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Omnipaedista (talk | contribs) sorry, I have to revert again; Botterweg14 undid your changes using a sound argument; this needs to be discussed further on the template's talk page |
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{{refimprove|date = January 2009}}
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▼
▲In [[modal logic]], a '''classical modal logic''' '''L''' is any modal logic containing (as axiom or theorem)
that is also [[Deductive closure|closed]] under the rule
▲<math>\Diamond A \equiv \lnot\Box\lnot A</math>
Alternatively, one can give a dual definition of '''L''' by which '''L''' is classical
▲<math> A \equiv B \vdash \Box A\equiv\Box B.</math>
▲Alternatively one can give a dual definition of '''L''' by which '''L''' is classical iff it contains (as axiom or theorem)
▲<math>\Box A \equiv \lnot\Diamond\lnot A</math>
and is closed under the rule
:<math>\frac{ A \
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.▼
{{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}}
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