Classical modal logic: Difference between revisions

Revision as of 16:19, 20 December 2020

Revision as of 17:20, 19 October 2022

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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>

that 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 [[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'''.