How to reject a counterfactual

According to D.K. Lewis (1973), would-couterfactuals and might-counterfactuals are duals. From this, it follows that the negation of a would-counterfactual is equivalent to the corresponding 'might-not'-counterfactual and that the negation of a might-counterfactual is equivalent to the corresponding 'would-not'- counterfactual. There are cases, however, where we seem to be entitled to accept the would-counterfactual and we are also equally entitled to accept the corresponding might-not-counterfactual and cases where we seem to be entitled to accept the might-counterfactual without being equally entitled to reject the corresponding would-not-counterfactual. In this paper, I will show that a distinction between two types of rejections for counterfactuals (p-rejection and s-rejection) and the recognition that might-not-counterfactuals may play the role of p-rejections (by an application to counterfactuals of the Lewisian approach to conversational scores) could explain why the problematic cases should not be seen as cases where the duality of would- and might-counterfactuals fails.