In
modal logic
Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields
it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
, 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" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
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
In logic, a normal modal logic is a set ''L'' of modal formulas such that ''L'' contains:
* All propositional tautology (logic), tautologies;
* All instances of the Kripke_semantics, Kripke schema: \Box(A\to B)\to(\Box A\to\Box B)
and it is closed ...
is regular and hence classical.
References
* Chellas, Brian.
Modal Logic: An Introduction'. Cambridge University Press, 1980.
Modal logic
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