You write a book. You believe every single sentence in that book. Yet you also write a preface in which you acknowledge that probably something you've said in the book is false. It seems that you believe each claim p1, p2, p3, ... pn, individually but you disbelieve the conjunction (p1 & p2 & p3 & ... & pn). But of course, it follows straightfowardly from p1, p2, p3, ... pn, that their conjunction is true. In denying it, you commit yourself to an inconsistent set of beliefs. We ordinarily think holding an inconsistent set of beliefs is irrational; yet your acknowledging the likelihood of error in the preface seems eminently rational. Hence, the paradox of the preface.
Much has been written about the paradox of the preface, but I want to focus on just one issue here: The challenge it poses for the idea, raised last Monday that we cannot have flatly contradictory beliefs. For if we accept what we might call the conjunctive principle of belief attribution -- the principle that someone who believes A and who believes B also believes A & B -- then it seems to follow that the preface writer has baldly contradictory beliefs: (p1 & p2 & p3 & ... & pn) [from repeated applications of the conjunction principle] and -(p1 & p2 & p3 & ... & pn) [from what he says in the preface].
The solution, I think, is to deny the conjunction principle. On a representational warehouse model of belief, according to which to believe something is to have representations of the right sort stored in an appropriate location in the mind, denying the conjunction principle invites the unsavory conclusion that in order to believe that I got in the car and drove to work I have to represent both "I got in the car" and "I drove to work" and "I got in the car and drove to work". (On the other hand, if the warehouse-representationalist accepts the conjunction principle, she risks sliding into the even more unsavory position of holding that we believe all the logical consequences of our beliefs.)
On a dispositional approach to belief, according to which to believe something is to act, cogitate, and feel in ways concordant with the truth of the proposition in question, there may be room to deny the conjunction principle, without dragging in a suite of redundant representations. The key is to notice that one needn't be absolutely consistently disposed to act in accord with some proposition P to count as believing that P. For example, one can believe in God despite passing fits of irreligiosity. One need only act appropriately generally speaking, most of the time, and when excusing conditions are not present. One need only match the profile of the full and complete believer to a certain degree. (For more on this, see my Phenomenal, Dispositional, Account of Belief.) And what comes in degrees doesn't conjoin.
To see this last point, consider the lottery paradox: It's approximately certain that Jean won't win and approximately certain that Bob won't win and approximately certain that Sanjay won't win, ..., but it doesn't follow that it's approximately certain that no one will win. The uncertainties compound with conjunction. So also, likewise, in the paradox of the preface: I come pretty close to matching the profile of a full and complete believer in each of p1, p2, p3, ... pn, considered individually, but it doesn't follow that I come at all close to matching the dispositional profile of a full believer in the conjunction (p1 & p2 & pn & ... & pn). This point is often made in terms of Bayesian degrees of belief; but I intend it here as a point of set theory, where the relevant sets are sets of dispositions. Having most of the elements of set A and most of the elements of set B does not necessarily imply that one has most of the elements of A+B.
(By the way, I believe it was Jay Rosenberg who first raised this as a puzzle for my rejection of baldly contradictory beliefs, at an APA meeting some years ago.)