Normal modal logic

In logic, 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: from A and A→B infer B;
• Necessitation rule: from A infer $\Box A$.

The modal logic satisfying exactly the above conditions is the most minimal normal modal logic called K. Most modal logics commonly used nowadays (in terms of having philosophical motivations), e.g. C. I. Lewis's S4 and S5, are extensions of K.