We propose a feature ranking method called dual dropout ranking (DDR) to identify the most discriminative linguistic features for Alzheimer's disease (AD) detection. The proposed DDR is based on a dual-net neural architecture that separates feature selection and recognition into two neural networks (operator and selector), which are alternatively and cooperatively trained to optimize the performance of both feature selection and AD recognition. The operator is trained on the features obtained from the selector to reduce classification loss. The selector is optimized to predict the operator's performance using as few selected features as possible. DDR ranks the features according to the probabilities that the corresponding features should be purged (or kept). The DDR and other feature ranking methods were evaluated on the AD ReSS dataset. Results show that the default linguistic feature set in ADReSS comprises many redundant features and that using feature ranking methods can improve the accuracy of AD recognition. Using the most discriminative feature subset (9 features) discovered by DDR, we obtain an FI score of 88.9% on the test set of ADReSS, which is 9.8 % (absolute) higher than what the default feature set can achieve.