  This paper studies implications of the consistency conditions among prior,
posteriors, and information sets on introspective properties of qualitative
belief induced from information sets. The main result reformulates the
consistency conditions as: (i) the information sets, without any assumption,
almost surely form a partition; and (ii) the posterior at a state is equal to
the Bayes conditional probability given the corresponding information set.
Implications are as follows. First, each posterior is uniquely determined.
Second, qualitative belief reduces to fully introspective knowledge in a
``standard'' environment. Thus, a care must be taken when one studies
non-veridical belief or non-introspective knowledge. Third, an information
partition compatible with the consistency conditions is uniquely determined by
the posteriors. Fourth, qualitative and probability-one beliefs satisfy truth
axiom almost surely. The paper also sheds light on how the additivity of the
posteriors yields negative introspective properties of beliefs.
