All standard epistemic logics legitimate something akin to the principle of closure, according to which knowledge is closed under competent deductive inference. And yet the principle of closure, particularly in its multiple premise guise, has a somewhat ambivalent status within epistemology. One might think that serious concerns about closure point us away from epistemic logic altogether—away from the very idea that the knowledge relation could be fruitfully treated as a kind of modal operator. This, however, need not be so. The (...) abandonment of closure may yet leave in place plenty of formal structure amenable to systematic logical treatment. In this paper we describe a family of weak epistemic logics in which closure fails, and describe two alternative semantic frameworks in which these logics can be modelled. One of these—which we term plurality semantics—is relatively unfamiliar. We explore under what conditions plurality frames validate certain much-discussed principles of epistemic logic. It turns out that plurality frames can be interpreted in a very natural way in light of one motivation for rejecting closure, adding to the significance of our technical work. The second framework that we employ—neighbourhood semantics—is much better known. But we show that it too can be interpreted in a way that comports with a certain motivation for rejecting closure. (shrink)

If what is known need not be closed under logical consequence, then a distinction arises between something's being known to be the case (by a specific agent) and its following from something known (to that subject). When each of these notions is represented by a sentence operator, we get a bimodal logic in which to explore the relations between the two notions.