Still thinking more about this logic where all the propositions are labelled over a boolean algebra, struggling to complete a heuristic analogy
trivial algebra : plain intuitionistic logic
commutative monoid : linear logic
monoid : ordered logic
boolean algebra : ???
(altough I'm not really putting all the pertinent information in this little table: e.g., it's not just the choice of the commutative monoid that gives linear logic, but also the choice of how you encode connectives into monoid operations. Still...)
Anyway it's so maddeningly close to looking like bunched logic, even though I know it's not quite, because it satisfies things like A & top |- A * A.
