  There has been a significant interest in extending various modal logics with
intersection, the most prominent examples being epistemic and doxastic logics
with distributed knowledge. Completeness proofs for such logics tend to be
complicated, in particular on model classes such as S5 like in standard
epistemic logic, mainly due to the undefinability of intersection of modalities
in standard modal logics. A standard proof method for the S5 case was outlined
in [8] and later explicated in more detail in [13], using an
"unraveling-folding method" case to achieve a treelike model to deal with the
problem of undefinability. This method, however, is not easily adapted to other
logics, due to the level of detail and reliance on S5. In this paper we propose
a simpler proof technique by building a treelike canonical model directly,
which avoids the complications in the processes of unraveling and folding. We
demonstrate the technique by showing completeness of the normal modal logics K,
D, T, B, S4 and S5 extended with intersection modalities. Furthermore, these
treelike canonical models are compatible with Fischer-Ladner-style closures,
and we combine the methods to show the completeness of the mentioned logics
further extended with transitive closure of union modalities known from PDL or
epistemic logic. Some of these completeness results are new.
