Abstract
General Epistemic Logics suffer from the problem of logical omniscience, which is that an agent's knowledge and beliefs are closed under implication. There have been many attempts to solve the problem of logical omniscience. However, according to our intuition, sometimes an agent's knowledge and beliefs are indeed closed under implication. Based on the notion of awareness, we introduce two kinds of negations: general negation and strong negation. Moreover, four kinds of implications, general implication, strong implication, weak implication, and semi-strong implication, are introduced to correspond with the two kinds of negations. In our logics of regular awareness, explicit beliefs are not necessarily closed under general implication, which means that agents are not logically omniscient. However, explicit beliefs are closed under strong implication and semi-strong implication, which captures an intuitive closure property of beliefs.
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