Graph Neural Networks (GNNs) have become an important graph feature learning paradigm that is extensively applied to graph inference tasks. However, GNNs still have limitations in some aspects such as representation capability, interpretability, etc. To this end, this paper introduces a novel hybrid Fuzzy-Logic Graph Neural Network (FL-GNN) that synergistically combines Fuzzy Neural Network (FNN) and GNN to effectively capture and aggregate local information flows within the graph. FL-GNN exhibits three distinctive features. First, we incorporate a specialized structure fuzzy rule to enhance FL-GNN’s graph inference capability, surpassing representative GNN models. Second, we augment the interpretability of FL-GNN by integrating analytical exploration methods from two perspectives: Fuzzy Inference System and Message Passing Algorithm (MPA). Lastly, we refine the structure of FL-GNN based on MPA to optimize its calculation complexity and consequently improve learning efficiency. Experimental results show that FL-GNN outperforms existing representative graph neural networks for graph inference tasks.