Overview
Standard Deontic Logic (SDL) interprets the normal modal system KD normatively: O\varphi means that \varphi is obligatory. It is monadic: the language has no primitive obligation conditional on a separate formula. Permissions are typically defined dually.
Syntax and Axioms
\varphi ::= p\mid\neg\varphi\mid(\varphi\land\psi)\mid O\varphi, \qquad P\varphi:=\neg O\neg\varphi.
SDL inherits classical propositional logic, modus ponens and necessitation for O. Its characteristic principles are
\mathrm{K}_O:\quad O(\varphi\to\psi)\to(O\varphi\to O\psi),
\mathrm{D}_O:\quad O\varphi\to P\varphi.
D excludes simultaneous obligations O\varphi\land O\neg\varphi at a world. SDL does not entail O\varphi\to\varphi: an obligation can be violated.
Serial Kripke Semantics
A model M=(W,R,V) has serial R:
\forall w\in W\,\exists v\in W\,Rwv, \qquad M,w\models O\varphi\iff\forall v(Rwv\to M,v\models\varphi).
The accessible worlds represent ideal or normatively acceptable alternatives relative to the current world; seriality ensures at least one such alternative. This is the same abstract frame class as KD, not a prescribed algorithm for choosing optimal alternatives.
Example and Limitations
O(\mathit{safe}) can hold even if \neg\mathit{safe} is true now. From O\varphi and O(\varphi\to\psi) one obtains O\psi (deontic closure under implication). This makes SDL simple and tractable to formalise, but it does not distinguish primary from contrary-to-duty obligations or explicitly represent priorities and exceptions. Chisholm-style cases highlight these limitations.
Decidability and Complexity
For the unrestricted propositional monomodal language, local satisfiability is PSPACE-complete, validity is PSPACE-complete, and SDL has the finite model property. Quantified extensions require additional semantic choices such as domain policies and cannot inherit these bounds automatically.
Relationships
KD is the corresponding normal modal point of the modal cube. Dyadic Deontic Logic makes the conditioning event part of the obligation operator; Deontic Defeasible Description Logic combines concept inclusions with norm- and expectation-based rankings.
References
- G. H. von Wright (1951). Deontic Logic. Mind 60(237), 1–15.
- Paul McNamara and Frederik Van De Putte. Deontic Logic. Stanford Encyclopedia of Philosophy. Overview.
- Patrick Blackburn, Maarten de Rijke, and Yde Venema (2001). Modal Logic. Cambridge University Press.