# 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:

and it is closed under:

- Detachment rule (
*modus ponens*): ; - Necessitation rule: implies .

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 extensions of **K**. However a number of deontic and epistemic logics, for example, are non-normal, often because they give up the Kripke schema.

## Common normal modal logics

The following table lists several common normal modal systems. The notation refers to the table at Kripke semantics § Common modal axiom schemata. Frame conditions for some of the systems were simplified: the logics are *complete* with respect to the frame classes given in the table, but they may *correspond* to a larger class of frames.

Name | Axioms | Frame condition |
---|---|---|

K | — | all frames |

T | T | reflexive |

K4 | 4 | transitive |

S4 | T, 4 | preorder |

S5 | T, 5 or D, B, 4 | equivalence relation |

S4.3 | T, 4, H | total preorder |

S4.1 | T, 4, M | preorder, |

S4.2 | T, 4, G | directed preorder |

GL, K4W | GL or 4, GL | finite strict partial order |

Grz, S4Grz | Grz or T, 4, Grz | finite partial order |

D | D | serial |

D45 | D, 4, 5 | transitive, serial, and Euclidean |

## References

- Alexander Chagrov and Michael Zakharyaschev,
*Modal Logic*, vol. 35 of Oxford Logic Guides, Oxford University Press, 1997.