We devise a shallow semantical embedding of Åqvist's dyadic deontic logic <strong>E</strong>&nbsp;in classical higher-order logic. This embedding is encoded in Isabelle/HOL, which turns this system into a proof assistant for deontic logic reasoning. The experiments with this environment provide evidence that this logic \textit{implementation} fruitfully enables interactive and automated reasoning at the meta-level and the object-level.

The LogiKEy workbench and dataset for ethical and legal reasoning is presented. This workbench simultaneously supports development, experimentation, assessment and deployment of formal logics and ethical and legal theories at different conceptual layers. More concretely, it comprises, in form of a dataset (Isabelle/HOL theory files), formal encodings of multiple deontic logics, logic combinations, deontic paradoxes and normative theories in the higher-order proof assistant system Isabelle/HOL. The data were acquired through application of the LogiKEy methodology, which supports experimentation with different normative theories, in different application scenarios, and which is not tied to specific logics or logic combinations. Our workbench consolidates related research contributions of the authors and it may serve as a starting point for further studies and experiments in flexible and expressive ethical and legal reasoning. It may also support hands-on teaching of non-trivial logic formalisms in lecture courses and tutorials. The LogiKEy methodology and framework is discussed in more detail in the companion research article titled “Designing Normative Theories for Ethical and Legal Reasoning: LogiKEy Framework, Methodology, and Tool Support” [5] .

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To design and engineer ethical and legal reasoners and responsible systems, Benzmüller, Parent and van der Torre introduce LogiKEy methodology based on the semantical embedding of deontic logics into classic higher-order logic. In this paper, we considerably extend the LogiKEy deontic logics and dataset using an algebraic approach. We develop theory of input/output operations for normative reasoning on top of Boolean algebras.

A shallow semantical embedding of a dyadic deontic logic by Carmo and Jones in classical higher-order logic is presented. This embedding is proven sound and complete, that is, faithful.The work presented here provides the theoretical foundation for the implementation and automation of dyadic deontic logic within off-the-shelf higherorder theorem provers and proof assistants.