Dyadic Deontic Logic

Origin. von Wright (1964), Hansson (1969). Conditional obligations: O(A/B). "A is obligatory given B." Handles contrary-to-duty paradoxes. Foundation for normative reasoning.

Models. Conditional obligation operator. O(A/B): A obligatory in B-contexts. Preference-based semantics. Avoids standard deontic paradoxes.

Formalism.

Dyadic operators: O(A/B): A is obligatory given B P(A/B): A is permitted given B F(A/B): A is forbidden given B

Relationship: P(A/B) ≡ ¬O(¬A/B) F(A/B) ≡ O(¬A/B)

Key axioms (system E): O(A/A) (identity) O(⊤/B) (something obligatory) O(A ∧ B/C) ↔ O(A/C) ∧ O(B/C) (AND) O(A/B ∨ C) → O(A/B) ∨ O(A/C) (OR)

Strengthening: NOT: O(A/B) → O(A/B ∧ C) (fails!) Allows: "normally obligatory, not in exception"

Contrary-to-duty: Standard deontic: O(¬steal), O(apologize|steal) conflicts. Dyadic: O(¬steal/⊤), O(apologize/steal) consistent.

Preference semantics: Worlds ordered by ideality. O(A/B): best B-worlds are A-worlds. ≤_B: preference among B-worlds.

Chisholm paradox solution: O(help/⊤): ought to help O(tell/help): if help, ought to tell O(¬tell/¬help): if not help, ought not tell ¬help: factual All consistent in dyadic system.

Symbols.

Symbol	Unicode	Name	Meaning
O(A/B)	—	Conditional ought	A obligatory given B
P(A/B)	—	Conditional permit	A permitted given B
F(A/B)	—	Conditional prohibition	A forbidden given B
≤	U+2264	Preference	Ideality ordering
≤_B	—	Restricted preference	Ordering among B-worlds
best(B)	—	Optimal set	Best B-worlds; O(A/B) iff best(B) ⊆ A
⊤	U+22A4	Verum	Unconditional case O(A/⊤)
⊥	U+22A5	Falsum	Vacuity condition ¬O(⊥/A)
⊆	U+2286	Inclusion	Truth condition for O(A/B)
≡	U+2261	Definitional equivalence	P(A/B) ≡ ¬O(¬A/B)
¬	U+00AC	Negation	Interdefinition of O, P and F
∧	U+2227	Conjunction	AND axiom; strengthened antecedent
∨	U+2228	Disjunction	OR axiom on antecedents
→	U+2192	Material conditional	Form of the axioms
↔	U+2194	Biconditional	AND axiom stated as an equivalence

Metatheory. The axioms above are those of a preferential consequence relation in the sense of Kraus, Lehmann and Magidor, read with the antecedent as premise: AND and OR hold, strengthening of the antecedent fails outright, and the restricted forms survive — cautious monotony, since best(B) ⊆ C forces best(B ∧ C) = best(B), and, when the betterness relation is total, rational monotony. The semantics is Hansson's (1969) and Lewis's (1973): O(A/B) holds when the best B-worlds are all A-worlds, over a preorder on worlds, with a limit assumption in the simplest systems to guarantee that best B-worlds exist. Spohn (1975) proved Hansson's DSDL3 complete for total preorders satisfying that assumption. Åqvist's E, F and G stratify the same idea, G being the strongest and corresponding to models with the limit assumption; completeness of E and F remained open until Parent (2015) settled it. Chisholm's four sentences are jointly satisfiable in these systems, and none follows from the others.

Applies to. Legal reasoning. AI ethics. Normative systems. Contract law. Exception handling.

Limitations. Consistency comes from surrendering factual detachment: O(A/B) together with B does not yield O(A), so the secondary duty is representable but never discharged into anything to do, and restoring unrestricted detachment restores Chisholm's inconsistency. Deontic detachment runs the other way and can license acting on a primary duty already violated. Prakken and Sergot's pragmatic oddity survives, since the violated primary obligation and the reparational one agglomerate into an obligation no one owes. The betterness ordering is primitive and fixed outside the logic, so which sub-ideal worlds count as best is stipulated. The limit assumption fails on infinite orderings without a smoothness condition, and dropping it loses the simplest completeness proofs. Nesting O inside O has no settled reading, and obligations still attach to propositions: no agent, no action, no time.