Deontic logic

The accessibility relation between possible world is interpreted as acceptability relations: ${\displaystyle v}$ is an acceptable world (viz. ${\displaystyle wRv}$) if and only if all the obligations in ${\displaystyle w}$ are fulfilled in ${\displaystyle v}$ (viz. ${\displaystyle (w\models OA)\to (v\models A)}$).

Alan R. Anderson (1959) shows how to define ${\displaystyle O}$ in terms of the alethic operator ${\displaystyle \Box }$ and a deontic constant (i.e. 0-ary modal operator) ${\displaystyle s}$ standing for some sanction (i.e. bad thing, prohibition, etc.): ${\displaystyle OA\equiv \Box (\lnot A\to s)}$. 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 ${\displaystyle \Box }$, Anderson's deontic logic only requires one additional axiom for the deontic constant ${\displaystyle s}$: ${\displaystyle \neg \Box s\equiv \Diamond \neg s}$, 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 (${\displaystyle \Box A\to A}$), it can be proved in Anderson's deontic logic that ${\displaystyle O(OA\to A)}$, which is not included in SDL. Anderson's deontic logic inevitably couples the deontic operator ${\displaystyle O}$ with the alethic operator ${\displaystyle \Box }$, which can be problematic in certain cases.

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), the, 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 ${\displaystyle \cup }$ and ${\displaystyle \cap }$, unary connective ${\displaystyle$ !} , and binary connectives ${\displaystyle f}$ and ${\displaystyle \infty }$.

* Mally read ${\displaystyle !A}$ as "A ought to be the case".
* He read ${\displaystyle AfB}$ as "A requires B" .
* He read ${\displaystyle A\infty B}$ as "A and B require each other."
* He read ${\displaystyle \cup }$ as "the unconditionally obligatory" .
* He read ${\displaystyle \cap }$ as "the unconditionally forbidden".

Mally defined ${\displaystyle f}$, ${\displaystyle \infty }$, and ${\displaystyle \cap }$ as follows:

Def. ${\displaystyle f.AfB=A\rightarrow !B}$
Def. ${\displaystyle \infty .A\infty B=(AfB)\&(BfA)}$
Def. ${\displaystyle \cap .\rightarrow \cap =\lnot \cup }$

Mally proposed five informal principles:

(i) If A requires B and if B requires 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:

I. ${\displaystyle \rightarrow ((AfB)\&(B\rightarrow C))\rightarrow (AfC)}$
II. ${\displaystyle \rightarrow ((AfB)\&(AfC))\rightarrow (Af(B\&C))}$
III. ${\displaystyle \rightarrow (AfB)\leftrightarrow$ !(A\rightarrow B)}
IV. ${\displaystyle \rightarrow \exists \cup$ !\cup }
V. ${\displaystyle \rightarrow \lnot (\cup f\cap )}$

From these axioms Mally deduced 35 theorems, many of which he rightly considered strange. Karl Menger showed that ${\displaystyle !A\leftrightarrow A}$ 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.[4] 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:

• It ought to be that Jones goes (to the assistance of his neighbors).
• It ought to be that if Jones goes, then he tells them he is coming.
• If Jones doesn't go, then he ought not tell them he is coming.
• Jones doesn't go