Axiom:Axioms of Deontic Logic

Definition

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} $

\((\text A 1)\)	$:$	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:	\(\ds O a \)	\(\ds \iff \)	\(\ds \neg \map P {\neg a} \)