Spin off

• Modal completeness of $\mathit{ILW}^*$.
  The main approach to a modal completeness result is always the same. However for every distinct logic a special ingredient is required to incorporate the frame condition of the added principle into the model construction. One new ingredient used for $\mathit{ILM_0}$ is given by lemmas 5.5 and 5.6. Another novelty is the local character of the construction method. Instead of defining the model in one blow and defining the S and R relations, we adhere to locally defining all entities to gradually build up the model. Dick de Jongh and Frank Veltman have given a completeness proof of $\mathit{ILW}^*$ in [dJV]. It might be the case that both ingredients of $\mathit{ILM_0}$ and $\mathit{ILW}$ can be combined resulting in a completeness proof of $\mathit{ILW}^*$. We conjecture $\mathit{ILW}^*$ to be decidable and modally complete. If both completeness proofs can be combined, the decidability will very likely be easier to establish than for $\mathit{ILM_0}$by itself.

• Essentially $\Sigma_1^0$-ness.
  In an article by Dick de Jongh and Duccio Pianigiani a theorem about essentially $\Sigma_1^0$-ness is proved. A sentence A in for example the language of Löbs logic is essentially $\Sigma^0_1$ in $\mathit{PA}$ if for any *, there is some $\sigma \in \Sigma^0_1$ such that ${\mathit PA}\vdash A^* \leftrightarrow \sigma$.
  Theorem 7.1 (Visser [Vis89], de Jongh [dJD90])   A sentence A in the language of Löbs logic is essentially $\Sigma^0_1$ iff A is provably equivalent to some formula of the form $\bigvee \hspace{-2.5mm} \bigvee_i \Box A_i$.

  Most likely this result can be extended to for example $\mathit{ILM}$. De Jongh uses the fact that $\sigma$ is $\Sigma^0_1$ in $\mathit{PA}$ iff ${\mathit PA} \vdash \alpha \rhd \beta \rightarrow \alpha \wedge \sigma \rhd \beta \wedge \sigma$ for all $\alpha$, $\beta$. Now suppose A is essentially $\Sigma^0_1$ but not provably equivalent to some $\bigvee \hspace{-2.5mm} \bigvee_i \Box A_i$. A model of $p\rhd q$ and $\neg (p\wedge A\rhd q\wedge A)$ is made. It seemed quite difficult to extend this method to $\mathit{ILM}$. The author and Rosalie Iemhoff have payed some effort to do so. With the construction method presented in this paper it might be possible to reduce the problem to an easier statement.