Extended modal logics, being invariant under various forms of bisimulation, turn out to be of particular interest, because we can consider properties expressed by these logics as process-invariants, instead of mere transition system properties. Altogether bisimulation can be seen as a bridge between Modal Logic, nonwell-founded Set Theory, and Process Theory, and one can fruitfully use this bridge to transfer results and techniques from one area to the others. In this dissertation we cross this bridge a number of times. We prove results in extended modal logics that can be used in Process Theory. This allows us to provide alternative definitions of classes of extended modal logics well-known by the modal logic community.
Interpolation is another desirable property that a well-behaved logic is supposed to have.
Since interpolation can be proved whenever a Gentzen-style sequent calculus without cut is available, a failure of interpolation can be seen as a signal that the logic cannot have such an elegant calculus. Moreover, interpolation has a nice consequence known as the Beth property, which says that implicit and explicit definitions in the logic coincide. Interpolation was also re-discovered in recent years by computer scientists from a more practical point of view, as a useful property in the context of modular databases. AF A consisting of infinitary modal formulae can be built. Other reasonable theories are known, whose study helped to understand the possible structure of such sets in more depth.