The beauty of thinking resides not only in finding answers but also in challenging already existing ones and creating intelligible networks of thoughts and ideas of many different perspectives. Understanding that the essence of critical thinking presents in the flows of questions and answers and aiming at a tool of augmenting it, we propose a new knowledge representation model based on bipartite graphs and a set of functions aided to explore those graphs. A bipartite graph has two disjoint vertex sets that are question and answer sets. Every edge connects a question to an answer holds an evidence for the answer and every edge connects an answer to a question holds an argument raised from the answer. The directional search mechanism is particularly designed to reuse reasoning flows in the debate graphs by a projecting function between the question and answer sets. We also introduce a framework placing the model and the search mechanism altogether in collaborative applications. Many philosophical debates are collected to demonstrate the advantages of the bigraph model besides its simplicity. The analysis and experiments on optimizing semantic search show a sufficient performance for a real-time application.