Three characterizability problems in deontic logic (Q2718309)

The deontic logics, \(H_m\) of this paper contain S5-alethic modal operators and a set of `systematic frame-constants', which represent `levels of perfection' among possible worlds in models for the systems. From these, various deontic concepts, especially conditional obligation and permission, as well as supererogation and permissible wrong-doing, can be defined. The three problems of the title are (1) What is the logic of supererogation and permissible wrong-doing generated from such definitions? This is introduced, but not discussed. (2) What is the logic of conditional obligation and permission that results from the given definitions within the family of logics containing frame constants? This problem is answered in terms of a family of dyadic deontic logics, \(G_m\), that contain postulates for the dyadic operators in combination with alethic modalities and frame constants, with the result that \(A\) is provable in \(G_m\) if and only if \(A\) is provable in \(H_m\) for any \(m\). (3) What is the logic of conditional obligation and permission that is common to all the family of systems \(H_m\) and so do not themselves contain frame constants? This question is given a partial answer in terms of a dyadic deontic logic \(G\) that lacks frame constants, with the result that \(A\) is provable in \(G\) only if \(A\) is provable in every \(G_m\), and hence, \(A\) is provable in \(G\) only if \(A\) is provable in every \(H_m\). But these are only `only if' results, not equivalences. The author proposes a method to strengthen the result by extending the expressive power of the language of \(H_m\) to include a novel `best' operator. The methods throughout are semantical, with models familiar to deontic logic.