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

contains all tautologies, and
The last condition means: if $A$ and $A\to B$ are in $\Lambda$, so is $B$ in $\Lambda$.
A normal modal logic is a modal logic $\Lambda$ that includes the law of distribution K (after Kripke):
$\square(A\to B)\to(\square A\to\square B)$ 
as an axiom schema, and obeying the rule of necessitation $RN$:
from $\vdash A$, we may infer $\vdash\square A$: if $A\in\Lambda$, then $\square A\in\Lambda$.
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:

4: $\square A\to\square\square A$

5: $\Diamond A\to\square\Diamond A$

D: $\square A\to\Diamond A$

T: $\square A\to A$

B: $A\to\square\Diamond A$

C: $\square(A\wedge\square B)\to\square(A\wedge B)$

M: $\square(A\wedge B)\to\square A\wedge\square B$

G: $\Diamond\square A\to\square\Diamond A$

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

W: $\square(\square A\to A)\to\square A$
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 $\Sigma_{1},\ldots,\Sigma_{n}$ is typically denoted
K$\mathbf{\Sigma_{1}\cdots\Sigma_{n}}$.
It is easy to see that K$\mathbf{\Sigma_{1}\cdots\Sigma_{n}}$ 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 $\Sigma_{i}$ 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 $\Lambda_{k}$ be the set of all wff’s deducible from deductions of lengths at most $k$. Then
K$\mathbf{\Sigma_{1}\cdots\Sigma_{n}}=\bigcup_{{i=1}}^{{\infty}}\Lambda_{i}$
Below are some of the most common normal modal logics:
name  D  T  B  S4  S5  GL  K4.3  S4.3 

notation  KD  KT  KTB  KT4  KT5  KW  K4L  KT4L 
Remarks

D is commonly used in the study of deontic logic (logic of obligation). Extensions of D such as KD4 and KD45 are used in the study of doxastic logic (logic of belief).

GL is known as provability logic, where $\square A$ means $A$ is provable in Peano arithmetic.

S4 and S5 are two of the Lewis’ 5 modal logical systems. They are commonly used in the study of epistemic logic (logic of knowledge). The modal logics S1, S2, and S3 are nonnormal.
Semantics
The dominant semantics for normal modal logic is the Kripke semantics, or relational semantics. More on this can be found here. A logic is sound in a class of frames if every theorem is valid in every frame in the class, and complete if any formula valid in every frame in the class is a theorem. When a logic $\Lambda$ is both sound and complete in a class $\mathcal{C}$ of frames, we say that $\mathcal{C}$ describes $\Lambda$.
The following table lists the logics K$\mathbf{\Sigma}$ and the corresponding sound and complete classes of (Kripke) frames:
$\Sigma$ in K$\mathbf{\Sigma}$  frame K$\mathbf{\Sigma}$ is sound in  frame K$\mathbf{\Sigma}$ is complete in 

4  transitive  transitive 
5  Euclidean  Euclidean 
D  serial  serial 
T  reflexive  reflexive 
B  symmetric  symmetric 
G  weakly directed  weakly directed 
L  weakly connected  weakly connected 
W  transitive and converse wellfounded  finite transitive and irreflexive 
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