## Monday, March 24, 2008

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
Philosophers from the Indian Mimamsa school to those of Ancient Greece have remarked on the formal logical relations of deontic concepts In his Elementa juris naturalis, Leibniz notes the logical relations between the licitum, illicitum, debitum, and indifferens are equivalent to those between the possible, impossible, necessarium, and contingens respectively.
Pre-History of Deontic Logic
Ernst Mally, a pupil of Alexius Meinong, was the first to propose a formal system of deontic logic in his Grundgesetze des Sollens and he founded it on the syntax of Whitehead's and Russell's propositional calculus. Mally's deontic vocabulary consisted of the logical constants U and ∩, unary connective !, and binary connectives f and ∞. * Mally read !A as "A ought to be the case". * He read A f B as "A requires B" . * He read A ∞ B as "A and B require each other." * He read U as "the unconditionally obligatory" . * He read ∩ as "the unconditionally forbidden". Mally defined f, ∞, and ∩ as follows:
Def. f. A f B = A → !B Def. ∞. A ∞ B = (A f B) & (B f A) Def. ∩. ∩ = ¬U Mally proposed five informal principles:
(i) If A requires B and if B then C, then A requires C. (ii) If A requires B and if A requires C, then A requires B and C. (iii) A requires B if and only if it is obligatory that if A then B. (iv) The unconditionally obligatory is obligatory. (v) The unconditionally obligatory does not require its own negation. He formalized these principles and took them as his axioms:

Mally's First Deontic Logic and von Wright's First Plausible Deontic Logic
In von Wright's first system, obligatoriness and permissibility were treated as features of acts. It was found not much later that a deontic logic of propositions could be given a simple and elegant Kripke-style semantics, and von Wright himself joined this movement. 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:
$O(A rightarrow B) rightarrow (OA rightarrow OB)$
$OA rightarrow PA$
In English, these axioms say, respectively:
FA, meaning it is forbidden that A, can be defined (equivalently) as $O lnot A$ or $lnot PA$.
The propositional system D can be extended to include quantifiers in a relatively straightforward way.

If it ought to be that A implies B, then if it ought to be that A, it ought to be that B;
If it ought to be that A, then it is permissible that A. Standard 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:
$O(mathrm{smoke} rightarrow mathrm{ashtray})$
$mathrm{smoke} rightarrow O(mathrm{ashtray})$
Under the first representation it is vacuously true that if you commit a forbidden act, then you ought to commit any other act, regardless of whether that second act was obligatory, permitted or forbidden (Von Wright 1956, cited in Aqvist 1994). Under the second representation, we are vulnerable to the gentle murder paradox, where the plausible statements if you murder, you ought to murder gently, you do commit murder and to murder gently you must murder imply the less plausible statement: you ought to murder.
Some deontic logicians have responded to this problem by developing dyadic deontic logics, which contain a binary deontic operators:
$O(A mid B)$ means it is obligatory that A, given B
$P(A mid B)$ 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.