Display Calculi for Logics with Relative Accessibility Relations

We define cut-free display calculi for knowledge logics where an indiscernibility relation is associated to each set of agents, and where agents decide the membership of objects using this indiscernibility relation. To do so, we first translate the knowledge logics into polymodal logics axiomatised by primitive axioms and then use Kracht's results on properly displayable logics to define the display calculi. Apart from these technical results, we argue that Display Logic is a natural framework to define cut-free calculi for many other logics with relative accessibility relations