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单词 NormalModalLogic
释义

normal modal logic


The study of modal logic is based on the concept of a logic, which is a set Λ of wff’s satisfying the following:

  • contains all tautologiesMathworldPlanetmath, and

  • is closed under modus ponensMathworldPlanetmath.

The last condition means: if A and AB are in Λ, so is B in Λ.

A normal modal logic is a modal logic Λ that includes the law of distribution K (after Kripke):

(AB)(AB)

as an axiom schemaMathworldPlanetmath, and obeying the rule of necessitation RN:

from A, we may infer A: if AΛ, then AΛ.

Normal modal logics are the most widely studied modal logics. The smallest normal modal logic is called K. Other normal modal logics are built from K by attaching wff’s as axiom schemas. Below is a list of schemas used to form some of the most common normal modal logics:

  • 4: AA

  • 5: AA

  • D: AA

  • T: AA

  • B: AA

  • C: (AB)(AB)

  • M: (AB)AB

  • G: AA

  • L: (AAB)(BBA)

  • W: (AA)A

For example, the normal modal logic D is the smallest normal modal logic containing D as its axiom schema.

Notation. The smallest normal modal logic containing schemas Σ1,,Σn is typically denoted

K𝚺𝟏𝚺𝐧.

It is easy to see that K𝚺𝟏𝚺𝐧 can be built from the “bottom up”: call a finite sequencePlanetmathPlanetmath of wff’s a deductionMathworldPlanetmathPlanetmath if each wff is either a tautology, an instance of Σi for some i, or as a result of an application of modus ponens or necessitation on earlier wff’s in the sequence. A wff is deducible from if it is the last member of some deduction. Let Λk be the set of all wff’s deducible from deductions of lengths at most k. Then

K𝚺𝟏𝚺𝐧=i=1Λi

Below are some of the most common normal modal logics:

Titlenormal modal logic
Canonical nameNormalModalLogic
Date of creation2013-03-22 19:33:38
Last modified on2013-03-22 19:33:38
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id16
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 03B45
Related topicDisjunctionProperty
Defineslaw of distribution
Definesnecessitation
DefinesK
Defineslogic
\\@unrecurse
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更新时间:2025/5/4 14:03:13