Abstract The Gödel translation provides an embedding of the intuitionistic logic $\mathsf {IPC}$ into the modal logic $\mathsf {Grz}$ , which then embeds into the modal logic $\mathsf {GL}$ via the splitting translation. Combined with Solovay’s theorem that $\mathsf {GL}$ is the modal logic of the provability predicate of Peano Arithmetic $\mathsf {PA}$ , both $\mathsf {IPC}$ and $\mathsf {Grz}$ admit provability interpretations. When attempting to ‘lift’ these results to the monadic extensions $\mathsf {MIPC}$ , $\mathsf {MGrz}$ , and $\mathsf {MGL}$ of these logics, the same techniques no longer work. Following a conjecture made by Esakia, we add an appropriate version of Casari’s formula to these monadic extensions (denoted by a ‘+’), obtaining that the Gödel translation embeds $\mathsf {M^{+}IPC}$ into $\mathsf {M^{+}Grz}$ and the splitting translation embeds $\mathsf {M^{+}Grz}$ into $\mathsf {MGL}$ . As proven by Japaridze, Solovay’s result extends to the monadic system $\mathsf {MGL}$ , which leads us to a provability interpretation of both $\mathsf {M^{+}IPC}$ and $\mathsf {M^{+}Grz}$ .

Abstract NNIL-formulas, introduced by Visser in 1983–1984 in a study of $\varSigma _1$-subsitutions in Heyting arithmetic, are intuitionistic propositional formulas that do not allow nesting of implication to the left. The first results about these formulas were obtained in a paper of 1995 by Visser et al. In particular, it was shown that NNIL-formulas are exactly the formulas preserved under taking submodels of Kripke models. Recently, Bezhanishvili and de Jongh observed that NNIL-formulas are also reflected by the colour-preserving monotonic maps of Kripke models. In the present paper, we first show how this observation leads to the conclusion that NNIL-formulas are preserved by arbitrary substructures not necessarily satisfying the topo-subframe condition. Then, we apply it to construct universal models for NNIL. It follows from the properties of these universal models that NNIL-formulas are also exactly the formulas that are reflected by colour-preserving monotonic maps. By using the method developed in constructing the universal models, we give a new direct proof that the logics axiomatized by NNIL-axioms have the finite model property.

With each superintuitionistic logic (si-logic for short), we associate its downward and upward subframizations and characterize them by means of Zakharyaschev’s canonical formulas, as well as by embedding si-logics into the extensions of the propositional lax logic |$\textsf{PLL}$|⁠. In an analogous fashion, with each si-logic, we associate its downward and upward stabilizations and characterize them by means of stable canonical formulas, as well as by embedding si-logics into extensions of the intuitionistic |$\textsf{S4}$|⁠.