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# Deontic logic

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 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, 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 ∞.
* Mally read !A as "A ought to be the case".
* He read A f B as "A requires B" .
* He read A ∞ B as "A and B require each other."
* He read U as "the unconditionally obligatory" .
* He read ∩ as "the unconditionally forbidden".
Mally defined f, ∞, and ∩ as follows:
Def. f. A f B = A → !B
Def. ∞. A ∞ B = (A f B) & (B f A)
Def. ∩. ∩ = ¬U
Mally proposed five informal principles:
(i) If A requires B and if B then 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. ((A f B) & (B → C)) → (A f C)
II. ((A f B) & (A f C)) → (A f (B & C))
III. (A f B) ↔ !(A → B)
IV. ∃U !U
V. ¬(U f ∩)

## Standard deontic logic

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-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:

$O\left(A rightarrow B\right) rightarrow \left(OA rightarrow OB\right)$
$OAtolnot Olnot A$

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 (i.e. not obligatory that not) A.

FA, meaning it is forbidden that A, can be defined (equivalently) as $O lnot A$ or $lnot PA$.

There are two main extensions of SDL that are usually considered. The first results by adding an alethic modal operator $Box$ in order to express the Kantian claim that "ought implies can":

$OA to Diamond A.$

where $DiamondequivlnotBoxlnot$. It is generally assumed that $Box$ 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:

$vdash Ato BRightarrow vdash OAto OB.$

If we introduce an intensional conditional operator then we can say that the starving ought to 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 $OAequiv O\left(A/top\right)$, where $top$ 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 $Box$ and a deontic constant (i.e. 0-ary modal operator) s standing for some sanction (i.e. bad thing, prohibition, etc.): $OAequivBox\left(lnot Ato s\right)$. Intuitively, the right side of the biconditional says that A's failing to hold necessarily (or strictly) implies a sanction.

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:

$O\left(mathrm\left\{smoke\right\} rightarrow mathrm\left\{ashtray\right\}\right)$
$mathrm\left\{smoke\right\} rightarrow O\left(mathrm\left\{ashtray\right\}\right)$

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:

$O\left(A mid B\right)$ means it is obligatory that A, given B
$P\left(A mid B\right)$ 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

Many other varieties of deontic logic have been developed, including non-monotonic deontic logics, paraconsistent deontic logics, and dynamic deontic logics.

## Jørgensen's Dilemma

Deontic logic faces Jørgensen's Dilemma. Norms cannot be true or false, but truth and truth values seem essential to logic. There are two possible answers:

• Deontic logic handles norm propositions, not norms;
• There might be alternative concepts to truth, e.g. validity or success, as it is defined in speech act theory.

## 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.