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. Consistency of obligation. 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. No obligation of impossibilities: ¬O⊥. Not "ought implies can" — P is deontic possibility, not ability.

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	Name	Meaning
D	—	Seriality axiom	□A → ◇A, i.e. OA → PA
O	—	Obligation	It is obligatory that
P	—	Permission	It is permitted that
F	—	Forbidden	FA := O¬A
□	U+25A1	Box	Necessity, read as obligation
◇	U+25C7	Diamond	Possibility, read as permission
R	—	Accessibility relation	Serial: no dead ends
⊥	U+22A5	Falsum	D as ¬O⊥
⊤	U+22A4	Verum	O⊤ from necessitation
∀	U+2200	Universal quantifier	Frame condition ∀w∃v wRv
∃	U+2203	Existential quantifier	Frame condition ∀w∃v wRv
¬	U+00AC	Negation	Interdefinition of O, P and F
∧	U+2227	Conjunction	Optionality PA ∧ P¬A
∨	U+2228	Disjunction	Ross's paradox
→	U+2192	Material conditional	Antecedent of K and D
⊊	U+228A	Proper inclusion	K ⊊ KD ⊊ KT

Metatheory. KD is sound and complete for serial frames. D is a Sahlqvist formula, so KD is canonical and seriality is elementary, defined by ∀w∃v wRv, and completeness follows from the canonical model with no separate argument. Filtration gives the finite model property with an exponential bound, so validity is decidable, and satisfiability is PSPACE-complete, as it is for K and T (Ladner, 1977), while S5 falls to NP-complete because a single cluster suffices there. Both inclusions are proper: K ⊊ KD, since K imposes no seriality, and KD ⊊ KT, since reflexivity implies seriality but not conversely — so T proves D and every extension of T contains it. Over K, D is equivalent to ¬O⊥ and to the consistency schema ¬(OA ∧ O¬A); it is exactly what fails on frames with dead ends, where □A holds vacuously for every A. KD is normal, so necessitation yields O⊤.

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

Limitations. D secures consistency and nothing else, and consistency is the smallest of the difficulties. Necessitation makes O⊤ a theorem and the K axiom carries obligation across entailment, so Ross's paradox and the Good Samaritan survive untouched, and Forrester's gentle murder still derives an unconditional obligation to murder. Chisholm's contrary-to-duty set has no KD formalization that is at once consistent and independent, so what one ought to do having already violated an obligation cannot be stated. The axiom is itself a substantive commitment: it makes OA ∧ O¬A unsatisfiable, so moral dilemmas are excluded by the logic rather than argued against. D does not say that ought implies can, since P is deontic possibility and not agent ability, and KD contains no notion of agency, action or time. Obligations attach to propositions and are monotonic — adding premises never withdraws one.