Within open, distributed and dynamic environments, agents frequently encounter and communicate with new agents and services that were previously unknown. However, to overcome the ontological heterogeneity which may exist within such environments, agents first need to reach agreement over the vocabulary and underlying conceptualisation of the shared domain, that will be used to support their subsequent communication. Whilst there are many existing mechanisms for matching the agents' individual ontologies, some are better suited to certain ontologies or tasks than others, and many are unsuited for use in a real-time, autonomous environment. Agents have to agree on which correspondences between their ontologies are mutually acceptable by both agents. As the rationale behind the preferences of each agent may well be private, one cannot always expect agents to disclose their strategy or rationale for communicating. This prevents the use of a centralised mediator or facilitator which could reconcile the ontological differences. The use of argumentation allows two agents to iteratively explore candidate correspondences within a matching process, through a series of proposals and counter proposals, i.e., arguments. Thus, two agents can reason over the acceptability of these correspondences without explicitly disclosing the rationale for preferring one type of correspondences over anoth