  Bayesian reasoning plays a significant role both in human rationality and in
machine learning. In this paper, we introduce transfinite modal logic, which
combines modal logic with ordinal arithmetic, in order to formalize Bayesian
reasoning semi-quantitatively. Technically, we first investigate some
nontrivial properties of ordinal arithmetic, which then enable us to expand
normal modal logic's semantics naturally and elegantly onto the novel
transfinite modal logic, while still keeping the ordinary definition of Kripke
models totally intact. Despite all the transfinite mathematical definition, we
argue that in practice, this logic can actually fit into a completely finite
interpretation as well. We suggest that transfinite modal logic captures the
essence of Bayesian reasoning in a rather clear and simple form, in particular,
it provides a perfect explanation for Sherlock Holmes' famous saying, "When you
have eliminated the impossible, whatever remains, however improbable, must be
the truth." We also prove a counterpart of finite model property theorem for
our logic.
