Standard Deontic Logic

Origin. von Wright (1951). Obligation and permission. Analogous to alethic modal. KD system. Foundation for normative reasoning.

Models. Kripke frames with seriality. Accessible worlds are "deontically perfect." Obligation = necessity, permission = possibility. Ideal world semantics.

Formalism.

Basic operators: Oφ: it is obligatory that φ Pφ: it is permitted that φ Fφ: it is forbidden that φ

Definitions: Pφ ≡ ¬O¬φ Fφ ≡ O¬φ ≡ ¬Pφ

Axioms (KD): K: O(φ → ψ) → (Oφ → Oψ) D: Oφ → Pφ (obligations are consistent — not "ought implies can")

D axiom: ¬(Oφ ∧ O¬φ) Obligations consistent. No dilemmas (controversial).

Seriality: ∀w∃v. wRv Always an ideal accessible.

Semantics: w ⊨ Oφ iff ∀v. wRv → v ⊨ φ Obligatory in all ideal worlds. w ⊨ Pφ iff ∃v. wRv ∧ v ⊨ φ Permitted in some ideal world.

Derived principles: O(φ ∧ ψ) ↔ Oφ ∧ Oψ Oφ → O(φ ∨ ψ) (Ross's paradox!) O⊤ (trivially obligatory)

Problems: Ross's paradox: O(mail) → O(mail ∨ burn) Contrary-to-duty: what if obligation violated? Good Samaritan: O(help if robbed)

Symbols.

Symbol	Unicode	Name	Meaning
O	—	Obligation	Ought
P	—	Permission	May
F	—	Forbidden	Must not
D	—	Seriality	Consistency
K	—	Distribution axiom	O(φ → ψ) → (Oφ → Oψ)
R	—	Accessibility relation	Serial: every world sees a deontic alternative
V	—	Violation constant	Anderson's sanction proposition
□	U+25A1	Box	Alethic necessity in the Anderson reduction
⊤	U+22A4	Verum	O⊤ is a theorem
⊨	U+22A8	Satisfaction	Truth at a world in a model
∀	U+2200	Universal quantifier	Over deontic alternatives
∃	U+2203	Existential quantifier	Seriality and permission
≡	U+2261	Definitional equivalence	Pφ ≡ ¬O¬φ
¬	U+00AC	Negation	Interdefinition of O, P and F
∧	U+2227	Conjunction	O agglomerates over it
∨	U+2228	Disjunction	Ross's paradox
→	U+2192	Material conditional	Antecedent of the K axiom
↔	U+2194	Biconditional	Agglomeration law for O
⊊	U+228A	Proper inclusion	K ⊊ KD ⊊ KT

Metatheory. SDL is the normal modal logic KD, sound and complete for serial frames. D is a Sahlqvist formula, so the system is canonical and seriality is elementary, defined by ∀w∃v wRv; filtration gives the finite model property, and satisfiability is PSPACE-complete, exactly as for K (Ladner, 1977). The proper inclusions are K ⊊ KD ⊊ KT, since reflexivity implies seriality but not conversely, and adding T destroys the deontic reading by validating Oφ → φ. Anderson (1958) and Kanger showed the system reducible: with a propositional constant V for violation and Oφ defined as □(¬φ → V), an alethic base yields exactly the theorems of KD. Necessitation is where the trouble starts — it makes O⊤ a theorem, and K then delivers O(φ ∨ ψ) from Oφ, which is Ross's paradox, and Oψ from Oφ whenever φ entails ψ, which is what drives the Good Samaritan.

Applies to. Legal reasoning. Ethics formalization. Normative AI. Compliance. Contract specification.

Limitations. The paradoxes are consequences of the axioms, not artefacts of careless formalizing. Monotonicity yields Ross's paradox and the Good Samaritan; necessitation makes every tautology obligatory. Chisholm's contrary-to-duty set is consistent and mutually independent as English, yet has no SDL formalization that is both consistent and independent, so the logic cannot say what one ought to do having already done wrong. Forrester's gentle murder derives an unconditional obligation to murder from the conditional obligation to murder gently, by monotonicity again. D excludes moral dilemmas by fiat, since Oφ ∧ O¬φ is unsatisfiable, so genuine conflicts of obligation cannot even be stated. Free choice permission fails — P(φ ∨ ψ) does not give Pφ ∧ Pψ — and adding it as an axiom collapses permission. There is no agent, no time and no defeasibility: obligations attach to propositions, and no addition of premises ever withdraws one.