Provability logic is a modal logic for studying properties of provability predicates, and Interpretability logic for studying interpretability between logical theories. Their natural models are GL-models and Veltman models, for which the accessibility relation is well-founded. That’s why the usual counterexample showing the necessity of finite image property in Hennessy-Milner theorem (see [1]) doesn’t exist for them. However, we show that the analogous condition must still hold, by constructing two GL-models with worlds in them that are modally equivalent but not (...) bisimilar, and showing how these GL-models can be converted to Veltman models with the same properties. In the process we develop some useful constructions: games on Veltman models, chains, and general method of transformation from GL-models/frames to Veltman ones. (shrink)

PA is Peano arithmetic. The formula $\operatorname{Interp}_\mathrm{PA}(\alpha, \beta)$ is a formalization of the assertion that the theory PA + α interprets the theory PA + β (the variables α and β are intended to range over codes of sentences of PA). We extend Solovay's modal analysis of the formalized provability predicate of PA, Pr PA (x), to the case of the formalized interpretability relation $\operatorname{Interp}_\mathrm{PA}(x, y)$ . The relevant modal logic, in addition to the usual provability operator `□', has a (...) binary operator ` $\triangleright$ ' to be interpreted as the formalized interpretability relation. We give an axiomatization and a decision procedure for the class of those modal formulas that express valid interpretability principles (for every assignment of the atomic modal formulas to sentences of PA). Our results continue to hold if we replace the base theory PA with Zermelo-Fraenkel set theory, but not with Gödel-Bernays set theory. This sensitivity to the base theory shows that the language is quite expressive. Our proof uses in an essential way earlier work done by A. Visser, D. de Jongh, and F. Veltman on this problem. (shrink)

Interpretability logic is an extension of provability logic. Veltman models and generalized Veltman models are two semantics for interpretability logic. We consider a connection between Veltman semantics and generalized Veltman semantics. We prove that for a complete image-finite generalized Veltman modelW there is a Veltman model W ′ that is bisimular to W.

We consider interpretability logic, a modal description of the interpretability predicate, and try to determine the most suitable notion of bisimulation for generalized Veltman semantics. In the first part of this paper we consider several notions of bisimulation and determine connections between them. In the second part we develop some further model theoretic properties for generalized Veltman semantics and consider difficulties that arise when studying quotient structures.

A generalized Veltman semantics developed by de Jongh is used to investigate correspondences between several extensions of intepretability logic . In this paper we present some new results on independences.

This paper is the first in a series of three related papers on modal methods in interpretability logics and applications. In this first paper the fundaments are laid for later results. These fundaments consist of a thorough treatment of a construction method to obtain modal models. This construction method is used to reprove some known results in the area of interpretability like the modal completeness of the logic IL. Next, the method is applied to obtain new results: the modal completeness (...) of the logic ILMo, and modal completeness of ILW*. (shrink)