Concealed questions (CQs) are determiner phrases that are naturally paraphrased as embedded questions: I know your age reads as I know what your age is. The availability of CQ readings of determiner phrases has recently been taken to depend on whether the DP head noun is relational (i.e., two-place, like age or capital) or sortal (one-place, like brick or city) (Barker 2016). This generalization is based on the observation that many definite DPs lack salient concealed question-readings when their head noun is an unmodified sortal. Relatedly, Frana (2013, 2017) argues that, while quantified DPs have multiple concealed question-readings when the head noun is relational, they have only one when the head noun is sortal. This remark brings together noted counterexamples to both generalizations, and argues on their basis for a pragmatic account of the availability of concealed question readings of determiner phrases. The proposed account refines the analysis of concealed questions developed by Aloni & Roelofsen (2011) through integrating it with (i) a standard analysis of the semantic distinction between relational and sortal nouns, and (ii) Barker's (2016) ideas about the role of salient sets of alternatives in licensing CQ interpretation. The resulting account is shown to deliver correct predictions for the counterexamples to both semantic generalizations, while simultaneously offering an explanation of the strong, yet imperfect, correlation of CQ readings of determiner phrases with a relational head noun.