Full Idea
If the Brouwersche principle, p ⊃ □◊p is adjoined to a standard quantified vesion of the weakest modal logic K, then one can prove both the Barcan principle, and its converse.
Gist of Idea
The two Barcan principles are easily proved in fairly basic modal logic
Source
Bob Hale (Necessary Beings [2013], 09.2)
Book Reference
Hale,Bob: 'Necessary Beings' [OUP 2013], p.204
A Reaction
The Brouwersche principle (that p implies that p must be possible) sounds reasonable, but the Barcan principles strike me as false, so something has to give. They are theorems of S5. Hale proposes giving up classical logic.