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.
"A Basic System of Congruential-to-Monotone Bimodal Logic and Two of Its Extensions." Notre Dame J. Formal Logic 37 (4) 602 - 612, Fall 1996. https://doi.org/10.1305/ndjfl/1040046144