Regular modal logic
Encyclopedia
In modal logic
, a regular modal logic L is a modal logic closed under the duality
of the modal operators:
and the rule
Every regular modal logic is classical
, and every normal modal logic
is regular and hence classical.
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
, a regular modal logic L is a modal logic closed under the duality
Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. As involutions sometimes have...
of the modal operators:
and the rule
Every regular modal logic is classical
Classical modal logic
In modal logic, a classical modal logic L is any modal logic containing the duality of the modal operators\Diamond A \equiv \lnot\Box\lnot Awhich is also closed under the rule A \equiv B \vdash \Box A\equiv\Box B....
, and every normal modal logic
Normal modal logic
In logic, a normal modal logic is a set L of modal formulas such that L contains:* All propositional tautologies;* All instances of the Kripke schema: \Box\toand it is closed under:...
is regular and hence classical.