Classical modal logic: Difference between revisions

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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 \equivleftrightarrow \lnot\Box\lnot A</math>

whichthat is also [[Deductive closure|closed]] under the rule

:<math>\frac{ A \equivleftrightarrow B \vdash }{\Box A\equivleftrightarrow \Box B}.</math>

Alternatively, one can give a dual definition of '''L''' by which '''L''' is classical iff[[if and only if]] it contains (as axiom or theorem)

:<math>\Box A \equivleftrightarrow \lnot\Diamond\lnot A</math>

and is closed under the rule

:<math>\frac{ A \equivleftrightarrow B \vdash }{\Diamond A\equivleftrightarrow \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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==References==

Chellas, Brian. ''[https://books.google.com/books?hl=en&lr=&id=YupiXWV5j6cC&oi=fnd&pg=PR7&dq=%22Modal+Logic:+An+Introduction%22+chellas&ots=0_wSpNFhE2&sig=E_wvBZnIl7EhESS32sa1EbCuznU#v=onepage&q=%22Classical%20modal%20logic%22&f=false Modal Logic: An Introduction]''. Cambridge University Press, 1980.

==Notes==

{{reflist}}

* Chellas, Brian. ''[https://books.google.com/books?hl=en&lr=&id=YupiXWV5j6cC&oi=fnd&pg=PR7&dqq=%22Modal+Logic:22Classical+Anmodal+Introductionlogic%22+chellas&otspg=0_wSpNFhE2&sig=E_wvBZnIl7EhESS32sa1EbCuznU#v=onepage&q=%22Classical%20modal%20logic%22&f=falsePR7 Modal Logic: An Introduction]''. Cambridge University Press, 1980.

==External links==

{{DEFAULTSORT:Classical Modal Logic}}

[[Category:Modal logic]]

{{logic-stub}}