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: ${\displaystyle \Box (A\to B)\to (\Box A\to \Box B)}$

and it is closed under:

• Detachment rule (modus ponens): ${\displaystyle A\to B,A\in L}$ implies ${\displaystyle B\in L}$;
• Necessitation rule: ${\displaystyle A\in L}$ implies ${\displaystyle \Box A\in L}$.

The smallest logic satisfying the above conditions is called K. Most modal logics commonly used nowadays (in terms of having philosophical motivations), e.g. C. I. Lewis's S4 and S5, are normal (and hence are extensions of K). However a number of deontic and epistemic logics, for example, are non-normal, often because they give up the Kripke schema.

Every normal modal logic is regular and hence classical.