Abstract: Since the advent of knowledge distillation, many researchers have been intrigued by the $\textit{dark knowledge}$ hidden in the soft labels generated by the teacher model. This prompts us to scrutinize the circumstances under which these soft labels are effective. Predominant existing theories implicitly require that the soft labels are close to the ground-truth labels. In this paper, however, we investigate whether biased soft labels are still effective. Here, bias refers to the discrepancy between the soft labels and the ground-truth labels. We present two indicators to measure the effectiveness of the soft labels. Based on the two indicators, we propose moderate conditions to ensure that, the biased soft label learning problem is both $\textit{classifier-consistent}$ and $\textit{Empirical Risk Minimization}$ (ERM) $\textit{learnable}$, which can be applicable even for large-biased soft labels. We further design a heuristic method to train Skillful but Bad Teachers (SBTs), and these teachers with accuracy less than 30\% can teach students to achieve accuracy over 90\% on CIFAR-10, which is comparable to models trained on the original data. The proposed indicators adequately measure the effectiveness of the soft labels generated in this process. Moreover, our theoretical framework can be adapted to elucidate the effectiveness of soft labels in three weakly-supervised learning paradigms, namely incomplete supervision, partial label learning and learning with additive noise. Experimental results demonstrate that our indicators can measure the effectiveness of biased soft labels generated by teachers or in these weakly-supervised learning paradigms.