Systematic Frame Constants in Defeasible Deontic Logic

Abstract

The present paper is concerned with the problem of providing adequate semantical foundations for certain systems of dyadic (“defeasible”, “non-monotonic”) deontic logic (logics of conditional obligation and permission), and of giving semantically sound and complete axiomatizations of those systems. These dyadic deontic logics are all extensions of the system DSDL3, which was proposed by Bengt Hansson in his well known pioneering paper (Hansson, 1969), where DSDL3 was characterized in purely semantical terms, without any attempt being made by him to characterize it in axiomatic or proof-theoretical terms. In section 2 infra we start out by studying an infinite hierarchy of systems of Alethic Modal Logic, to which we intend to add definitions of dyadic deontic operators — this is in the spirit of the familiar Andersonian reduction of deontic to alethic modal logic (Anderson, 1956; Anderson, 1958). In section 3, then, we go on to deal with an infinite hierarchy of dyadic deontic logics, which we prove to be representable in the former hierarchy of alethic systems. This done in section 4, which forms the bulk of the paper. Finally, in the concluding section 5, we announce a result on a certain “core” system of dyadic deontic logic, the detailed proof of which will have to be deferred to another occasion.