Argument mining has started to yield early results in automatic analysis of text to produce representations of reason-conclusion structures. This paper addresses for the first time the question of automatically extracting such structures from dialogical settings of argument. More specifically, we introduce theoretical foundations for dialogical argument mining as well as show the initial implementation in a software for dialogue processing, and the application in corpus analysis. We combine analysis of illocutionary structure with structured argumentation frameworks as our scaffolding, and apply a combination of statistical and grammatically based analytical techniques.