Deontic logic

Deontic logic is the field of philosophical 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".
Etymology.
The term "deontic" is derived from the Ancient Greek δέον "déon" (gen.: δέοντος "déontos"), meaning "that which is binding or proper."
Standard deontic logic.
In Georg Henrik von Wright's first system, obligatoriness and permissibility were treated as features of "acts". Soon after this, it was found 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:
In English, these axioms say, respectively:
"FA", meaning it is forbidden that "A", can be defined (equivalently) as formula_3 or formula_4.
There are two main extensions of SDL that are usually considered. The first results by adding an alethic modal operator formula_5 in order to express the Kantian claim that "ought implies can":
where formula_7. It is generally assumed that formula_5 is at least a KT operator, but most commonly it is taken to be an S5 operator.
The other main extension results by adding a "conditional obligation" operator O(A/B) read "It is obligatory that A given (or conditional on) B". Motivation for a conditional operator is given by considering the following ("Good Samaritan") case. It seems true that the starving and poor ought to be fed. But that the starving and poor are fed implies that there are starving and poor. By basic principles of SDL we can infer that there ought to be starving and poor! The argument is due to the basic K axiom of SDL together with the following principle valid in any normal modal logic:
If we introduce an intensional conditional operator then we can say that the starving ought to be fed "only on the condition that there are in fact starving": in symbols O(A/B). But then the following argument fails on the usual (e.g. Lewis 73) semantics for conditionals: from O(A/B) and that A implies B, infer OB.
Indeed, one might define the unary operator O in terms of the binary conditional one O(A/B) as formula_10, where formula_11 stands for an arbitrary tautology of the underlying logic (which, in the case of SDL, is classical). Similarly Alan R. Anderson (1959) shows how to define O in terms of the alethic operator formula_5 and a deontic constant (i.e. 0-ary modal operator) s standing for some sanction (i.e. bad thing, prohibition, etc.): formula_13. Intuitively, the right side of the biconditional says that A's failing to hold necessarily (or strictly) implies a sanction.
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:
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 (1) "if you murder, you ought to murder gently", (2) "you do commit murder", and (3) "to murder gently you must murder" imply the less plausible statement: "you ought to murder". Others argue that "must" in the phrase "to murder gently you must murder" is a mistranslation from the ambiguous English word (meaning either "implies" or "ought"). Interpreting "must" as "implies" does not allow one to conclude "you ought to murder" but only a repetition of the given "you murder". Misinterpreting "must" as "ought" results in a perverse axiom, not a perverse logic. With use of negations one can easily check if the ambiguous word was mistranslated by considering which of the following two English statements is equivalent with the statement "to murder gently you must murder": is it equivalent to "if you murder gently it is forbidden not to murder" or "if you murder gently it is impossible not to murder" ?
Some deontic logicians have responded to this problem by developing dyadic deontic logics, which contain binary deontic operators:
(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.
History.
Early deontic logic.
Philosophers from the Indian Mimamsa school to those of Ancient Greece have remarked on the formal logical relations of deontic concepts and philosophers from the late Middle Ages compared deontic concepts with alethic ones.
In his "Elementa juris naturalis" (written between 1669 and 1671), Gottfried Wilhelm Leibniz notes the logical relations between the "licitum" (permitted), the "illicitum" (prohibited), the "debitum" (obligatory), the, and the "indifferens" (facultative) are equivalent to those between the "possibile", the "impossibile", the "necessarium", and the "contingens" respectively.
Mally's first deontic logic and von Wright's first "plausible" 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" (1926) 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 defined f, ∞, and ∩ as follows:
Mally proposed five informal principles:
He formalized these principles and took them as his axioms:
From these axioms Mally deduced 35 theorems, many of which he rightly considered strange. Karl Menger showed that !A ↔ A is a theorem and thus that the introduction of the ! sign is irrelevant and that A ought to be the case if A is the case. After Menger, philosophers no longer considered Mally's system viable. Gert Lokhorst lists Mally's 35 theorems and gives a proof for Menger's theorem at the Stanford Encyclopedia of Philosophy under "Mally's Deontic Logic".
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 use the term "deontic" in English to refer to this kind of logic although Mally published the German paper "Deontik" in 1926.) 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.
G. H. von Wright did not base his 1951 deontic logic on the syntax of the propositional calculus as Mally had done, but was instead influenced by alethic modal logics, which Mally had not benefited from. In 1964, von Wright published "A New System of Deontic Logic", which was a return to the syntax of the propositional calculus and thus a significant return to Mally's system. (For more on von Wright's departure from and return to the syntax of the propositional calculus, see "Deontic Logic: A Personal View" and "A New System of Deontic Logic", both by Georg Henrik von Wright.) G. H. von Wright's adoption of the modal logic of possibility and necessity for the purposes of normative reasoning was a return to Leibniz.
Although von Wright's system represented a significant improvement over Mally's, it raised a number of problems of its own. For example, "Ross's paradox" applies to von Wright's deontic logic, allowing us to infer from "It is obligatory that the letter is mailed" to "It is obligatory that either the letter is mailed or the letter is burned", which seems to imply it is permissible that the letter is burned. The "Good Samaritan paradox" also applies to his system, allowing us to infer from "It is obligatory to nurse the man who has been robbed" that "It is obligatory that the man has been robbed". Another major source of puzzlement is "Chisholm's paradox". There is no formalisation in von Wright's system of the following claims that allows them to be both jointly satisfiable and logically independent:
Jørgensen's dilemma.
Deontic logic faces Jørgensen's dilemma.
This problem is best seen as a trilemma. 
The following three claims are incompatible:
Responses to this problem involve rejecting one of the three premises.
The input/output logics reject the first premise. 
They provide inference mechanism on elements without presupposing that these elements have truth-values. 
Alternatively, one can deny the second premise. One way to do this is to distinguish between the norm itself and a proposition about the norm.
According to this response, only the proposition about the norm has a truth-value. 
Finally, one can deny the third premise. But this is to deny that there is a logic of norms worth investigating.