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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.

Contents 1 History of Deontic Logic 1.1 Pre-History of Deontic Logic 1.2 Mally's First Deontic Logic and Georg Henrick von Wright's First Plausible Deontic Logic 2 Standard Deontic Logic 2.1 Problems for SDL 2.1.1 Joergensen's Dilemma 2.1.2 Ross's Paradoxes 2.1.3 The Logical Necessity of Obligations Problem 2.1.4 The Good Samaritan Paradox 2.1.5 Satre's Dilemma and Plato's Dilemma 2.1.6 Contrary-to-Duty (or Chisholm's) Paradox 2.2 Notes

[edit] History of Deontic Logic

[edit] Pre-History of Deontic Logic

Philosophers from the Mimansa school of India 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] Leibniz was the first to attempt systematizing normative reasoning according to alethic modalities. 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.

[edit] Mally's First Deontic Logic and Georg Henrick 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 did so along the lines of Whitehead's and Russell's propositional calculus.^[3] 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 ∩)

From these axioms Mally deduced 35 theorems, many of which he considered strange but acceptable. 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 iff 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.^[5] (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.^[6]) Von Wright's logic of 1951 was a return to the comparison of Leibniz between deontic concepts and those of necessity and possibility. It was also a departure from the syntax of the sentential calculus, which Mally used. The ought and should of von Wright in 1951 was the ought and should of actions, not situations. 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.

In 1964, von Wright published A New System of Deontic Logic,^[7] which was return to the syntax of the propositional calculus that was in accordance with the general opinion of the time. For more on von Wright's departure and return to the syntax of the propositional calculus, see Deontic Logic: A Personal View^[8] and A New System of Deontic Logic,^[7] both by Georg Henrick von Wright.

[edit] Standard Deontic Logic

In von Wright's first system, obligatoriness (O) and permissibility (P) 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 as follows:

A0. All tautologies over SDL
A1. O(A → B) → (OA → OB)
A2. OA → ~O~A
R1. If ⊢ A and ⊢ A → B then ⊢ B
R2. If ⊢ A then ⊢ OA

A1 says that if it ought to be the case that A implies B, then if it ought to be the case that A, then it ought to be the case that B. A2 says that if it ought to be the case that A, then A is permissible

[edit] Problems for SDL

[edit] Joergensen's Dilemma

Deontic logic is the logic of norms, the logical relations between what is permissible and obligatory. A popular opinion of late is that normative judgmets are not descriptive and thus cannot be true or false. The question arises, how can we have a logic of statements that are neither true nor false.

One solution is that (i) norms are fiated. E.g. some authority asserts OA, and that evaluative judgments is considered stipulated. And (ii) once stipulated, normative statements are just descritions of those stipulations. Thus, normative statents are factual because they are descriptions of what has been stipulated. There remains disagreement about whether the normative statements of (i) can be factual. One camp supposes that because the statement is a stipulation, it is not a description of anything and thus cannot be true or false. The opposing camp believes the statement creates the very situation it purports to describe.^[9]

[edit] Ross's Paradoxes

The following two oddities were proposed by Ross.^[10] The first is Ross's paradox:

(1) OA
(2) O(A ∨ B)

2 follows from 1 by the rules of the predicate calculus and R2. 1 could be interpreted as, saving kittens is obligated. 2 could be inerpreted, saving kittens or killing them is obligated. The second is The Free Choice Permission Paradox:

(1) P(A ∨ B)
(2) PA & PB

P(A ∨ B) → (PA & PB) is not a theorem of SDL. 1 might be interpreted as, It is permissible to save kittens or pet them. 2 could be interpreted, it is permissible to save kittens and it is permissible to pet them. Both of these oddities spring from the counterintuitive element of logic and it is not clear that what they really say is counterintuitive.

[edit] The Logical Necessity of Obligations Problem

From R2, we know that if ⊢ p, then ⊢ Op. This ensures that something is obligatory, even if only the logical truths, which is strange. If there was a world without rational agents, there would still be obligations.

[edit] The Good Samaritan Paradox

Prior suggested the paradox in 1958. OB(A & B) → OBB follows from the rules of the predicate calculus and R2. So if A and B are obligated then B is obligatory. The difficulty of Prior's paradox is more obvious with an interpretation. Suppose it ought to be the case that people on a plane stop a terrorist who is hijacking the train. Plausibly, OB(people stop the hijacker & the hijacker hijacks the plane). But it follows from this that OB(the hijacker hijacks the plane. Certainly it is not obligatory that the hijacker hijacks the plane. A much discussed variant of this paradox is The Paradox of Epistemic Obligation.^[11]

[edit] Satre's Dilemma and Plato's Dilemma

Satre's Dilemma is that in deontic logic, two contradictory duties when formalized become logically contradictory. Plato's Dilemma is that there may be two obligations we cannot satisfy, say taking a child to school every day and curing cancer. In deontic logic, these actions are mutually exclusive and result in a contradiction. However, in ordinary life, we have ways of choosing one action over another.

[edit] Contrary-to-Duty (or Chisholm's) Paradox

Chisholm's Paradox^[12] is the most important of these and has spurred the most controversy and secured the place of deontic as a non-normal modal logic. consider the following four cases:

(1) It ought to be that A. OA.
(2) It ought to be that if A, then B. O(A → B).
(3) If not A, then not B. ¬A → O¬B.
(4) Jones doesn't go. ¬A.

From (2) by A1, we get OA → OB.We get OB by modus ponens from (1). and then from (1) by MP, we get OB; but by MP alone we get O¬b from (3) and (4). By the rules of sentential logic, we get ¬(OB → ~O~B), contradicting R2. So, 1-4 seem consistent but according to the laws of PDL, they are not.

[edit] Notes

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