In this paper we study 1. the frame-theory of certain bimodal provability logics involving the reflection principle and we study2. certain specific bimodal logics with a provability predicate for a subtheory of Peano arithmetic axiomatized by a non-standardly finite number of axioms.

In this paper the system IL for relative interpretability is studied.

A Kripke-style semantics developed by de Jongh and Veltman is used to investigate relations between several extensions of interpretability logic, IL.

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)

The problem of Uniqueness and Explicit Definability of Fixed Points for Interpretability Logic is considered. It turns out that Uniqueness is an immediate corollary of a theorem of Smoryński.

This paper contains a completeness proof for the system ILW, a rather bewildering axiom system belonging to the family of interpretability logics. We have treasured this little proof for a considerable time, keeping it just for ourselves. Johan’s ftieth birthday appears to be the right occasion to get it out of our wine cellar.

Formulas of the propositional modal language with the unary modal operators □, Σ1, 1, Σ2, 2,… are considered as schemata of sentences of arithmetic , where □A is interpreted as “A is PA-provable”, ΣnA as “A is PA-equivalent to a Σn-sentence” and nA as “A is PA-equivalent to a Boolean combination of Σn-sentences”. We give an axiomatization and show decidability of the sets of the modal formulas which are schemata of: PA-provable, true arithmetical sentences.

This paper is a presentation of astatus quæstionis, to wit of the problemof the interpretability logic of all reasonablearithmetical theories.We present both the arithmetical side and themodal side of the question.Dedicated to Dick de Jongh on the occasion of his 60th birthday.

This paper contains a careful derivation of principles of Interpretability Logic valid in extensions of I0+1.

The problem of Uniqueness and Explicit Definability of Fixed Points for Interpretability Logic is considered. It turns out that Uniqueness is an immediate corollary of a theorem of Smoryski.