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In [[modal logic]], a '''classical modal logic''' '''L''' is any modal logic containing (as axiom or theorem) the [[duality (mathematics)|duality]] of the modal operators
:<math>\Diamond A \leftrightarrow \lnot\Box\lnot A</math>
that is also [[Deductive closure|closed]] under the rule
:<math>\frac{ A \leftrightarrow B }{\Box A\leftrightarrow \Box B}.</math>
Alternatively, one can give a dual definition of '''L''' by which '''L''' is classical [[if and only if]] it contains (as axiom or theorem)
:<math>\Box A \leftrightarrow \lnot\Diamond\lnot A</math>
and is closed under the rule
:<math>\frac{ A \leftrightarrow B }{\Diamond A\leftrightarrow \Diamond B}.</math>
The weakest classical system is sometimes referred to as '''E''' and is [[normal modal logic|non-normal]]. Both [[algebraic semantics (mathematical logic)|algebraic]] and [[neighborhood semantics]] characterize familiar classical modal systems that are weaker than the weakest normal modal logic '''K'''.
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