Argument mining is natural language processing technology aimed at identifying arguments in text. Furthermore, the approach is being developed to identify the premises and claims of those arguments, and to identify the relationships between arguments including support and attack relationships. In this paper, we assume that an argument map contains the premises and claims of arguments, and support and attack relationships between them, that have been identified by argument mining. So from a piece of text, we assume an argument map is obtained automatically by natural language processing. However, to understand and to automatically analyse that argument map, it would be desirable to instantiate that argument map with logical arguments. Once we have the logical representation of the arguments in an argument map, we can use automated reasoning to analyze the argumentation (e.g. check consistency of premises, check validity of claims, and check the labelling on each arc corresponds with the logical arguments). We address this need by using classical logic for representing the explicit information in the text, and using default logic for representing the implicit information in the text. In order to investigate our proposal, we consider some specific options for instantiation. 1 Commonsense knowledge is knowledge normally known by everyone, i.e. universally known (as opposed to local knowledge). For instance, it is commonsense knowledge that if drop an egg on the floor, it will probably break, or if you tell your friends a funny joke, they will probably laugh. For a review of commonsense knowledge representation and reasoning see [Dav17]. In contrast, common knowledge is knowledge that is known by a subset of people? For example, if two old friends are talking and one of them refers a past event that they had experienced together, without explicitly recalling it, they may both have implicit but common knowledge of the event which would be a context known only to relatively few people (non-universal knowledge).