Until recently, little work has been dedicated to the representation and interchange of informal, semi-structured arguments of the type found in natural language prose and dialogue. To redress this, the research community recently initiated work towards an Argument Interchange Format (AIF). The AIF aims to facilitate the exchange of semi-structured arguments among different argument analysis and argumentation-support tools. In this paper, we present a Description Logic ontology for annotating arguments, based on a new reification of the AIF and founded in Walton's theory of argumentation schemes. We demonstrate how this ontology enables a new kind of automated reasoning over argument structures, which complements classical reasoning about argument acceptability. In particular, Web Ontology Language reasoning enables significantly enhanced querying of arguments through automatic scheme classifications, instance classification, inference of indirect support in chained argument structures, and inference of critical questions. We present the implementation of a pilot Web-based system for authoring and querying argument structures using the proposed ontology.