Classical modal logic: Difference between revisions

Revision as of 13:56, 20 December 2007

Latest revision as of 14:46, 1 March 2024

(30 intermediate revisions by 20 users not shown)

Line 1:

{{refimprove|date = January 2009}}

In [[modal logic]], a '''classical modal logic''' '''L''' is any modal logic containing (as axiom or theorem)

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

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

and being closed under the rule

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 of '''L''' by which '''L''' is classical iff it contains (as axiom or theorem)

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 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'''.

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.

== References ==

[[Category:Logic]]

{{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.

{{DEFAULTSORT:Classical Modal Logic}}

[[Category:Modal logic]]

{{logic-stub}}