Epistemic closure under deductive inference: what is it and can we afford it?

1. KNOWLEDGE BY INFERENCE I believe many things, but I also know I’m wrong about some of them. There is nothing irrational, bad, or paradoxical about my epistemic situation. In fact, it seems that there’s something wrong with subjects who don’t know, or at least strongly believe, that they are mistaken about one or more of their beliefs. This mundane and seemingly innocent observation, nonetheless, is of great consequence for the question of whether the sets of propositions that are believed or known are closed under certain logical operations. Standardly understood, a set is closed under a logical operation if and only if the result of the logical operation will be a member of the set. Specifically, since belief and knowledge are closely connected to truth and since the set of all true propositions is surely closed under logical operations that preserve truth, knowledge and belief will be closed under any operation that is salve veritate. Thus, the set of all known propositions would be closed under deduction. The observation above, however, shows that this is not the case. Although the truth of propositions believed or known is certainly closed under