How to Combine Ordering and Minimizing in a Deontic Logic Based on Preferences

In this paper we propose a semantics for dyadic deontic logic with an explicit preference ordering between worlds, representing diierent degrees of ideality. We argue that this ideality ordering can be used in two ways to evaluate formulas , which we call ordering and minimizing. Ordering uses all preference relations between relevant worlds, whereas minimizing uses the most preferred worlds only. We show that ordering corresponds to strengthening of the antecedent , and minimizing to weakening of the consequent. Moreover, we show that in some cases ordering and minimizing have to be combined to obtain certain desirable conclusions, and that this can only be done in a so-called two-phase deontic logic. In the rst phase, the preference ordering is constructed, and in the second phase the ordering is used for minimization. If these two phases are not distinguished, then counterintuitive conclusions follow.