It is generally agreed that poverty cannot be faithfully represented by a single monetary measure. Poverty involves many different aspects of life and people can be poor in different ways, to different degrees and from several points of view. For this reason, fuzzy multidimensional indicators have been proposed to measure material deprivation. However, the statistical procedures used to build such indicators do not seem to be fully consistent. In this paper we show how partially ordered set theory can help overcome such consistency problems, providing a sound basis for a multidimensional and fuzzy analysis of poverty. In particular, we propose new criteria to define fuzzy poverty membership functions and give a fuzzy generalization of the notion of quantile, that can prove useful in analyzing and comparing poverty data.