  Recent success of Bayesian methods in neuroscience and artificial
intelligence gives rise to the hypothesis that the brain is a Bayesian machine.
Since logic, as the laws of thought, is a product and practice of the human
brain, it leads to another hypothesis that there is a Bayesian algorithm and
data-structure for logical reasoning. In this paper, we give a Bayesian account
of entailment and characterize its abstract inferential properties. The
Bayesian entailment is shown to be a monotonic consequence relation in an
extreme case. In general, it is a sort of non-monotonic consequence relation
without Cautious monotony or Cut. The preferential entailment, which is a
representative non-monotonic consequence relation, is shown to be maximum a
posteriori entailment, which is an approximation of the Bayesian entailment. We
finally discuss merits of our proposals in terms of encoding preferences on
defaults, handling change and contradiction, and modeling human entailment.
