Deontic logic is a form of modal logic used to describe and reason about obligation and permission.

Deontic logic adds the modal operator <math>O<math> to the language of propositional logic. <math>O\phi<math> is read as it is obligatory that <math>\phi<math>. In the semantics of modal logic, the operator behaves analogously to the <math>\Box<math> (necessity) operator of alethic modal logic; the relation <math>R<math> between possible worlds is taken to be partial order that orders worlds by moral 'virtue:' <math>vRw<math> holds iff <math>w<math> is morally better than <math>v<math>.

The <math>O<math> operator has two duals: <math>P<math> (permission), defined as <math>P\phi\leftrightarrow \lnot O\lnot\phi<math> (similar to <math>\Diamond<math> in other modal systems); and <math>F<math> (forbidden), defined as <math>F\phi\leftrightarrow O\lnot\phi<math>.

There are however some significant differences between alethic and deontic logic. It is accepted (as an axiom) that <math>\Box\phi\to\phi<math>, but obviously it cannot be accepted that <math>O\phi\to\phi<math>. It is also true that <math>\phi\to\Diamond\phi<math>, but it is false that <math>\phi\to P\phi<math>.

This raises the question of how similar are alethic and deontic logics and the question whether deontic logic should actually follow the modal (alethic) model.

Deontic logic faces also the Joergensen's Dilemma. Norms cannot be true or false, but truth and truth values seem essential to logic. There are two possible answers:

• Deontic logic handles norm propositions, not norms;
• There might be alternative concepts to truth, e.g. validity or success.

See also

• Alexius Meinong
• Georg_Henrik_von_Wright
• Deontology
• Norm (philosophy)

External link

• Mally's Deontic Logic (Stanford Encyclopedia of Philosophy) (http://plato.stanford.edu/entries/mally-deontic)