Deep neural networks have increased the accuracy of automatic segmentation, however, their accuracy depends on the availability of a large number of fully segmented images. Methods to train deep neural networks using images for which some, but not all, regions of interest are segmented are necessary to make better use of partially annotated datasets. In this paper, we propose the first axiomatic definition of label-set loss functions that are the loss functions that can handle partially segmented images. We prove that there is one and only one method to convert a classical loss function for fully segmented images into a proper label-set loss function. Our theory also allows us to define the leaf-Dice loss, a label-set generalization of the Dice loss particularly suited for partial supervision with only missing labels. Using the leaf-Dice loss, we set a new state of the art in partially supervised learning for fetal brain 3D MRI segmentation. We achieve a deep neural network able to segment white matter, ventricles, cerebellum, extra-ventricular CSF, cortical gray matter, deep gray matter, brainstem, and corpus callosum based on fetal brain 3D MRI of anatomically normal fetuses or with open spina bifida. Our implementation of the proposed label-set loss functions is available at https://github.com/LucasFidon/label-set-loss-functions
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