Normal modal logic


In logic, a normal modal logic is a set L of modal formulas such that L contains:
and it is closed under:
The smallest logic satisfying the above conditions is called K. Most modal logics commonly used nowadays, e.g. C. I. Lewis's S4 and S5, are normal. 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.

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 sound and complete with respect to the frame classes given in the table, but they may correspond to a larger class of frames.
NameAxiomsFrame condition
K—all frames
TTreflexive
K44transitive
S4T, 4preorder
S5T, 5 or D, B, 4equivalence relation
S4.3T, 4, Htotal preorder
S4.1T, 4, Mpreorder and
S4.2T, 4, Gdirected preorder
GL, K4WGL or 4, GLfinite strict partial order
Grz, S4GrzGrz or T, 4, Grzfinite partial order
DDserial
D45D, 4, 5transitive, serial, and Euclidean