The aim of this paper

We will present a method for establishing modal modal completeness results which uncouples the completeness result from the finite model property. This method is first applied to $\mathit{ILM}$. So a new proof of the modal completeness of $\mathit{ILM}$ is provided. There is an option built in in this construction method to also obtain the finite model property and hence the decidability of a certain logic. By this option we obtain the decidability of $\mathit{ILM}$ in chapter 4. This construction method is then applied in chapter 5 to obtain the completeness of $\mathit{ILM_0}$. In this case however, we did not succeed in proving decidability. In chapter 6 we present a new modal principle and use generalized Veltman semantics to subject it to a modal study.