Impossibility Results for Rational Belief

There are two ways of representing rational belief: qualitatively as yes‐or‐no belief, and quantitatively as degrees of belief. Standard rationality conditions are: (i) consistency and logical closure, for qualitative belief, (ii) satisfaction of the probability axioms, for quantitative belief, and (iii) a relationship between qualitative and quantitative beliefs in accordance with the Lockean thesis. In this paper, it is shown that these conditions are inconsistent with each of three further rationality conditions: fallibilism, open‐mindedness, and invariance under independent conceptual expansions. Restrictions of the Lockean thesis that have been suggested in the literature cannot remove the inconsistency. In the outlook we discuss two alternative ways of dealing with this problem: restricting conjunctive closure or going for a dual system account.