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:

• contains all tautologies, and

• 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 $⊢A$, we may infer $⊢\mathrm{\square }A$: 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:

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

• 5: $\mathrm{◇}A\to \mathrm{\square }\mathrm{◇}A$

• D: $\mathrm{\square }A\to \mathrm{◇}A$

• T: $\mathrm{\square }A\to A$

• B: $A\to \mathrm{\square }\mathrm{◇}A$

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

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

• G: $\mathrm{◇}\mathrm{\square }A\to \mathrm{\square }\mathrm{◇}A$

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

• 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${𝚺}_{\mathrm{𝟏}}\mathbf{}\mathrm{\cdots }\mathbf{}{𝚺}_{𝐧}$.

It is easy to see that K${𝚺}_{\mathrm{𝟏}}\mathbf{}\mathrm{\cdots }\mathbf{}{𝚺}_{𝐧}$ 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${𝚺}_{\mathrm{𝟏}}\mathbf{}\mathrm{\cdots }\mathbf{}{𝚺}_{𝐧}\mathrm{=}{\mathrm{\bigcup }}_{i\mathrm{=}\mathrm{1}}^{\mathrm{\infty }}{\mathrm{\Lambda }}_i$

Below are some of the most common normal modal logics:

Title	normal modal logic
Canonical name	NormalModalLogic
Date of creation	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