  Bosbach states represent a way of probabilisticly evaluating the formulas
from various (commutative or non-commutative) many-valued logics. They are
defined on the algebras corresponding to these logics with values in $[0,1]$.
Starting from the observation that in the definition of Bosbach states there
intervenes the standard MV-algebra structure of $[0,1]$, in this paper we
introduce Bosbach states defined on residuated lattices with values in
residuated lattices. We are led to two types of generalized Bosbach states,
with distinct behaviours. The properties of generalized Bosbach states, proven
in the paper, may serve as an algebraic foundation for developping some
probabilistic many-valued logics.
  {\bf Keywords}: Bosbach states, residuated lattices, MV-algebras, $s$-Cauchy
completion, metric completion.
  {\bf MSC 2010}: Primary 06F35. Secondary 06D35.
