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 (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

Metatheory. Sound and complete for serial frames. Decidable. NP-complete satisfiability. D logic.

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

Limitations. Paradoxes (Ross, CTD). No agency. No time. Monotonic. Idealization issues.