Since the seminal work of [Hin62], classical epistemic logic (CEL) and its applications are undermined by the so-called problem of logical omniscience (PLO). Indeed, this problem is only one instance of logical monotony, a strong idealization that is common to most of actual epistemic models. The purpose of this paper is, following [Dub91] and [Dub02], to defend a prooforiented family of solutions to the (PLO). The main feature of this family is the replacement of classical base logic by a cognitively more realistic base logic. Two major weak epistemic logics of this family are discussed: an intuitionistic epistemic logic (IEL) and a linear epistemic logic (LEL).