Classical modal logic

In modal logic, a classical modal logic L is any modal logic containing (as axiom or theorem)

$\Diamond A\equiv \lnot \Box \lnot A$

and being closed under the rule

$A\equiv B\vdash \Box A\equiv \Box B.$

Alternatively one can give a dual definition of L by which L is classical iff it contains (as axiom or theorem)

$\Box A\equiv \lnot \Diamond \lnot A$

and is closed under the rule

$A\equiv B\vdash \Diamond A\equiv \Diamond B.$

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.

References

Chellas, Brian. Modal Logic: An Introduction. Cambridge University Press, 1980.