In Medicare question-answering (QA) tasks, the need for effective systems is pivotal in delivering accurate responses to intricate medical queries. However, existing approaches often struggle to grasp the intricate logical structures and relationships inherent in medical contexts, thus limiting their capacity to furnish precise and nuanced answers. In this work, we address this gap by proposing a novel Abstractive QA system MedLogic-AQA that harnesses first-order logic-based rules extracted from both context and questions to generate well-grounded answers. Through initial experimentation, we identified six pertinent first-order logical rules, which were then used to train a Logic-Understanding (LU) model capable of generating logical triples for a given context, question, and answer. These logic triples are then integrated into the training of MediLogic-AQA, enabling reasoned and coherent reasoning during answer generation. This distinctive fusion of logical reasoning with abstractive question answering equips our system to produce answers that are logically sound, relevant, and engaging. Evaluation with respect to both automated and human-based demonstrates the robustness of MedLogic-AQA against strong baselines. Through empirical assessments and case studies, we validate the efficacy of MedLogic-AQA in elevating the quality and comprehensiveness of answers in terms of reasoning as well as informativeness.