Editing Deontic logic (section)

=== Anderson's deontic logic ===
Alan R. Anderson (1959) shows how to define <math>O</math> in terms of the alethic operator <math>\Box</math> and a deontic constant (i.e. 0-ary modal operator) <math>s</math> standing for some sanction (i.e. bad thing, prohibition, etc.): <math>OA\equiv\Box(\lnot A\to s)</math>. Intuitively, the right side of the biconditional says that A's failing to hold necessarily (or strictly) implies a sanction.

In addition to the usual modal axioms (necessitation rule '''N''' and distribution axiom '''K''') for the alethic operator <math>\Box</math>, Anderson's deontic logic only requires one additional axiom for the deontic constant <math>s</math>: <math>\neg \Box s\equiv \Diamond \neg s</math>, which means that there is alethically possible to fulfill all obligations and avoid the sanction. This version of the Anderson's deontic logic is equivalent to '''SDL'''.

However, when modal axiom '''T''' is included for the alethic operator (<math>\Box A\to A</math>), it can be proved in Anderson's deontic logic that <math>O(OA \to A)</math>, which is not included in '''SDL'''. Anderson's deontic logic inevitably couples the deontic operator <math>O</math> with the alethic operator <math>\Box</math>, which can be problematic in certain cases.