Classical modal logic
which is also closed under the rule
Alternatively one can give a dual definition of L by which L is classical if it contains (as axiom or theorem)
and is closed under the rule
The weakest classical system is sometimes referred to as E and is non-normal. Both algebraic and neighborhood semantics characterize familiar classical modal systems that are weaker than the weakest normal modal logic K.
Chellas, Brian. Modal Logic: An Introduction. Cambridge University Press, 1980.