Outlier detection is an essential capability in safety-critical applications of supervised visual recognition. Most of the existing methods deliver best results by encouraging standard closed-set models to produce low-confidence predictions in negative training data. However, that approach conflates prediction uncertainty with recognition of the negative class. We therefore reconsider direct prediction of K+1 logits that correspond to K groundtruth classes and one outlier class. This setup allows us to formulate a novel anomaly score as an ensemble of in-distribution uncertainty and the posterior of the outlier class which we term negative objectness. Now outliers can be independently detected due to i) high prediction uncertainty or ii) similarity with negative data. We embed our method into a dense prediction architecture with mask-level recognition over K+2 classes. The training procedure encourages the novel K+2-th class to learn negative objectness at pasted negative instances. Our models outperform the current state-of-the art on standard benchmarks for image-wide and pixel-level outlier detection with and without training on real negative data.