The many faces of interpretability
Evan Goris, Joost J. Joosten

Abstract:
In this paper we discus work in progress on interpretability
logics. We show how semantical considerations have allowed us to
formulate non-trivial principles about formalized interpretability. In
particular we falsify the conjecture about the nature of the
interpretability logic of all reasonable arithmetical theories. We
consider this an interesting example of how purely semantical
considerations give new non-trivial facts about syntactical and
arithmetical notions.  In addition we give some apparatus that allows
us to push 'global' semantical properties into more 'local'
syntactical ones. With this apparatus, the rather wild behavior of the
different interpretability logics are nicely formulated in a single
notion that expresses their differences in a uniform way.