The new situation

In this paper we have seen the logic $\mathit{ILM_0}$ to be modally complete. The decidability is unknown so far. Furthermore a new logic $\mathit{ILP_0}$ is introduced which is seen to be modally incomplete. It remains unknown whether $\mathit{ILW}^*$ is modally complete or not. The author and Dick de Jongh conjecture $\mathit{ILW}^*$ to be modally complete, but the logic ${\mathit ILP_0W^*}$ to be modally incomplete, and to have $\mathit{GIL}$ somewhere in between ${\mathit ILP_0W^*}$ and the meet of $\mathit{ILM}$ and $\mathit{ILP}$. $\mathit{GIL}$ is assumed to be stronger than ${\mathit ILP_0W^*}$ since this logic is likely to be incomplete and the situation that $\mathit{GIL}$ is incomplete is to ghastly to imagine. On the other hand $\mathit{GIL}$ cannot be the meet of $\mathit{ILM}$ and $\mathit{ILP}$ because this does not have the disjunction property: $\vdash \Box A \vee \Box B \Rightarrow \vdash A$ or $\ \vdash B$. Theorem 6.3 tells us that the new situation is as depicted below.