Normal modal logic

2022-10-19T17:10:03Z

2A02:1812:110C:DC00:7485:BAD8:F6B3:110:

In [[logic]], a '''normal [[modal logic]]''' is a set ''L'' of modal formulas such that ''L'' contains:
* All propositional [[tautology (logic)|tautologies]];
* All instances of the [[Saul_Kripke|Kripke]] schema: <math>\Box(A\to B)\to(\Box A\to\Box B)</math>
and it is closed under:
* Detachment rule (''[[modus ponens]]''): <math> A\to B, A \in L</math> implies <math> B \in L</math>;
* Necessitation rule: <math> A \in L</math> implies <math>\Box A \in L</math>.

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 (modal logic)|S5]], are normal (and hence are extensions of '''K'''). However a number of [[deontic logic|deontic]] and [[epistemic logic]]s, for example, are non-normal, often because they give up the Kripke schema.

Every normal modal logic is [[regular modal logic|regular]] and hence [[classical modal logic|classical]].

== 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>

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

{{Logic-stub}}

[[Category:Modal logic]]

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Normal modal logic