正规模态逻辑

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在逻辑中，正规模态逻辑是模态公式的集合 Ldisplaystyle LL，Ldisplaystyle LL 包含


  • 所有命题重言式，

  • 所有满足 Kripke 模式的实例: ◻(A→B)→(◻A→◻B)displaystyle Box (Ato B)to (Box Ato Box B)Box(Ato B)to(Box AtoBox B)，

并且 Ldisplaystyle LL 闭合于


  • 分拆规则（肯定前件）: A→B,A⊢Bdisplaystyle Arightarrow B,Avdash Bdisplaystyle Arightarrow B,Avdash B，

  • 必然性规则: 从 ⊢Adisplaystyle vdash Adisplaystyle vdash A 推出 ⊢◻Adisplaystyle vdash Box Adisplaystyle vdash Box A。

最小化的满足上述条件的逻辑叫做 K。大多数如今常用的模态逻辑（指有哲学动机的）如C. I. 刘易斯的S4与S5皆为在K基础之上的扩展。然而也有一部分如道义逻辑与认识逻辑是非正规的，因为它们舍弃了Kripke模式。



常见的模态逻辑


下表给出了一些常见的模态逻辑系统。表中的标记可参见 Kripke 语义 § 常见模态公理模式。 某些系统的框架条件要求被简化了，它们在给定的框架类中完备，但是可能对应一个更大的框架类。









































名称公理框架条件
K
—
所有框架
T
T
自反
K4
4
传递
S4
T, 4

预序

S5
T, 5 或 D, B, 4

等价关系
S4.3
T, 4, H
完全预序
S4.1
T, 4, M
预序, ∀w∃u(wRu∧∀v(uRv⇒u=v))displaystyle forall w,exists u,(w,R,uland forall v,(u,R,vRightarrow u=v))displaystyle forall w,exists u,(w,R,uland forall v,(u,R,vRightarrow u=v))
S4.2
T, 4, G

有向预序
GL
GL or 4, GL
有穷严格偏序
Grz, S4Grz
Grz or T, 4, Grz
有穷偏序
D
D
serial
D45
D, 4, 5
传递，全序且欧拉


参见


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


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