Self Provers and $Σ_1$ Sentences

Abstract: This paper from 2012 is the second in a series of three papers. All three papers deal with interpretability logics and related matters. In the first paper a construction method was exposed to obtain models of these logics. Using this method, we obtained some completeness results, some already known, and some new. In this paper, we will set the construction method to work to obtain more results. First, the modal completeness of the logic ${\textbf{IL}}({\sf M})$ is proved using the construction method. This is not a new result, but by using our new proof we can obtain new results. Among these new results are some admissible rules for ${\textbf{IL}}({\sf M})$ and ${\textbf{GL}}$. Moreover, the new proof will be used to classify all the essentially $\Delta_1$ and also all the essentially $\Sigma_1$ formulas of ${\textbf{IL}}({\sf M})$. Closely related to essentially $\Sigma_1$ sentences are the so-called \emph{self provers}. A self-prover is a formula $\varphi$ which implies its own provability, that is $\varphi \to \Box \varphi$. Each formula $\varphi$ will generate a self prover $\varphi \wedge \Box \varphi$. We will use the construction method to characterize those sentences of ${\textbf{GL}}$ that generate a self prover that is trivial in the sense that it is $\Sigma_1$.