Classical modal logic

In modal logic, a classical modal logic L is any modal logic containing (as axiom or theorem) the duality of the modal operators
formula_1
that is also closed under the rule
formula_2
Alternatively one can give a dual definition of L by which L is classical if and only if it contains (as axiom or theorem)
formula_3
and is closed under the rule
formula_4
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.
Every regular modal logic is classical, and every normal modal logic is regular and hence classical.