D Modal Logic

Origin. The D axiom (□p → ◇p) as seriality; von Wright's Deontic Logic (1951) is the source of the reading, Hintikka (1957) of the modal treatment, and Kripke's serial-frame semantics (1963) of the system. D = K + D, and Standard Deontic Logic is its normal extension. Serial frames. Ought implies can. Minimal deontic. Foundation of obligation logic.

Models. Serial accessibility. No dead ends. Obligation/permission. Deontic possibility always exists.

Formalism.

Language: □ as O (obligation). ◇ as P (permission). OA: A is obligatory. PA: A is permitted.

Axioms: K: O(A → B) → (OA → OB). D: OA → PA. Equivalently: ¬O⊥.

D axiom readings: OA → ¬O¬A. Obligation implies permission. Ought implies can. No obligation of impossibilities.

Frame condition: Seriality: ∀w∃v.wRv. Every world accesses some world. No moral dead ends. Ideals always exist.

Contrast with T: T: □A → A (truth). D: □A → ◇A (consistency). D weaker than T. Obligations not always fulfilled.

Extensions: KD4: deontic + transitive. KD45: + Euclidean. Standard Deontic Logic (SDL) variants.

Deontic operators: OA: obligatory. PA := ¬O¬A: permitted. FA := O¬A: forbidden. Optional: PA ∧ P¬A.

Paradoxes: Ross paradox: O(mail) → O(mail ∨ burn). Gentle murder. Good Samaritan. D doesn't solve all.

Symbols.

Symbol	Unicode	Meaning
D	—	seriality axiom
O	—	obligation
P	—	permission
F	—	forbidden

Metatheory. Serial frames. Deontic consistency. Weaker than T. Completeness.

Applies to. Deontic logic. Normative reasoning. Legal logic. Ethics.

Limitations. Paradoxes remain. Conflicting obligations. Contrary-to-duty. Too simple for real norms.