We study the logic of neighbourhood models with pointwise intersection, as a
means to characterize multi-modal logics. Pointwise intersection takes us from
a set of neighbourhood sets Ni​ (one for each member i of a set
G, used to interpret the modality □i​) to a new neighbourhood set
NG​, which in turn allows us to interpret the operator □G​.
Here, X is in the neighbourhood for G if and only if X equals the
intersection of some Y={Yi​∣i∈G}. We show that the
notion of pointwise intersection has various applications in epistemic and
doxastic logic, deontic logic, coalition logic, and evidence logic. We then
establish sound and strongly complete axiomatizations for the weakest logic
characterized by pointwise intersection and for a number of variants, using a
new and generally applicable technique for canonical model construction.Comment: Submitted to Advances in Modal Logic 201