Classical modal logic: Difference between revisions

Revision as of 02:05, 23 November 2010

Latest revision as of 14:46, 1 March 2024

(14 intermediate revisions by 10 users not shown)

Line 3:

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'''.

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==

{{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.

==External links==

{{DEFAULTSORT:Classical Modal Logic}}

[[Category:Modal logic]]

{{logic-stub}}

[[pt:Lógica modal clássica]]