  Natural language explanations have become a proxy for evaluating explainable
and multi-step Natural Language Inference (NLI) models. However, assessing the
validity of explanations for NLI is challenging as it typically involves the
crowd-sourcing of apposite datasets, a process that is time-consuming and prone
to logical errors. To address existing limitations, this paper investigates the
verification and refinement of natural language explanations through the
integration of Large Language Models (LLMs) and Theorem Provers (TPs).
Specifically, we present a neuro-symbolic framework, named Explanation-Refiner,
that augments a TP with LLMs to generate and formalise explanatory sentences
and suggest potential inference strategies for NLI. In turn, the TP is employed
to provide formal guarantees on the logical validity of the explanations and to
generate feedback for subsequent improvements. We demonstrate how
Explanation-Refiner can be jointly used to evaluate explanatory reasoning,
autoformalisation, and error correction mechanisms of state-of-the-art LLMs as
well as to automatically enhance the quality of human-annotated explanations of
variable complexity in different domains.
