  The interpretability logic of a mathematical theory describes the structural
behavior of interpretations over that theory. Different theories have different
logics. This paper from 2011 revolves around the question what logic describes
the behavior that is present in all theories with a minimum amount of
arithmetic; the intersection over all such theories so to say. We denote this
target logic by ${\textbf{IL}}({\rm All})$. In this paper we present a new
principle $\sf R$ in ${\textbf{IL}}({\rm All})$. We show that $\sf R$ does not
follow from the logic ${\textbf{IL}}{\sf P_0W^*}$ that contains all previously
known principles. This is done by providing a modal incompleteness proof of
${\textbf{IL}}{\sf P_0W^*}$: showing that $\sf R$ follows semantically but not
syntactically from ${\textbf{IL}}{\sf P_0W^*}$. Apart from giving the
incompleteness proof by elementary methods, we also sketch how to work with
so-called Generalized Veltman Semantics as to establish incompleteness. To this
extent, a new version of this Generalized Veltman Semantics is defined and
studied. Moreover, for the important principles the frame correspondences are
calculated. After the modal results it is shown that the new principle $\sf R$
is indeed valid in any arithmetically theory. The proof employs some elementary
results on definable cuts in arithmetical theories.
