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/B) → O(A/B) (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
≤	U+2264	Preference	Ideality ordering
⊤	U+22A4	Tautology	Unconditional

Metatheory. Various systems (E, F, G). Preference semantics. Avoids paradoxes. Decidable (propositional).

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

Limitations. Multiple competing systems. Semantics debates. Detachment problems. Iteration complex.