  Harvey Friedman shows that, over Peano Arithmetic, the consistency statement
for a finitely axiomatised theory $A$ can be characterised as the weakest
statement $C$ over Peano Arithmetic such that ${\sf PA}+C$ interprets $A$. We
study which base theories $U$ have the property that, for any finitely
axiomatised $A$, there is a weakest $C$ such that $U+C$ interprets $A$. We call
such theories Friedman-reflexive.
  We show that a very weak theory, Peano Corto, is Friedman-reflexive. We do
not get the usual consistency statements here, but bounded, cut-free or
Herbrand consistency statements.
  We prove a characterisation theorem for Friedman-reflexive sequential
theories. We provide an example of a Friedman-reflexive sequential theory that
substantially differs from the paradigm cases of Peano Arithmetic and Peano
Corto.
  The consistency-like statements provided by a Friedman-reflexive base $U$ can
be used to define a provability-like notion for a finitely axiomatised $A$ that
interprets $U$ via an interpretation $K$ of $U$ in $A$. We call the modal
logics based on this idea \emph{interpreter logics}. These logics satisfy the
L\"ob Conditions. We provide conditions for when these logics extend {\sf S}4,
{\sf K}45, and L\"ob's Logic. We show that, if either $U$ or $A$ is sequential,
then the condition for extending L\"ob's Logic is fulfilled. Moreover, if our
base theory $U$ is sequential and if, in addition, its interpreters can be
effectively found, we prove Solovay's Theorem. This holds even if the
provability-like operator is not necessarily representable by a predicate of
G\"odel numbers.
  At the end of the paper, we briefly discuss how successful the
coordinate-free approach is.
