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With the increasing popularity of Graph Neural Networks (GNNs) for predictive tasks on graph structured data, research on their explainability is becoming more critical and achieving significant progress. Although many methods are proposed to explain the predictions of GNNs, their focus is mainly on “how to generate explanations.” However, other important research questions like “whether the GNN explanations are inaccurate,” “what if the explanations are inaccurate,” and “how to adjust the model to generate more accurate explanations” have gained little attention. Our previous GNN Explanation Supervision (GNES) framework demonstrated effectiveness on improving the reasonability of the local explanation while still keep or even improve the backbone GNNs model performance. In many applications instead of per sample explanations, we need to find global explanations which are reasonable and faithful to the domain data. Simply learning to explain GNNs locally is not an optimal solution to a global understanding of the model. To improve the explainability power of the GNES framework, we propose the Global GNN Explanation Supervision (GGNES) technique which uses a basic trained GNN and a global extension of the loss function used in the GNES framework. This GNN creates local explanations which are fed to a Global Logic-based GNN Explainer, an existing technique that can learn the global Explanation in terms of a logic formula. These two frameworks are then trained iteratively to generate reasonable global explanations. Extensive experiments demonstrate the effectiveness of the proposed model on improving the global explanations while keeping the performance similar or even increase the model prediction power.