One of the central challenges in artificial intelligence is reasoning under partial observability, where key values are missing but essential for understanding and modeling the system. This paper presents a neuro-symbolic framework for latent rule discovery and missing value imputation. In contrast to traditional latent variable models, our approach treats missing grounded values as latent predicates to be inferred through logical reasoning. By interleaving neural representation learning with symbolic rule induction, the model iteratively discovers—both conjunctive and disjunctive rules—that explain observed patterns and recover missing entries. Crucially, the inferred values not only fill in gaps in the data but also serve as supporting evidence for further rule induction and inference—creating a feedback loop in which imputation and rule mining reinforce one another. Using coordinate gradient descent, the system learns these rules end-to-end, enabling interpretable reasoning over incomplete data. Experiments on both synthetic and real-world datasets demonstrate that our method effectively imputes missing values while uncovering meaningful, human-interpretable rules that govern the system dynamics.