Definition:Deontic Logic

Definition

Deontic logic is the logic of obligation and permissibility.

Act

The atoms of deontic logic are known as acts.

They can be used to model things that can wilfully be done, with the tacit understanding that human beings are the ones doing those things.

Obligation

Obligation is one of the modes of deontic logic.

It can be used to model acts that must be performed.

That is, such acts are obligatory, or mandatory.

An act $a$ is denoted as being obligatory by:

$O a$

Permissibility

Permissibility is one of the modes of deontic logic.

It can be used to model acts that may be performed.

That is, such acts are permitted, or allowed.

An act $a$ is denoted as being permitted by:

$P a$

Axioms of Deontic Logic

The axioms of deontic logic are as follows:

\((\text A 1)\)	$:$		an obligatory act is permitted:			\(\ds O a \)	\(\ds \implies \)	\(\ds P a \)
\((\text A 2)\)	$:$		permissibility distributes over disjunction:			\(\ds \map P {a \lor b} \)	\(\ds \implies \)	\(\ds P a \lor P b \)
\((\text A 3)\)	$:$		it is not permissible to not perform an obligatory act:			\(\ds O a \)	\(\ds \iff \)	\(\ds \neg \map P {\neg a} \)

Also see

• Results about deontic logic can be found here.

Historical Note

Deontic logic was first developed by Georg Henrik von Wright in the $1950$s.

Since then many alternative systems have been proposed.

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): deontic logic