Dyadic Deontic Logic in HOL : Faithful Embedding and Meta-Theoretical Experiments

A shallow semantical embedding of a dyadic deontic logic by Carmo and Jones in classical higher-order logic is presented. The embedding is proven sound and complete, that is, faithful. This result provides the theoretical foundation for the implementation and automation of dyadic deontic logic within off-the-shelf higher-order theorem provers and proof assistants. To demonstrate the practical relevance of our contribution, the embedding has been encoded in the Isabelle/HOL proof assistant. As a result a sound and complete (interactive and automated) theorem prover for the dyadic deontic logic of Carmo and Jones has been obtained. Experiments have been conducted which illustrate how the exploration and assessment of meta-theoretical properties of the embedded logic can be supported with automated reasoning tools integrated with Isabelle/HOL.