Modal matters in interpretability logics

Abstract

In this paper we expose a method for building models for interpretability logics. The method can be compared to the method of taking unions of chains in classical model theory. Many applications of the method share a common part. We isolate this common part in a main lemma. Doing so, many of our results become applications of this main lemma. We also briefly describe how our method can be generalized to modal logics with a different signature. With the general method, we prove completeness for the interpretability logics IL, ILM, ILM0 and ILW*. We also apply our method to obtain a classification of the essential Σ1-sentences of essentially reflexive theories. We briefly comment on such a classification for finitely axiomatizable theories. As a digression we proof some results on self-provers. Towards the end of the paper we concentrate on modal matters con- cerning IL(All), the interpretability logic of all reasonable arithmetical theories. We prove the modal incompleteness of the logic ILW*P0. We put forward a new principle R, and show it to be arithmetically sound in any reasonable arithmetical theory. Finally we make some general remarks on the logics ILRW and IL(All).