It can be used to formalize imperative logic, or directive modality in natural languages.
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
The term deontic is derived from the Ancient Greek: δέον, romanized: déon (gen.: δέοντος, déontos), meaning "that which is binding or proper."
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
In practical situations, obligations are usually assigned in anticipation of future events, in which case alethic possibilities 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.
Motivation for a conditional operator is given by considering the following ("Good Samaritan") case.
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.
stands for an arbitrary tautology of the underlying logic (which, in the case of SDL, is classical).
Intuitively, the right side of the biconditional says that A's failing to hold necessarily (or strictly) implies a sanction.
, which means that there is alethically possible to fulfill all obligations and avoid the sanction.
An important problem of deontic logic is that of how to properly represent conditional obligations, e.g.
Philosophers from the Indian Mimamsa school to those of Ancient Greece have remarked on the formal logical relations of deontic concepts[1] and philosophers from the late Middle Ages compared deontic concepts with alethic ones.
[2] 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), and the indifferens (facultative) are equivalent to those between the possibile, the impossibile, the necessarium, and the contingens respectively.
[3] 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
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.
[4] After Menger, philosophers no longer considered Mally's system viable.
(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.
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.
(For more on von Wright's departure from and return to the syntax of the propositional calculus, see Deontic Logic: A Personal View[citation needed] and A New System of Deontic Logic[citation needed], 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.
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".
There is no formalisation in von Wright's system of the following claims that allows them to be both jointly satisfiable and logically independent: Several extensions or revisions of Standard Deontic Logic have been proposed over the years, with a view to solve these and other puzzles and paradoxes (such as the Gentle Murderer and Free choice permission).
The following three claims are incompatible: Responses to this problem involve rejecting one of the three premises.