Deontic Logic
Mathematical Logic Spring 2012 Kelly Moser
Deontic Logic
Deals with:
Obligation
What we ought to do Permission
What we are allowed to do Forbiddance
What we must not do
History
From Ancient Greek
”That which is binding or proper” Philosophers
Greece, India Middle Ages Ernst Mally, an Austrian
1926 First formal system Syntax based on propositional calculus
History: Mally's Deontic Logic
((A f B) & (B→C))→(A f C)
If A requires B and if B requires C, then A requires C ((A f B) & (A f C))→(A f (B&C))
If A requires B and if A requires C, then A requires B and C (A f B) ↔!(A→B)
A requires B if and only if it is obligatory that if A then B There exists U !U
The unconditionally obligatory is obligatory ¬(U f ∩)
The unconditionally obligatory does not require its own negation
History: Mally's Deontic Logic
Proof: !A ↔A A ought to be the case iff A is the case ! is irrelevant Lead to downfall of Mally's system New System
G.H. VonWright in 1951 First to use term ”deontic” Switched syntax in 1964
Deontic Logic
Forbidden: not permitted
Theft is not permitted Theft is forbidden Obligation: negation of the act is forbidden
It is forbidden to disobey the law It is obligatory to obey the law We ought to do that which we are not allowed not to do
Deontic Logic
(Morally) indifferent
An act and its negation are both permitted People over 21: allowed to drink, also allowed to not drink (Morally) incompatible
Conjuction of the two acts is forbidden Giving a promise AND not keeping it
SDL: Syntax Used
P(A)
Act A is permitted O(A)
Act A is obligatory Equivalent to ¬(P(¬A)) F(A) (sometimes used)
Act A is forbidden Equivalent to O(¬A) or ¬P(A)
SDL: Syntax Used
(P(A)) & (P(¬A))
Act A is (morally) indifferent ¬P(A&B)
Acts A and B are (morally) incompatible O(A→B)
Performance of A commits us to perform B
SDL: Axioms
O(A→B)→(OA→OB)
If it ought to be that A implies B, then if it ought to be that A, it ought to be that B PA→¬O¬A
If A is permissible, then it is not the case that it ought not to be that A
SDL: Axioms
O(A→B)→(OA→OB)
If it ought to be that A implies B, then if it ought to be that A, it ought to be that B PA→¬O¬A
If A is permissible, then it is not the case that it ought not to be that A
O(OA → A)
It is obligatory that obligations be fulfilled
Extensions
Andersonian-Kangerian reduction
□ necessary ◊ possible □A→A A→◊A “d” for “all (relevant) normative demands are met” O(A) =□(d → A) P(A) =◊(d & A) Conditional Obligation
O(A/B)
It is obligatory that A given B Issues
Are deontic propositions ”relative” to a moral code?
What is obligatory within one moral code may be forbidden in another What if x permits y to do A?
Adds complexity Logical Necessity of Obligations
Nothing is obligatory SDL gives a contradiction
Issues
Free Choice Permission Paradox
(1) You may either sleep on the sofa-bed or sleep on the guest room bed.
P(S^G) (2) You may sleep on the sofa-bed and you may sleep on the guest room bed.
P(S) & P(G)
Issues
Free Choice Permission Paradox
(1) You may either sleep on the sofa-bed or sleep on the guest room bed.
P(S^G) (2) You may sleep on the sofa-bed and you may sleep on the guest room bed.
P(S) & P(G) Does P(S^G)→(P(S) & P(G)) ? If so, if something is permissible, everything is
Issues
Conflict of Obligations
(1) It is obligatory that I now meet Joe (because I promised by friend Rebecca that I would do so). (2) It is obligatory that I now do not meet Joe (because I promised my friend Sally that I would not). What do I do?
Issues
Plato's Dilemma
(1) I'm obligated to meet you for a light lunch at noon. (2) I'm obligated to rush my choking child to the hospital at noon.
Of course, (2) takes precedence over (1) How to know which obligations override which others?
Issues
The Must versus Ought Dilemma
(1) Although you can skip the meeting, you ought to attend.
Urmson's Puzzle—Indifference versus Optionality
(1) It is optional that you attend the meeting, but not a matter of indifference that you do so.
Issues
Gentle Murder Paradox
(1) if you murder, you ought to murder gently, (2) you do commit murder (3) to murder gently you must murder
Together, these imply: you ought to murder
Resources & Further Reading
AQVIST, L. Introduction to Deontic Logic and the Theory of Normative Systems. Bibliopolis, 1987. BROWN, J. ”Moral Theory and the Ought-Can Principle” Mind, Vol. 86, No. 342, p. 206-223. GOLDMAN, H. ”David Lewis's Semantics for Deontic Logic” Mind, Vol. 86, No. 342, p. 242-248. JACKSON, F. ”On the Semantics and Logic of Obligation” Mind, Vol. 94, No. 374, p. 177-195. Stanford Encylopedia of Philosophy [Online] VonWRIGHT, G.H. ”Deontic Logic” Mind, Vol. 60, No. 237, p. 1-15.