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:

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):


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


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


Below are some of the most common normal modal logics:

Title normal modal logic
Canonical name NormalModalLogic
Date of creation 2013-03-22 19:33:38
Last modified on 2013-03-22 19:33:38
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 16
Author CWoo (3771)
Entry type Definition
Classification msc 03B45
Related topic DisjunctionProperty
Defines law of distribution
Defines necessitation
Defines K
Defines logic