  We propose a doxastic \L ukasiewicz logic \textbf{B\L} that is sound and
complete with respect to the class of Kripke-based models in which atomic
propositions and accessibility relations are both infinitely valued in the
standard MV-algebra [0,1]. We also introduce some extensions of \textbf{B\L}
corresponding to axioms \textbf{D}, \textbf{4}, and \textbf{T} of classical
epistemic logic. Furthermore, completeness of these extensions are established
corresponding to the appropriate classes of models.
