Reactive standard deontic logic

We introduce a reactive variant of SDL (standard deontic logic): SDLR1 (reactive standard deontic logic). Given a Kripkean view on the semantics of SDL in terms of directed graphs where arrows → represent the accessibility relation between worlds, reactive models add two elements: arrows → are labelled as “active” or “inactive”, and double arrows ։ connect arrows, e.g., (x1 → x2) ։ (x3 → x4). The idea is that passing through x1 → x2 activates a switch represented by ։ which inverts the label of x3 → x4 and hence activates resp. deactivates this arrow. This allows to introduce two modalities: is the usual KD-modality of SDL and operates on the Kripkean graph where all labels and double arrows are ignored, while ⊘ takes them into account. We demonstrate that RSDL1 allows for an intuitive interpretation of ‘ought’. The logic can handle contrary-to-duty cases such as several instantiations of the Chisholm set in a paradox-free way by means of using double arrows and annotations to block and give access to ideal worlds. 1 Standard Deontic Logic and its Problems A logic with modality is KD modality if we have the axioms K0 All substitution instances of classical tautologies K1 (p ∧ q) ≡ ( p ∧ q) K2 ⊢ A ⇒ ⊢ A D ¬ ⊥ It is complete for frames of the form (S ,R, a) where S , ⊘ is a set of possible worlds, a ∈ S ,R ⊆ S × S is serial (i.e., for all x there exists a y such that xRy). Standard Deontic Logic SDL is a KD modality O. We read u Op as saying p holds in all ideal worlds relative to u, i.e. for all t we have: if uRt then t p. So the set of ideal worlds relative to u is the set I(u) = {t | uRt}.