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

• is closed under modus ponens.

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):

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:

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 non-normal.