Reactive standard deontic logic
Michael Gabbay, Christian Straßer · Journal of Logic and Computation · 2012
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.