Classical modal logic

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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.

[edit] References

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

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