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. It can be used to formalize imperative logic, or directive modality in natural languages. 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", which is defined as formula_1. Note that in natural language, the statement "You may go  to the zoo OR the park" should be understood as formula_2 instead of formula_3, as both options are permitted by the statement; See Hans Kamp's paradox of free choice for more details. When there are multiple agents involved in the domain of discourse, the deontic modal operator can be specified to each agent to express their individual obligations and permissions. For example, by using a subscript formula_4 for agent formula_5, formula_6 means that "It is an obligation for agent formula_5 (to bring it about/make it happen) that formula_8". Note that formula_8 could be stated as an action by  another agent; One example is "It is an obligation for Adam that Bob doesn't crash the car", which would be represented as formula_10, where B="Bob doesn't crash the car". Etymology. The term "deontic" is derived from the (gen.: ), 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_14 or formula_15. There are two main extensions of SDL that are usually considered. The first results by adding an alethic modal operator formula_16 in order to express the Kantian claim that "ought implies can": where formula_18. It is generally assumed that formula_16 is at least a  KT operator, but most commonly it is taken to be an S5 operator. In practical situations, obligations are usually assigned in anticipation of future events, in which case alethic possiblities can be hard to judge; Therefore, obligation assignments may be performed under the assumption of different conditions on different branches of timelines in the future, and past obligation assignments may be updated due to unforeseen developments that happened along the timeline. 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_21, where formula_22 stands for an arbitrary tautology of the underlying logic (which, in the case of SDL, is classical). Semantics of standard deontic logic. The accessiblity relation between possible world is interpreted as "acceptibility" relations: formula_23 is an acceptable world (viz. formula_24) if and only if all the obligations in  formula_25 are fulfilled in formula_23 (viz. formula_27). Anderson's deontic logic. Alan R. Anderson (1959) shows how to define formula_28 in terms of the alethic operator formula_16 and a deontic constant (i.e. 0-ary modal operator) formula_30 standing for some sanction (i.e. bad thing, prohibition, etc.): formula_31. 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 formula_16, Anderson's deontic logic only requires one additional axiom for the deontic constant formula_30 : formula_34, 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 (formula_35), it can be proved in Anderson's deontic logic that formula_36, which is not included in SDL. Anderson's deontic logic inevitably couples the deontic operator formula_28 with the alethic operator formula_16, which can be problematic in certain cases. 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.