Classical modal logic
Encyclopedia
In modal logic
, a classical modal logic L is any modal logic containing (as axiom or theorem) the duality
of the modal operators
which is also closed under the rule
Alternatively one can give a dual definition of L by which L is classical iff it contains (as axiom or theorem)
and is closed under the rule
The weakest classical system is sometimes referred to as E and is non-normal
. Both 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.
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 classical modal logic L is any modal logic containing (as axiom or theorem) 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
which is also closed under the rule
Alternatively one can give a dual definition of L by which L is classical iff it contains (as axiom or theorem)
and is closed under the rule
The weakest classical system is sometimes referred to as E and is non-normal
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:...
. Both algebraic
Algebraic semantics
An programming language theory, the algebraic semantics of a programming language is a form of axiomatic semantics based on algebraic laws for describing and reasoning about program semantics in a formal manner....
and neighborhood semantics
Neighborhood semantics
Neighborhood semantics, also known as Scott-Montague semantics, is a formal semantics for modal logics. It is a generalization, developed independently by Dana Scott and Richard Montague, of the more widely known relational semantics for modal logic...
characterize familiar classical modal systems that are weaker than the weakest normal modal logic K.
Every regular modal logic
Regular modal logic
In modal logic, a regular modal logic L is a modal logic closed under the duality of the modal operators:\Diamond A \equiv \lnot\Box\lnot Aand the rule\to C \vdash \to\Box C....
is classical, 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.