Overview
Dyadic Deontic Logic (DDL) is not a single uniquely fixed calculus. It comprises systems that represent conditional obligation by a primitive operator O(\varphi\mid\psi): given \psi, it ought to be that \varphi. The reference interpretation here is the Hansson–Lewis/Åqvist preference-based family, especially Åqvist's system E and its relatives. Other DDLs, such as Carmo–Jones systems, have different semantics and must be treated separately.
Syntax
For propositional formulas \varphi,\psi one adds the binary connective
O(\varphi\mid\psi).
The antecedent \psi is inside the deontic operator. In general O(\varphi\mid\psi) is not interchangeable with \psi\to O\varphi or O(\psi\to\varphi).
Preference-Based Semantics
A preferential model (W,\succeq,V) ranks worlds according to comparative ideality. Fixing a convention in which \operatorname{Best}_{\succeq}(X) selects the most preferred worlds in X, a common truth clause is
M,w\models O(\varphi\mid\psi) \iff \operatorname{Best}_{\succeq}(\llbracket\psi\rrbracket_M) \subseteq\llbracket\varphi\rrbracket_M.
This simplified global-preference clause fits systems with world-independent preference; context-indexed or selection-function variants differ. The underlying order, smoothness / limit assumptions, and treatment of an impossible antecedent must be fixed before drawing proof-theoretic conclusions. In a nonempty-minimum semantics, one often imposes a condition ensuring that relevant nonempty antecedent sets have best elements.
Worked Contrary-to-Duty Example
Represent a primary duty to submit a report and a reparative duty after failure by
O(\mathit{submit}\mid\top),\qquad O(\mathit{notify}\mid\neg\mathit{submit}).
The second norm evaluates the best worlds among the non-submission worlds, not the globally best worlds. Consequently, it does not immediately cancel the primary duty. Whether factual detachment from \neg\mathit{submit} produces a monadic obligation is a further question, not automatic in all DDLs.
Proof Theory, Properties, and Complexity
Preference semantics can validate characteristic conditional-obligation principles, but their precise axiomatizations depend on the chosen DDL. The simple optimal-world clause alone is insufficient to justify a universal statement of decidability or complexity for every system called DDL. For rigorous complexity claims, first specify whether one means system E, one of its extensions, a Carmo–Jones system, or another DDL, and specify the underlying frame/order class.
Relationships
SDL is monadic KD and does not itself express this dyadic distinction. Deontic Defeasible Description Logic instead expresses norms as defeasible concept inclusions and adds separate rankings for expected and normatively preferable individuals.
References
- Bengt Hansson (1969). An Analysis of Some Deontic Logics. Noûs 3(4), 373–398.
- David Lewis (1973). Counterfactuals. Blackwell.
- Xavier Parent et al. (2024). Conditional Normative Reasoning as a Fragment of HOL. Journal of Applied Non-Classical Logics. DOI.
- Paul McNamara and Frederik Van De Putte. Deontic Logic. Stanford Encyclopedia of Philosophy. Overview.