Classical modal logic: Difference between revisions

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In [[modal logic]], a '''classical modal logic''' '''L''' is any modal logic closed under either

In [[modal logic]], a '''classical modal logic''' '''L''' is any modal logic closedcontaining under(as eitheraxiom or theorem) the [[duality (mathematics)|duality]] of the modal operators

<math>\Diamond A \equiv \lnot\Box\lnot A</math>

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

and

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

<math> A \equiv B \vdash \Box A\equiv\Box B.</math>

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

Alternatively one can give a dual definition '''L''' by which '''L''' is classical iff

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 \equiv \lnot\Diamond\lnot A</math>

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

and

and is closed under the rule

<math> A \equiv B \vdash \Diamond A\equiv\Diamond B.</math>

:<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|algebraic]] and [[neighborhood semantics]] characterize familiar classical modal systems that are weaker than the weakest normal modal logic '''K'''.

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'''.

== References ==

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

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

[[Category:Logic]]

== References ==

{{reflist}}

* Chellas, Brian. ''[https://books.google.com/books?id=YupiXWV5j6cC&q=%22Classical+modal+logic%22&pg=PR7 Modal Logic: An Introduction]''. Cambridge University Press, 1980.

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[[Category:Modal logic]]

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