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

Metatheory. SDL is decidable (equivalent to modal KD). Complete with respect to serial Kripke frames. The Ross paradox: from O(mail the letter), SDL derives O(mail the letter ∨ burn it). The Good Samaritan paradox: O(help the victim) seems to require the existence of a victim. Contrary-to-duty paradoxes (Chisholm, 1963): when you've already violated an obligation, what becomes obligatory? SDL struggles with these; various extensions address them.

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. SDL conflates ideal and actual: Oφ → Pφ fails if contradictory obligations exist (normative conflicts). The paradoxes (Ross, Good Samaritan, contrary-to-duty) suggest SDL doesn't fully capture normative reasoning. No distinction between violating an obligation and the obligation ceasing to exist. Dyadic/conditional approaches help but add complexity. Deontic logic describes norms but not their source — it's formal structure, not normative theory. The is/ought gap: deontic logic operates in the "ought" realm and cannot bridge to facts without additional machinery.