Deontic logic
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'''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 ==
=== Pre-History of Deontic Logic ===
Philosophers from the [[India]]n [[Mimamsa|Mimamsa school]] to those of [[Ancient Greece]] have remarked on the formal logical relations of deontic concepts<ref name=Greece>Huisjes, C. H., 1981, "Norms and logic," Thesis, University of Groningen.</ref> and philosophers from the late [[Medieval philosophy|Middle Ages]] compared deontic concepts with alethic ones.<ref name=ones>Knuuttila, Simo, 1981, “The Emergence of Deontic Logic in the Fourteenth Century,” in New Studies in Deontic Logic, Ed. Hilpinen, Risto, pp. 225-248, University of Turku, Turku, Finland: D. Reidel Publishing Company.</ref> 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.
=== 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'' 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 ∞.<br />
: * Mally read !A as "A ought to be the case".<br />* He read A f B as "A requires B" .<br />* He read A ∞ B as "A and B require each other."<br />* He read U as "the unconditionally obligatory" .<br />* He read ∩ as "the unconditionally forbidden".
Mally defined f, ∞, and ∩ as follows:
: Def. f. A f B = A → !B<br />Def. ∞. A ∞ B = (A f B) & (B f A)<br />Def. ∩. ∩ = ¬U
Mally proposed five informal principles:
: (i) If A requires B and if B then C, then A requires C.<br />(ii) If A requires B and if A requires C, then A requires B and C.<br />(iii) A requires B if and only if it is obligatory that if A then B.<br />(iv) The unconditionally obligatory is obligatory.<br />(v) The unconditionally obligatory does not require its own negation.
He formalized these principles and took them as his axioms:
: I. ((A f B) & (B → C)) → (A f C)<br />II. ((A f B) & (A f C)) → (A f (B & C))<br />III. (A f B) ↔ !(A → B)<br />IV. ∃U !U<br />V. ¬(U f ∩)
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.<ref name=A>Menger, Karl, 1939, "A logic of the doubtful: On optative and imperative logic," in Reports of a Mathematical Colloquium, 2nd series, 2nd issue, pp. 53-64, Notre Dame, Indiana: Indiana University Press.</ref> 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 [http://plato.stanford.edu/ Stanford Encyclopedia of Philosophy] under [http://plato.stanford.edu/entries/mally-deontic/ ''Mally's Deontic Logic''].
The first plausible system of deontic logic was proposed by [[Georg Henrik von Wright|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 logic]]s, 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 significantly 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.
== Standard deontic logic ==
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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 semantics|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:
: <math>O(A \rightarrow B) \rightarrow (OA \rightarrow OB)</math>
: <math>OA \rightarrow PA</math>
In English, these axioms say, respectively:
* 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.
''FA'', meaning it is forbidden that ''A'', can be defined (equivalently) as <math>O \lnot A</math> or <math>\lnot PA</math>.
The propositional system '''D''' can be extended to include [[quantifiers]] in a relatively straightforward way.
== Dyadic deontic logic ==
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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:
: <math>O(\mathrm{smoke} \rightarrow \mathrm{ashtray})</math>
: <math>\mathrm{smoke} \rightarrow O(\mathrm{ashtray})</math>
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:
: <math>O(A \mid B)</math> means ''it is obligatory that A, given B''
: <math>P(A \mid B)</math> 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.
== Other variations ==
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Many other varieties of deontic logic have been developed, including [[Non-monotonic logic|non-monotonic]] deontic logics, [[Paraconsistent logic|paraconsistent]] deontic logics, and [[Dynamic logic|dynamic]] deontic logics.
== Jørgensen's Dilemma ==
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Deontic logic faces Jørgensen's Dilemma. [[norm (philosophy)|Norms]] cannot be true or false, but truth and [[Logical value|truth values]] seem essential to logic. There are two possible answers:
* Deontic logic handles norm [[proposition]]s, not norms;
* There might be alternative concepts to truth, ''e.g.'' [[validity]] or success, as it is defined in [[speech act]] theory.
==See also==
* [[Modal logic]]
* [[Imperative logic]]
* [[Norm (philosophy)]]
==Notes==
{{reflist}}
== Resources ==
* [[Lennart Åqvist]], 1994, "Deontic Logic" in D. Gabbay and F. Guenthner, ed., ''Handbook of Philosophical Logic: Volume II Extensions of Classical Logic''. Kluwer.
*Hilpinen, Risto, 2001, "Deontic Logic," in Goble, Lou, ed., ''The Blackwell Guide to Philosophical Logic''. Blackwell.
* [[G. H. von Wright]], 1951. "Deontic logic," ''Mind 60'': 1-15.
==External links==
*McNamara, [http://plato.stanford.edu/entries/logic-deontic/ Deontic Logic], [[Stanford Encyclopedia of Philosophy]], 2006.
*Lokhorst, [http://plato.stanford.edu/entries/mally-deontic Mally's Deontic Logic], Stanford Encyclopedia of Philosophy, 2004.
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{{Logic}}
[[Category:Modal logic]]
[[es:Lógica deóntica]]
[[fr:Logique déontique]]
[[it:Logica deontica]]
[[ja:義務論理]]
[[ru:Деонтическая логика]]
[[sk:Deontická logika]]
[[fi:Deonttinen logiikka]]