Lockean thesis (necessary): S ought to believe p only if S has a rational high credence in p.
According to this version of the thesis, rational belief and rational low credence is impossible, but one can have a rational high credence without being required to believe. In lottery and statistical evidence cases, there are proposi- tions we should have a high credence in but should not believe. However, there is a problem for this version of the Lockean thesis: the preface paradox (Christensen, 2004; Douven & Uffink, 2003; Easwaran & Fitelson, 2015; Makinson, 1965; Pollock, 1986; Ryan, 1991; Worsnip, 2015). Suppose you write a well-researched book, and you should believe every claim in your book. Again, by closure, you then should believe the conjunction of the claims in your book: claim 1 and claim 2 and…claim n. Nevertheless, you might reasonably write in the preface of your book, “while I spent countless hours researching this book, I’m fallible; it is unlikely that all the claims in this book are true.” The large conjunction of all your book’s claims has a low probability. According to the necessity version of the Lockean thesis, you ought not believe it. Again, we have a contradiction. The preface paradox can be solved by endorsing merely a sufficiency version of the Lockean thesis:
Do people see any problems with this?