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 a dagger connect arrows, e.g. (x(1) -> x(2)) a dagger (x(3) -> x(4)). The idea is that passing through x(1) -> x(2) activates a switch represented by a dagger that inverts the label of x(3) -> x(4) and hence activates respectively deactivates this arrow. This allows to introduce two modalities: a- 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