Deontic logic is the field of logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts. Typically, a deontic logic uses OA to mean it is obligatory that A, (or it ought to be (the case) that A), and PA to mean it is permitted (or permissible) that A. The term deontic is derived from the ancient Greek déon, meaning, roughly, that which is binding or proper.

History

The central intuition behind deontic logic, first articulated by the German philosopher and mathematician Gottfried Leibniz in the 17th century, is that what is obligatory is what is necessary for a good person to do, and what is permitted is what is possible for a good person to do. Thus, the logic of obligation and permission is analogous to the logic of necessity and possibility.

The first formal system of deontic logic was proposed by Ernst Mally in the 1920s. While Mally's intentions were good, it turned out that his system entailed that a proposition is true just in case it is obligatory, or in symbols:

$OA\; \backslash leftrightarrow\; A$.

This is, of course, extremely counterintuitive.

The first plausible system of deontic logic was proposed by G. H. von Wright in his paper "Deontic Logic" in the philosophical journal Mind in 1951. (Von Wright was also the first to actually use the term "deontic" to refer to this kind of logic.) Since the publication of von Wright's seminal paper, many philosophers and computer scientists have investigated and developed systems of deontic logic. Nevertheless, to this day deontic logic remains one of the most controversial and least agreed-upon areas of logic.

Standard deontic logic

In von Wright's first system, obligatoriness and permissibility were treated as features of acts. It was later found that a deontic logic of propositions could be given a simple and elegant Kripke-style semantics. The deontic logic so specified came to be known as "standard deontic logic," often referred to as SDL, KD, or simply D. It can be axiomatized by adding the following axioms to a standard axiomatization of classical propositional logic:

$O(A\; \backslash rightarrow\; B)\; \backslash rightarrow\; (OA\; \backslash rightarrow\; OB)$

$OA\; \backslash rightarrow\; PA$

In English, these axioms say, respectively:

• If it ought to be that A implies B, and it ought to be that A, then it ought to be that B;
• If it ought to be that A, then it is permissible that A.

FA, meaning it is forbidden that A, can be defined (equivalently) as $O\; \backslash lnot\; A$ or $\backslash lnot\; PA$.

The propositional system D can be extended to include quantifiers in a relatively straightforward way.

Dyadic deontic logic

An important problem of deontic logic is that of how to properly represent conditional obligations, e.g. If you smoke (s), then you ought to use an ashtray (a). It is not clear that either of the following representations is adequate:

$O(smoke\; \backslash rightarrow\; ashtray)$

$smoke\; \backslash rightarrow\; Oashtray$

Some deontic logicians have responded to this problem by developing dyadic deontic logics, which contain a binary deontic operators:

O(A/B) means it is obligatory that A, given B

P(A/B) means it is permissible that A, given B.

(The notation is modeled on that used to represent conditional probability.) Dyadic deontic logic escapes some of the problems of standard (unary) deontic logic, but it is subject to some problems of its own.

Other variations

Many other varieties of deontic logic have been developed, including non-monotonic deontic logics, paraconsistent deontic logics, and dynamic deontic logics.

Joergensen's Dilemma

Deontic logic faces 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

• Modal logic
• Norm (philosophy)

Resources

• Hilpinen, R. 2001. Deontic logic. In L. Goble, ed., The Blackwell Guide to Philosophical Logic. Oxford: Blackwell, 2001.
• von Wright, G. H. 1951. Deontic logic. Mind 60: 1-15.

External links

• McNamara, Deontic Logic, Stanford Encyclopedia of Philosophy, 2006.
• Lokhorst, Mally's Deontic Logic, Stanford Encyclopedia of Philosophy, 2004.

Modal logic

Deontische Logik | Lógica deóntica | Logique déontique | Logica deontica