Refinements of provability and consistency principles for the second incompleteness theorem

This paper continues the author’s previous study \cite{Kura20}, showing that several weak principles inspired by non-normal modal logic suffice to derive various refined forms of the second incompleteness theorem. Among the main results of the present paper, we show that the set $\{\mathbf{E},\mathbf{C},\mathbf{D}\mathbf{3}\}$ suffices to establish the unprovability of the consistency statement $\mathrm{\neg }{\mathrm{P}\mathrm{r}}_T(0=1)$. We also prove that the set $\{{\mathbf{E}}^{\mathrm{U}},\mathbf{C}{\mathbf{B}}_{\mathbf{\exists }}\}$ yields formalized ${\mathrm{\Sigma }}_1$-completeness.