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

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

which is also 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.

Every regular modal logic is classical, and every normal modal logic is regular and hence classical.

References

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

Notes

    External links


    This article is issued from Wikipedia - version of the Sunday, March 17, 2013. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.