Argument mining is a natural language processing technology aimed at identifying the explicit premises and claims of arguments in text, and the support and attack relationships between them. To better understand, and automatically analyse, the argument maps that are output from argument mining, it would be desirable to instantiate the arguments in the argument map with logical arguments. However, most real-world arguments are enthymemes (i.e. some of the premises and/or claim are implicit), which need to be decoded (i.e. the implicit aspects need to be identified). A key challenge is to decode enthymemes so as to respect the support and attack relationships in the argument map. addressing the problem of identifying the missing premises and/or claim, and discerning the relationships between them. To address this, we present a novel framework, based on default logic, for representing arguments including enthymemes. We show how decoding an enthymeme means identifying the default rules that are implicit in the premises and claims. We then show how choosing a decoding of the enthymemes in an argument map can be formalized as an optimization problem, and that a solution can be obtained using MaxSAT solvers.