normal modal logic
The study of modal logic is based on the concept of a logic, which is a set $\mathrm{\Lambda}$ of wff’s satisfying the following:

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contains all tautologies^{}, and

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is closed under modus ponens^{}.
The last condition means: if $A$ and $A\to B$ are in $\mathrm{\Lambda}$, so is $B$ in $\mathrm{\Lambda}$.
A normal modal logic is a modal logic $\mathrm{\Lambda}$ that includes the law of distribution K (after Kripke):
$$\mathrm{\square}(A\to B)\to (\mathrm{\square}A\to \mathrm{\square}B)$$ 
as an axiom schema^{}, and obeying the rule of necessitation $RN$:
from $\u22a2A$, we may infer $\u22a2\mathrm{\square}A$: if $A\in \mathrm{\Lambda}$, then $\mathrm{\square}A\in \mathrm{\Lambda}$.
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:

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4: $\mathrm{\square}A\to \mathrm{\square}\mathrm{\square}A$

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5: $\mathrm{\u25c7}A\to \mathrm{\square}\mathrm{\u25c7}A$

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D: $\mathrm{\square}A\to \mathrm{\u25c7}A$

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T: $\mathrm{\square}A\to A$

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B: $A\to \mathrm{\square}\mathrm{\u25c7}A$

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C: $\mathrm{\square}(A\wedge \mathrm{\square}B)\to \mathrm{\square}(A\wedge B)$

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M: $\mathrm{\square}(A\wedge B)\to \mathrm{\square}A\wedge \mathrm{\square}B$

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G: $\mathrm{\u25c7}\mathrm{\square}A\to \mathrm{\square}\mathrm{\u25c7}A$

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L: $\mathrm{\square}(A\wedge \mathrm{\square}A\to B)\vee \mathrm{\square}(B\wedge \mathrm{\square}B\to A)$

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W: $\mathrm{\square}(\mathrm{\square}A\to A)\to \mathrm{\square}A$
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 ${\mathrm{\Sigma}}_{1},\mathrm{\dots},{\mathrm{\Sigma}}_{n}$ is typically denoted
K${\mathbf{\Sigma}}_{\mathrm{\U0001d7cf}}\mathbf{}\mathrm{\cdots}\mathbf{}{\mathbf{\Sigma}}_{\mathbf{n}}$.
It is easy to see that K${\mathbf{\Sigma}}_{\mathrm{\U0001d7cf}}\mathbf{}\mathrm{\cdots}\mathbf{}{\mathbf{\Sigma}}_{\mathbf{n}}$ can be built from the “bottom up”: call a finite sequence^{} of wff’s a deduction^{} if each wff is either a tautology, an instance of ${\mathrm{\Sigma}}_{i}$ 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 ${\mathrm{\Lambda}}_{k}$ be the set of all wff’s deducible from deductions of lengths at most $k$. Then
K${\mathbf{\Sigma}}_{\mathrm{\U0001d7cf}}\mathbf{}\mathrm{\cdots}\mathbf{}{\mathbf{\Sigma}}_{\mathbf{n}}\mathrm{=}{\mathrm{\bigcup}}_{i\mathrm{=}\mathrm{1}}^{\mathrm{\infty}}{\mathrm{\Lambda}}_{i}$
Below are some of the most common normal modal logics:
Title  normal modal logic 

Canonical name  NormalModalLogic 
Date of creation  20130322 19:33:38 
Last modified on  20130322 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 
\@unrecurse 