Large language models (LLMs) have achieved impressive human-like performance across various reasoning tasks. However, their mastery of underlying inferential rules still falls short of human capabilities. To investigate this, we propose a logic scaffolding inferential rule generation framework, to construct an inferential rule base, ULogic, comprising both primitive and compositional rules across five domains. Our analysis of GPT-series models over a rule subset reveals significant gaps in LLMs' logic understanding compared to human performance, especially in compositional and structural complex rules with certain bias patterns. We further distill these rules into a smaller-scale inference engine for flexible rule generation and enhancing downstream reasoning. Through a multi-judger evaluation, our inference engine proves effective in generating accurate, complex and abstract conclusions and premises, and improve various commonsense reasoning tasks. Overall, our work sheds light on LLMs' limitations in grasping inferential rule and suggests ways to enhance their logical reasoning abilities~\footnote{Code and data are available at \url{https://github.com/SiyuanWangw/ULogic}.}.

When learning a language, people can quickly expand their understanding of the unknown content by using compositional skills, such as from two words "go" and "fast" to a new phrase "go fast." In recent work of Lake and Baroni (2017), modern Sequence-to-Sequence(seq2seq) Recurrent Neural Networks (RNNs) can make powerful zero-shot generalizations in specifically controlled experiments. However, there is a missing regarding the property of such strong generalization and its precise requirements. This paper explores this positive result in detail and defines this pattern as the synonymous generalization, an ability to recognize an unknown sequence by decomposing the difference between it and a known sequence as corresponding existing synonyms. To better investigate it, I introduce a new environment called Colorful Extended Cleanup World (CECW), which consists of complex commands paired with logical expressions. While demonstrating that sequential RNNs can perform synonymous generalizations on foreign commands, I conclude their prerequisites for success. I also propose a data augmentation method, which is successfully verified on the Geoquery (GEO) dataset, as a novel application of synonymous generalization for real cases.