Abstract Enthymemes are arguments that are not fully articulated, often omitting a connection between premise and conclusion but sometimes also other information that is crucial for their interpretation. This implicitness poses challenges for the analysis and evaluation of argumentative discourse. We use the concept of “argument form” as employed in the argument classification framework of the Periodic Table of Arguments to address this issue. By developing an algorithmic procedure grounded in this concept, we provide a method for explicating missing statements and connections condensed in enthymemes. Our approach contributes to understanding the pragmatics of argumentation, as it offers a formal framework for analysing how the interpretation of implicit elements in argumentation arises from apparent non-sequiturs. The algorithmic procedure we developed can function as a guideline for human annotation of argumentative discourse and is also suitable for implementation in (AI-assisted) annotation software for argument mining.