Deontic Logic

Origin. Georg Henrik von Wright's "Deontic Logic" (1951) founded the modern field. Ernst Mally made earlier attempts (1926) but his system had paradoxes. Extends modal logic to obligation and permission. Developed further by Chisholm, Føllesdal, Hilpinen, and the legal logic community.

Models. Obligation, permission, and prohibition. What ought to be the case? What is permitted? What is forbidden? The logic of norms, duties, and normative reasoning. Core question: given a set of norms, what actions are obligatory, permitted, or forbidden?

Formalism.

Syntax: Propositional logic extended with deontic operators:

Oφ: it is obligatory that φ (φ ought to be the case)
Pφ: it is permitted that φ (φ is allowed)
Fφ: it is forbidden that φ (φ is prohibited)

Interdefinability: Pφ ≡ ¬O¬φ; Fφ ≡ O¬φ ≡ ¬Pφ

Standard Deontic Logic (SDL): System KD for the O operator:

O(φ → ψ) → (Oφ → Oψ) (K axiom)
Oφ → Pφ (D axiom: ought implies permitted, equivalently ¬(Oφ ∧ O¬φ))

Semantics: Kripke models where worlds represent "deontically ideal" or "normatively acceptable" states. M, w ⊨ Oφ iff φ is true in all deontically accessible worlds from w. The D axiom requires every world access at least one ideal world (no normative dead ends).

Conditional obligations: O(ψ | φ) — it is obligatory that ψ given φ. Dyadic deontic logic handles contrary-to-duty obligations.

Symbols.

Symbol	Unicode	Name	Meaning
O	—	Obligation	It ought to be that
P	—	Permission	It is permitted that
F	—	Prohibition	It is forbidden that
⊘	U+2298	Forbidden (alt)	Alternative prohibition symbol
→	U+2192	Conditional	Material conditional
⇒	U+21D2	Ought-conditional	Conditional obligation
O(ψ \| φ)	—	Dyadic obligation	ψ obligatory given φ
K	—	Distribution axiom	O(φ → ψ) → (Oφ → Oψ)
D	—	Seriality axiom	Oφ → Pφ
R	—	Accessibility relation	Serial: every world sees a deontic alternative
V	—	Violation constant	Anderson's sanction proposition
φ	U+03C6	Greek phi	Formula variable
ψ	U+03C8	Greek psi	Second formula variable
⊨	U+22A8	Satisfaction	M, w ⊨ Oφ
□	U+25A1	Box	Alethic necessity in the Anderson reduction
⊤	U+22A4	Verum	O⊤ is a theorem
∀	U+2200	Universal quantifier	Frame condition ∀w∃v wRv
∃	U+2203	Existential quantifier	Frame condition ∀w∃v wRv
≡	U+2261	Definitional equivalence	Pφ ≡ ¬O¬φ
¬	U+00AC	Negation	Interdefinition of O, P and F
∧	U+2227	Conjunction	Consistency schema ¬(Oφ ∧ O¬φ)
∨	U+2228	Disjunction	Ross's paradox
⊊	U+228A	Proper inclusion	K ⊊ KD ⊊ KT

Metatheory. Standard Deontic Logic is the normal modal logic KD, sound and complete for serial frames. D is a Sahlqvist formula, so KD 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 inclusions K ⊊ KD ⊊ KT are proper, and adding T destroys the deontic reading, because Oφ → φ says every obligation is met. 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. The paradoxes follow from the axioms rather than from careless formalizing — necessitation makes O⊤ a theorem, monotonicity gives O(φ ∨ ψ) from Oφ, which is Ross's paradox, and drives the Good Samaritan, and Chisholm's four sentences have no KD rendering that is at once consistent and independent. Dyadic systems in the manner of Hansson (1969) and Lewis (1973) answer the last; Parent (2015) proved completeness for Åqvist's E and F.

Applies to. Legal reasoning and legal AI. Ethics and normative theory. Specification of system policies. Access control and authorization. Agent programming (norms for autonomous agents). Business rules. Compliance checking.

Limitations. D makes Oφ ∧ O¬φ unsatisfiable, so conflicts of obligation are excluded by the logic rather than argued against: the axiom is a substantive ethical commitment in the guise of a consistency requirement. Obligations attach to propositions, with no agent, no action and no time, so what ought to be the case is not distinguished from what someone ought to do, nor violating an obligation from its lapsing. Monotonicity yields Ross's paradox and the Good Samaritan, and Forrester's gentle murder derives an unconditional obligation to murder from a conditional obligation to murder gently. Free choice permission fails — P(φ ∨ ψ) does not give Pφ ∧ Pψ — and adding it as an axiom collapses permission. Nothing is defeasible: no addition of premises ever withdraws an obligation. And the accessibility relation is given, so which worlds count as deontically ideal is stipulated rather than derived from any set of norms.