Editing Normal modal logic (section)

== Common normal modal logics ==
<onlyinclude>The following table lists several common normal modal systems. </onlyinclude>The notation refers to the table at [[Kripke semantics#Common modal axiom schemata|Kripke semantics § Common modal axiom schemata]]. 
<onlyinclude>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.

{| class="wikitable"
! Name !! Axioms !! Frame condition
|-
! id="K" | K
| —
| all frames
|-
! T
| T
| reflexive
|-
! K4
| 4
| transitive
|-
! S4
| T, 4
| [[preorder]]
|-
! [[S5 (modal logic)|S5]]
| T, 5 or D, B, 4
| [[equivalence relation]]
|-
! S4.3
| T, 4, H
| [[total preorder]]
|-
! S4.1
| T, 4, M
| preorder, <math>\forall w\,\exists u\,(w\,R\,u\land\forall v\,(u\,R\,v\Rightarrow u=v))</math>
|-
! S4.2
| T, 4, G
| [[directed set|directed]] preorder
|-
! [[provability logic|GL]], K4W
| GL or 4, GL
| finite [[strict order|strict partial order]]
|-
! Grz, S4Grz
| Grz or T, 4, Grz
| finite [[partial order]]
|-
! D
| D
| [[serial relation|serial]]
|- 
! D45
| D, 4, 5
| transitive, serial, and Euclidean
|}
</onlyinclude>