  Beklemishev introduced an ordinal notation system for the Feferman-Sch\"utte
ordinal $\Gamma_0$ based on the autonomous expansion of provability algebras.
In this paper we present the logic $\textbf{BC}$ (for Bracket Calculus). The
language of $\textbf{BC}$ extends said ordinal notation system to a strictly
positive modal language. Thus, unlike other provability logics, $\textbf{BC}$
is based on a self-contained signature that gives rise to an ordinal notation
system instead of modalities indexed by some ordinal given a priori. The
presented logic is proven to be equivalent to $\textbf{RC}_{\Gamma_0}$, that
is, to the strictly positive fragment of $\textbf{GLP}_{\Gamma_0}$. We then
define a combinatorial statement based on $\textbf{BC}$ and show it to be
independent of the theory $\textbf{ATR}_0$ of Arithmetical Transfinite
Recursion, a theory of second order arithmetic far more powerful than Peano
Arithmetic.
