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 Karl Menger 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. .