In the widely used class of multidimensional poverty measures introduced by Alkire and Foster (2011), dimension-specific weights combined with a single cut-off parameter play a fundamental role in identifying who is multidimensionally poor. This paper revisits how these parameters are understood, revealing they do not uniquely characterise who is identified as poor and that the weights do not reliably reflect each dimensions' relative importance. Drawing on insights from Boolean algebra, I demonstrate that the set of 'minimum deprivation bundles' constitutes an intuitive and unique characterization of Alkire-Foster identification functions. This provides a formal foundation for various analytical innovations, namely: a novel poverty decomposition based only on the unique properties of each identification function; and metrics of dimensional power, which capture the effective importance or 'value' of each dimension across all possible combinations of deprivations. These insights are illustrated using deprivation data from Mozambique and applying various identification functions, including a close replica of the international MPI (multidimensional poverty index).