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This is a largely expository paper in which the following simple idea is pursued. Take the truth value of a formula to be the set of agents that accept the formula as true. This means we work with an arbitrary (finite) Boolean algebra as the truth value space. When this is properly formalized, complete modal tableau systems exist, and there are natural versions of bisimulations that behave well from an algebraic point of view. There remain significant problems concerning the proper (...) 

In this paper, we advocate the usage of the family of Heytingvalued modal logics, introduced by M. Fitting, by presenting a simple 3valued modal language and axiomatizing an interesting 3valued logic of belief. We give two simple bisimulation relations for the modal language, one that respects nonfalsity and one that respects the truth value. The doxastic logic axiomatized, apart from being interesting in its own right for KR applications, it comes with an underlying 3valued propositional logic which is a syntactic (...) 

This is a largely expository paper in which the following simple idea is pursued. Take the truth value of a formula to be the set of agents that accept the formula as true. This means we work with an arbitrary Boolean algebra as the truth value space. When this is properly formalized, complete modal tableau systems exist, and there are natural versions of bisimulations that behave well from an algebraic point of view. There remain significant problems concerning the proper formalization, (...) 