  We investigate modal logics of high probability having two unary modal
operators: an operator $K$ expressing probabilistic certainty and an operator
$B$ expressing probability exceeding a fixed rational threshold $c\geq\frac
12$. Identifying knowledge with the former and belief with the latter, we may
think of $c$ as the agent's betting threshold, which leads to the motto "belief
is willingness to bet." The logic $\mathsf{KB.5}$ for $c=\frac 12$ has an
$\mathsf{S5}$ $K$ modality along with a sub-normal $B$ modality that extends
the minimal modal logic $\mathsf{EMND45}$ by way of four schemes relating $K$
and $B$, one of which is a complex scheme arising out of a theorem due to
Scott. Lenzen was the first to use Scott's theorem to show that a version of
this logic is sound and complete for the probability interpretation. We
reformulate Lenzen's results and present them here in a modern and accessible
form. In addition, we introduce a new epistemic neighborhood semantics that
will be more familiar to modern modal logicians. Using Scott's theorem, we
provide the Lenzen-derivative properties that must be imposed on finite
epistemic neighborhood models so as to guarantee the existence of a probability
measure respecting the neighborhood function in the appropriate way for
threshold $c=\frac 12$. This yields a link between probabilistic and modal
neighborhood semantics that we hope will be of use in future work on modal
logics of qualitative probability. We leave open the question of which
properties must be imposed on finite epistemic neighborhood models so as to
guarantee existence of an appropriate probability measure for thresholds
$c\neq\frac 12$.
