Deontic Action Logic

Origin. Von Wright combined action and deontic logic (1960s). Segerberg, Meyer, and others developed dynamic deontic logic (1980s-90s). Obligations about actions, not just states. "It is obligatory to do α." Foundation for normative multi-agent systems.

Models. Norms about actions. Standard deontic: O(φ) — obligatory that state φ. Action deontic: O(α) — obligatory to perform action α. Combines dynamic logic [α]φ with deontic operators. Distinguishes doing from refraining.

Formalism.

Action deontic operators:

O(α): obligatory to do α
P(α): permitted to do α
F(α): forbidden to do α

Dynamic + deontic:

[α]φ: after all α-executions, φ
O([α]φ): obligatory that doing α leads to φ
[α]O(φ): after doing α, φ becomes obligatory

Meyer's dynamic deontic logic: Violation predicate V. O(α) ≡ [ᾱ]V (not doing α leads to violation) P(α) ≡ ¬[α]V (doing α doesn't lead to violation) F(α) ≡ [α]V (doing α leads to violation)

Action negation: ᾱ: not doing α (doing something else or nothing). α ∪ ᾱ: doing anything.

Contrary-to-duty: O(α) ∧ [ᾱ]O(β): should do α, but if not, then should do β. Handles reparational obligations.

STIT connection: [i stit]φ + O(φ) in branching frameworks. Agent-relative obligations.

Symbols.

Symbol	Unicode	Name	Meaning
O	—	Obligation	Must do
P	—	Permission	May do
F	—	Forbidden	Must not do
[α]	—	After α	Dynamic modality
ᾱ	—	Not doing α	Complement action
V	—	Violation	Norm breach
∪	U+222A	Choice	Action choice

Metatheory. Decidable for propositional base. Avoids some deontic paradoxes. Contrary-to-duty handled. Complexity varies by fragment. Complete axiomatizations exist. Connects to input/output logic.

Applies to. Legal reasoning. Normative multi-agent systems. Robot ethics. Contract formalization. Compliance checking. Business rules. AI safety.

Limitations. Action negation semantically subtle. Multiple competing frameworks. Contrary-to-duty still debated. Time and action interaction complex. Tool support limited. Philosophical debates unresolved.