In
modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
, a classical modal logic L is any modal logic containing (as axiom or theorem) the
duality of the modal operators
that is also
closed under the rule
Alternatively, one can give a dual definition of L by which L is classical
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
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 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
Kripke sem ...
characterize familiar classical modal systems that are weaker than the weakest normal modal logic K.
Every
regular modal logic In modal logic, a regular modal logic is a modal logic containing (as axiom or theorem) the duality of the modal operators:
\Diamond A \leftrightarrow \lnot\Box\lnot A
and closed under the rule
\frac.
Every normal modal logic In logic, a norma ...
is classical, and every
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(A\to B)\to(\Box A\to\Box B)
and it is closed under:
* Detachment rule (''modus po ...
is regular and hence classical.
References
* Chellas, Brian.
Modal Logic: An Introduction'. Cambridge University Press, 1980.
Modal logic
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