Faithful Logic Embeddings in HOL -- Deep and Shallow
This incremental approach benefits logic education, research, and application by providing a conceptual framework for interactive and automated reasoning tools.
The paper tackles the problem of embedding non-classical logics in classical higher-order logic by presenting a method for simultaneous deep and shallow embeddings, enabling flexible theorem proving, counterexample finding, and automated faithfulness proofs, illustrated with a simple propositional modal logic.
Deep and shallow embeddings of non-classical logics in classical higher-order logic have been explored, implemented, and used in various reasoning tools in recent years. This paper presents a method for the simultaneous deployment of deep and shallow embeddings of various degrees in classical higher-order logic. This enables flexible, interactive and automated theorem proving and counterexample finding at meta and object level, as well as automated faithfulness proofs between these logic embeddings. The method is beneficial for logic education, research and application and is illustrated here using a simple propositional modal logic. However, this approach is conceptual in nature and not limited to this simple logic context.