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 → that inverts the label of x₃ → x₄ and hence activates respectively 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 Ø 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.