief, and the paradoxes that arise when one tries to connect them. It examines the lottery paradox, where high probability for each ticket's losing seems to license believing each will lose yet not that all will, and the preface paradox, where an author rationally believes each claim in a book yet believes the book contains some error. It surveys responses that reject a threshold view, deny conjunctive closure, or dispense with full belief. Bayesian credence is treated in a companion topic.

Core questions

• Is full belief reducible to having a sufficiently high credence?
• Why do the lottery and preface paradoxes threaten a threshold view of belief?
• Should rational belief be closed under conjunction?
• Can epistemology dispense with full belief in favour of credences?

Key theories

The lottery paradox

Kyburg notes that if high probability suffices for rational belief, then in a large fair lottery one may believe of each ticket that it will lose, yet conjoining these beliefs yields the belief that no ticket wins, contradicting the known fact that one will.

The preface paradox

Makinson observes that a careful author may rationally believe each individual claim in their book while also rationally believing, as prefaces often state, that the book surely contains at least one error, so a set of individually rational beliefs is jointly inconsistent.

Separating belief from credence

Foley and others argue that the epistemology of full belief and the epistemology of degrees of belief are distinct projects, so that the threshold linking them must be denied or qualified, and conjunctive closure for rational belief abandoned.

History

Kyburg introduced the lottery paradox in 1961 to argue against requiring deductive consistency and closure of rational belief, and Makinson's 1965 preface paradox reinforced the point with an everyday example. The paradoxes became central to debates over whether full belief reduces to high credence, prompting work like Foley's that treats categorical and graded belief as governed by different norms.

Debates

Whether rational belief is closed under conjunction

Defenders of closure must reject a simple probability threshold for belief, since the lottery and preface cases show threshold belief plus closure breeds inconsistency, while those who keep the threshold abandon closure; how to relate full belief and credence without paradox remains open.

Key figures

• Henry Kyburg
• David Makinson
• Richard Foley

Seminal works

• kyburg1961
• makinson1965

Frequently asked questions

What is the lottery paradox?

In a fair lottery with very many tickets, the probability that any given ticket loses is extremely high, so a threshold view of belief says you may rationally believe of each ticket that it will lose. But conjoining all these beliefs gives the belief that no ticket wins, which you know to be false, producing a paradox.

How does the preface paradox differ from the lottery paradox?

Both pit individually rational beliefs against joint consistency, but the preface paradox uses an ordinary case: an author who believes each claim in their book yet, knowing they are fallible, also believes the book contains some mistake. It shows the tension does not depend on artificial lottery setups.