A second nutrition transition seems to emerge towards more plant-based diets, curving meat consumption in developed countries in the beginning of the XXI century. This shift suggests that logical arguments tend to influence more and more individuals towards vegetarian diets. This paper proposes a methodology to model a network of arguments around vegetarian diets by an abstract argumentation approach. Each argument, formalized by a node is connected with other arguments by arrows formalizing relationships between arguments. Thanks to this methodology we were able to formalize an argument network about vegetarian diets and identify the foremost importance of health argument compared with ethical or other type of arguments. This methodology also identified key arguments due to their high centrality in being challenged or challenging other arguments. These first results from this argument network construction suggest that any controversy surrounding vegetarian diets may polarize around such high centrality arguments about health. Even though ethical arguments appear of low importance in our network, the key issue of the necessity of animal products for human health may be essential for ethical choices towards vegetarian diets.