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In this paper, we present an experiment to identify emotions in tweets. Unlike previous studies, which typically use the six basic emotion classes defined by Ekman, we classify emotions according to a set of eight basic bipolar emotions defined by Plutchik (Plutchik's "wheel of emotions"). This allows us to treat the inherently multi-class problem of emotion classification as a binary problem for four opposing emotion pairs. Our approach applies distant supervision, which has been shown to be an effective way to overcome the need for a large set of manually labeled data to produce accurate classifiers. We build on previous work by treating not only emoticons and hashtags but also emoji, which are increasingly used in social media, as an alternative for explicit, manual labels. Since these labels may be noisy, we first perform an experiment to investigate the correspondence among particular labels of different types assumed to be indicative of the same emotion. We then test and compare the accuracy of independent binary classifiers for each of Plutchik's four binary emotion pairs trained with different combinations of label types. Our best performing classifiers produce results between 75-91%, depending on the emotion pair; these classifiers can be combined to emulate a single multi-label classifier for Plutchik's eight emotions that achieves accuracies superior to those reported in previous multi-way classification studies.