No intelligent decision support system functions even remotely without knowing the preferences of the user. A major problem is that the way average users think about and formulate their preferences does not match the utility-based quantitative frameworks currently used in decision support systems. For the average user qualitative models are a better fit. This paper presents an argumentation-based framework for the modelling of, and automated reasoning about multi-issue preferences of a qualitative nature. The framework presents preferences according to the lexicographic ordering that is well-understood by humans. The main contribution of the paper is that it shows how to reason about preferences when only incomplete information is available. An adequate strategy is proposed that allows reasoning with incomplete information and it is shown how to incorporate this strategy into the argumentation-based framework for modelling preferences.