 Since belief in an obvious contradiction is a paradigm example of irrationality, Kyburg poses a dilemma: either reject agglomeration or reject rules that license belief for a probability of less than one. (Martin Smith 2016, 186–196) warns that even a probability of one leads to joint inconsistency for a lottery that has infinitely many tickets.) Kyburg rejects agglomeration. He promotes toleration of joint inconsistency (having beliefs that cannot all be true together) to avoid belief in contradictions. Reason forbids us from believing a proposition that is necessarily false but permits us to have a set of beliefs that necessarily contains a falsehood. Henry Kyburg’s choice was soon supported by the discovery of a companion paradox.