Completeness and decidability results for a logic of contrary-to-duty conditionals

This article has two parts. In Part I, we briefly outline the analysis of ‘contrary-to-duty’ obligation sentences presented in our 2002 handbook chapter ‘Deontic logic and contrary-to-duties’, with a focus on the intuitions that motivated the basic formal-logical moves we made. We also explain that the present account of the theory differs in two significant respects from the earlier version, one terminological, the other concerning the way the constituent modalities interconnect. Part II is the principal contribution of this article, in which we show that it is possible to define a complete and decidable axiomatization for the Carmo and Jones logic, a problem that was still open. The axiomatization includes two new inference rules; we illustrate their use in proofs, and show that on the basis of this axiomatization we can recover all the axioms and rules considered in ‘Deontic logic and contrary-to-duties’, and used there in the analysis of contrary-to-duty conditional scenarios.